Research Article On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations ElHassan ElJaoui and Said Melliani Department of Mathematics, University of Sultan Moulay Slimane, P.O. Box 523, 23000 Beni Mellal, Morocco Correspondence should be addressed to ElHassan ElJaoui; [email protected] Received 31 October 2015; Revised 20 December 2015; Accepted 3 January 2016 Academic Editor: Najeeb A. Khan Copyright © 2016 E. ElJaoui and S. Melliani. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability. 1. Introduction Wu introduced in [1] the improper fuzzy Riemann integral and presented some of its elementary properties; then he studied numerically this kind of integrals. is notion was exploited by certain researchers to study fuzzy differential equations (FDEs) of first or second order utilizing fuzzy Laplace transform, namely, by Allahviranloo and Ahmadi in [2], then by Salahshour et al. (see [3, 4]), and by ElJaoui et al. in [5]. e objective of this paper is to study the improper fuzzy Riemann integrals by establishing some important results about the continuity and the differentiability of a fuzzy improper integral depending on a given parameter. ese results are then employed to prove some fuzzy Laplace transform’s properties, which we use to solve fuzzy partial differential equations (FPDEs). e organization of the remainder of this work is as follows. Section 2 is reserved for preliminaries. In Section 3, the main results are proved and new properties of fuzzy Laplace transform are investigated. en, in Section 4, the procedure for solving first-order FPDEs by fuzzy Laplace transform is proposed. Section 5 deals with some numerical examples. In Section 6, we present conclusion and a further research topic. 2. Preliminaries By (R) we meant the set of all nonempty compact convex subsets of R, which is endowed with the usual addition and scalar multiplication. Denote (see [6]) = { : R → [0,1] | verifies (1) – (4) below}, (1) where (1) is normal; that is, ∃ ∈ R for which () = 1, (2) is convex in the fuzzy sense, (3) is upper semicontinuous, (4) the closure of its support supp = { ∈ R | () > 0} is compact. For 0<≤1, [] = { ∈ R | () ≥ } denotes the -level set of ∈. en, it is obvious that [] ∈ (R) for all ∈, 0≤≤ 1, and [ 1 + 2 ] = [ 1 ] + [ 2 ] , [] = [] . (2) Let : × → [0, ∞) be a function which is defined by the identity ( 1 , 2 )= sup 0≤≤1 ([ 1 ] , [ 2 ] ), (3) Hindawi Publishing Corporation International Journal of Differential Equations Volume 2016, Article ID 7246027, 8 pages http://dx.doi.org/10.1155/2016/7246027
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![Page 1: Research Article On Fuzzy Improper Integral and Its ...downloads.hindawi.com/journals/ijde/2016/7246027.pdf · Wu introduced in [] the improper fuzzy Riemann integral and presented](https://reader035.cupdf.com/reader035/viewer/2022071021/5fd5401e6ba5386cb526b789/html5/thumbnails/1.jpg)
Research ArticleOn Fuzzy Improper Integral and Its Application forFuzzy Partial Differential Equations
ElHassan ElJaoui and Said Melliani
Department of Mathematics University of Sultan Moulay Slimane PO Box 523 23000 Beni Mellal Morocco
Correspondence should be addressed to ElHassan ElJaoui eljaouihassgmailcom
Received 31 October 2015 Revised 20 December 2015 Accepted 3 January 2016
Academic Editor Najeeb A Khan
Copyright copy 2016 E ElJaoui and S Melliani This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We establish some important results about improper fuzzy Riemann integrals we prove some properties of fuzzy Laplacetransforms which we apply for solving some fuzzy linear partial differential equations of first order under generalized Hukuharadifferentiability
1 Introduction
Wu introduced in [1] the improper fuzzy Riemann integraland presented some of its elementary properties then hestudied numerically this kind of integrals
This notion was exploited by certain researchers to studyfuzzy differential equations (FDEs) of first or second orderutilizing fuzzy Laplace transform namely by Allahviranlooand Ahmadi in [2] then by Salahshour et al (see [3 4]) andby ElJaoui et al in [5]
The objective of this paper is to study the improper fuzzyRiemann integrals by establishing some important resultsabout the continuity and the differentiability of a fuzzyimproper integral depending on a given parameter
These results are then employed to prove some fuzzyLaplace transformrsquos properties which we use to solve fuzzypartial differential equations (FPDEs)
The organization of the remainder of this work is asfollows Section 2 is reserved for preliminaries In Section 3the main results are proved and new properties of fuzzyLaplace transform are investigated Then in Section 4 theprocedure for solving first-order FPDEs by fuzzy Laplacetransform is proposed Section 5 deals with some numericalexamples In Section 6 we present conclusion and a furtherresearch topic
2 Preliminaries
By 119875119888(R) we meant the set of all nonempty compact convex
subsets of R which is endowed with the usual addition andscalar multiplication Denote (see [6])
119864 = 120583 R 997888rarr [0 1] | 120583 verifies (1) ndash (4) below (1)
where(1) 120583 is normal that is exist119905 isin R for which 120583(119905) = 1(2) 120583 is convex in the fuzzy sense(3) 120583 is upper semicontinuous(4) the closure of its support supp 120583 = 119905 isin R | 120583(119905) gt 0
is compactFor 0 lt 120572 le 1 [120583]120572 = 119905 isin R | 120583(119905) ge 120572 denotes the 120572-levelset of 120583 isin 119864
Then it is obvious that [120583]120572 isin 119875119888(R) for all 120583 isin 119864 0 le 120572 le
1 and[1205831+ 1205832]120572
= [1205831]120572
+ [1205832]120572
[119896120583]120572
= 119896 [120583]120572
(2)
Let119863 119864 times 119864 rarr [0infin) be a function which is defined by theidentity
119863(1205831 1205832) = sup0le120572le1
119889 ([1205831]120572
[1205832]120572
) (3)
Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2016 Article ID 7246027 8 pageshttpdxdoiorg10115520167246027
2 International Journal of Differential Equations
where 119889 is the Hausdorff distance defined in 119875119888(R)Then it is
clear that (119864119863) is a complete metric space (for more detailsabout the metric119863 see [7])
Definition 1 (see [2]) One defines a fuzzy number V inparametric form as a couple (V V) of mappings V(120572) and V(120572)0 le 120572 le 1 verifying the following properties
(1) V(120572) is bounded increasing left continuous in ]0 1]and right continuous at 0
(2) V(120572) is bounded decreasing left continuous in ]0 1]and right continuous at 0
(3) V(120572) le V(120572) for all 0 le 120572 le 1
The following identity holds true (see [8])
119863(1205831 1205832) = sup0le120572le1
max 100381610038161003816100381610038161003816120583120572
1
minus 120583120572
2
1003816100381610038161003816100381610038161003816100381610038161003816120583120572
1minus 120583120572
2
1003816100381610038161003816 (4)
Theorem 2 (see [1]) One considers a fuzzy valued function119865(119909) = (119865(119909 120572) 119865(119909 120572)) defined on [119886infin[ Suppose that forall fixed 120572 isin [0 1] the crisp functions 119865(119909 120572) 119865(119909 120572) areintegrable on [119886 119887] for every 119887 ge 119886 and that there exist twopositive constants 119870(120572) and 119870(120572) such that int119887
119886
|119865(119909 120572)|119889119909 le
119870(120572) and int119887119886
|119865(119909 120572)|119889119909 le 119870(120572) for every 119887 ge 119886 Then 119865(119909)is fuzzy Riemann integrable (in the sense of Wu) on [119886infin[ itsimproper fuzzy integral intinfin
119886
119865(119909)119889119909 isin 119864 and
int
infin
119886
119865 (119909) 119889119909 = (int
infin
119886
119865 (119909 120572) 119889119909 int
infin
119886
119865 (119909 120572) 119889119909) (5)
For 1205831 1205832isin 119864 if there exists an element 120583
3isin 119864 such that
1205833= 1205831+ 1205832 then 120583
3is called the Hukuhara difference of 120583
1
and 1205832 which we denote by 120583
1⊖ 1205832
Definition 3 (see [2]) A mapping 119865 (119886 119887) rarr 119864 is said to bestrongly generalized differentiable at 119909 isin (119886 119887) if there exists1198651015840
(119909) isin 119864 such that
(i) for all ℎ gt 0 being very small there exist 119865(119909 + ℎ) ⊖119865(119909) 119865(119909) ⊖ 119865(119909 minus ℎ) and the limits
limℎrarr0+
119865 (119909 + ℎ) ⊖ 119865 (119909)
ℎ= limℎrarr0+
119865 (119909) ⊖ 119865 (119909 minus ℎ)
ℎ
= 1198651015840
(119909)
(6)
or
(ii) for all ℎ gt 0 being very small there exist 119865(119909)⊖119865(119909+ℎ) 119865(119909 minus ℎ) ⊖ 119865(119909) and the limits
limℎrarr0+
119865 (119909) ⊖ 119865 (119909 + ℎ)
(minusℎ)= limℎrarr0+
119865 (119909 minus ℎ) ⊖ 119865 (119909)
(minusℎ)
= 1198651015840
(119909)
(7)
or(iii) for all ℎ gt 0 being very small there exist 119865(119909 + ℎ) ⊖
119865(119909) 119865(119909 minus ℎ) ⊖ 119865(119909) and the limits
limℎrarr0+
119865 (119909 + ℎ) ⊖ 119865 (119909)
ℎ= limℎrarr0+
119865 (119909 minus ℎ) ⊖ 119865 (119909)
(minusℎ)
= 1198651015840
(119909)
(8)
or(iv) for all ℎ gt 0 being very small there exist 119865(119909)⊖119865(119909+
ℎ) 119865(119909) ⊖ 119865(119909 minus ℎ) and the limits
limℎrarr0+
119865 (119909) ⊖ 119865 (119909 + ℎ)
(minusℎ)= limℎrarr0+
119865 (119909) ⊖ 119865 (119909 minus ℎ)
ℎ
= 1198651015840
(119909)
(9)
The next theorem permits us to consider only case (i) orcase (ii) of Definition 3 almost everywhere in the domain ofthe mappings studied
Theorem4 (see [9]) If119865 (119886 119887) rarr 119864 is a strongly generalizeddifferentiable function on (119886 119887) in the sense of Definition 3 (iii)or (iv) then 1198651015840(119909) isin R for each 119909 isin (119886 119887)
Theorem 5 (see eg [10]) We consider a fuzzy function 119865 R rarr 119864 which is represented by 119865(119909) = (119865(119909 120572) 119865(119909 120572)) forall 120572 isin [0 1]
(1) If 119865 is (i)-differentiable then the crisp functions119865(119909 120572) and 119865(119909 120572) are differentiable and 1198651015840(119909) =(1198651015840
(119909 120572) 1198651015840
(119909 120572))(2) If 119865 is (ii)-differentiable then the crisp functions119865(119909 120572) and 119865(119909 120572) are differentiable and 1198651015840(119909) =(1198651015840
(119909 120572) 1198651015840
(119909 120572))
Definition 6 (see [2]) If 119865 [0infin[rarr 119864 is a continuousmapping such that 119890minus119904119909119865(119909) is fuzzy Riemann integrableon [0infin[ then intinfin
0
119890minus119904119909
119865(119909)119889119909 is called the fuzzy Laplacetransform of 119865 which one denotes by
L [119865 (119909)] = intinfin
0
119890minus119904119909
119865 (119909) 119889119909 119904 gt 0 (10)
Denote by L(119896(119909)) the classical Laplace transform of acrisp function 119896(119909) and then
L [119865 (119909)] = (L (119865 (119909 120572)) L (119865 (119909 120572))) (11)
Theorem 7 (see [2]) Let 119865 [0infin[rarr 119864 be a fuzzy valuedfunction and 1198651015840 its derivative on [0infin[ Then if 119865 is (i)-differentiable
L [1198651015840 (119909)] = 119904L [119865 (119909)] ⊖ 119865 (0) (12)
or if 119865 is (ii)-differentiable
L [1198651015840 (119909)] = (minus119865 (0)) ⊖ (minus119904) L [119865 (119909)] (13)
provided that the Laplace transforms of 119865 and 1198651015840 exist
International Journal of Differential Equations 3
3 Continuity and Differentiability ofFuzzy Improper Integral
In this section 119868 denotes one of the intervals ] minus infin 119887] or[119887infin[ or ]minusinfininfin[ where 119887 isin R 119869 denotes another intervaland 119860 is a nonempty subset of R
Lemma 8 Let 1198911(119905) 1198912(119905) be two fuzzy valued functions
which are fuzzy Riemann integrable on 119868 in the sense of Wu
(see [1]) such that the real function119863(1198911(119905) 1198912(119905)) is integrable
on 119868 and then
119863(int119868
1198911(119905) 119889119905 int
119868
1198912(119905) 119889119905) le int
119868
119863(1198911(119905) 1198912(119905)) 119889119905 (14)
Proof From identity (4) we have
119863(int119868
1198911(119905) 119889119905 int
119868
1198912(119905) 119889119905) = sup
0le120572le1
max 1003816100381610038161003816100381610038161003816int119868
1198911
(119905 120572) minus 1198912
(119905 120572) 119889119905
1003816100381610038161003816100381610038161003816
1003816100381610038161003816100381610038161003816int119868
1198911(119905 120572) 119889119905 minus 119891
2(119905 120572) 119889119905
1003816100381610038161003816100381610038161003816
le sup0le120572le1
max int119868
1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816119889119905 int119868
100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816119889119905
le sup0le120572le1
int119868
max 1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816 119889119905
le int119868
sup0le120572le1
max 1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816 119889119905 = int
119868
119863(1198911(119905) 1198912(119905)) 119889119905
(15)
Theorem 9 Let 119865(119909 119905) 119860 times 119868 rarr 119864 be a fuzzy functionsatisfying the following conditions
(1198671) For all 119909 isin 119860 119905 997891rarr 119865(119909 119905) is continuous on 119868
(1198672) For each 119905 isin 119868 119909 997891rarr 119865(119909 119905) is continuous on 119860 sub R
(1198673) For all 120572 isin [0 1] there exist a couple of nonnegativecontinuous crisp functions 120593
120572(119905) and 120595
120572(119905) which are
integrable on 119868 verifying for all 119909 isin 119860 119905 isin 1198681003816100381610038161003816119865 (119909 119905 120572)
1003816100381610038161003816 le 120593120572 (119905)
10038161003816100381610038161003816119865 (119909 119905 120572)
10038161003816100381610038161003816le 120595120572(119905)
(16)
Therefore the fuzzy mapping 120601(119909) = int119868
119865(119909 119905)119889119905 is continuouson 119860
Proof Let 119909 isin 119860 and let 119909119896infin
119896=1be a sequence of elements of
119860 which converges to 119909 as 119896 rarr infin For 119896 isin N 119905 isin 119868 and120572 isin [0 1] we have
119865 (119909119896 119905 0) le 119865 (119909
119896 119905 120572) le 119865 (119909
119896 119905 1)
119865 (119909119896 119905 1) le 119865 (119909
119896 119905 120572) le 119865 (119909
119896 119905 0)
(17)
Thus1003816100381610038161003816119865 (119909119896 119905 120572)
1003816100381610038161003816 le max 1003816100381610038161003816119865 (119909119896 119905 1)1003816100381610038161003816 1003816100381610038161003816119865 (119909119896 119905 0)
1003816100381610038161003816
le max 1205930(119905) 1205931(119905) = 119892 (119905)
10038161003816100381610038161003816119865 (119909119896 119905 120572)
10038161003816100381610038161003816le max 10038161003816100381610038161003816119865 (119909119896 119905 1)
1003816100381610038161003816100381610038161003816100381610038161003816119865 (119909119896 119905 0)
10038161003816100381610038161003816
le max 1205950(119905) 1205951(119905) = ℎ (119905)
(18)
By tending 119896 rarr infin and using assumption (1198672) we obtain
1003816100381610038161003816119865 (119909 119905 120572)1003816100381610038161003816 le max 120593
0(119905) 1205931(119905) = 119892 (119905)
10038161003816100381610038161003816119865 (119909 119905 120572)
10038161003816100381610038161003816le max 120595
0(119905) 1205951(119905) = ℎ (119905)
(19)
Therefore
119863(119865 (119909119896 119905) 119865 (119909 119905))
= sup0le120572le1
max 1003816100381610038161003816119865 (119909119896 119905 120572) minus 119865 (119909 119905 120572)1003816100381610038161003816
10038161003816100381610038161003816119865 (119909119896 119905 120572) minus 119865 (119909 119905 120572)
10038161003816100381610038161003816
119863 (119865 (119909119896 119905) 119865 (119909 119905)) le 2 (119892 (119905) + ℎ (119905))
(20)
From (1198671) and (119867
3) we deduce that the mappings 119892(119905) ℎ(119905)
and119863(119865(119909119896 119905) 119865(119909 119905)) are all integrable on 119868
On the other hand we get the following inequality fromLemma 8
119863(int119868
119865 (119909119896 119905) 119889119909 int
119868
119865 (119909 119905) 119889119909)
le int119868
119863(119865 (119909119896 119905) 119865 (119909 119905)) 119889119909
(21)
That is
119863(120601 (119909119896) 120601 (119909)) le int
119868
119863(119865 (119909119896 119905) 119865 (119909 119905)) 119889119909 (22)
By assumption (1198672) we have119863(119865(119909
119896 119905) 119865(119909 119905)) rarr 0 as 119896 rarr
infin
4 International Journal of Differential Equations
So by the dominated convergence theoremint119868
119863(119865(119909119896 119905) 119865(119909 119905))119889119909 rarr 0 as 119896 rarr infin
From inequality (22) we deduce that 120601(119909119896) rarr 120601(119909) as
119896 rarr infinConsequently 120601 is continuous on 119860
Lemma 10 One considers two fuzzy valued functions 1198911(119905)
1198912(119905) 119868 rarr 119864 which are fuzzy Riemann integrable on 119868 (in the
sense of Wu) such that 1198911(119905) ⊖ 119891
2(119905) exists for all 119905 isin 119868 then
1198911(119905) ⊖ 119891
2(119905) is fuzzy Riemann integrable on 119868 the Hukuhara
difference int119868
1198911(119905)119889119905 ⊖ int
119868
1198912(119905)119889119905 is well defined and
int119868
(1198911(119905) ⊖ 119891
2(119905)) 119889119909 = int
119868
1198911(119905) 119889119905 ⊖ int
119868
1198912(119905) 119889119905 (23)
Proof Let 119896(119905) = 1198911(119905) ⊖ 119891
2(119905) that is 119891
1(119905) = 119891
2(119905) + 119896(119905) It
is clear that there exist positive constants 119870(120572 1198911) 119870(120572 119891
1)
119870(120572 1198912) and119870(120572 119891
2) such that for all 119886 le 119887 in 119868 we have
int
119887
119886
1003816100381610038161003816100381610038161198911
(119905 120572)100381610038161003816100381610038161003816119889119905 le 119870 (120572 119891
1)
int
119887
119886
100381610038161003816100381610038161198911(119905 120572)
10038161003816100381610038161003816119889119905 le 119870 (120572 119891
1)
int
119887
119886
1003816100381610038161003816100381610038161198912
(119905 120572)100381610038161003816100381610038161003816119889119905 le 119870 (120572 119891
2)
int
119887
119886
100381610038161003816100381610038161198912(119905 120572)
10038161003816100381610038161003816119889119905 le 119870 (120572 119891
2)
(24)
Hence
int
119887
119886
1003816100381610038161003816119896 (119905 120572)1003816100381610038161003816 119889119905 = int
119887
119886
1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816119889119905
le 119870 (120572 1198911) + 119870 (120572 119891
2)
(25)
and similarly
int
119887
119886
10038161003816100381610038161003816119896 (119905 120572)
10038161003816100381610038161003816119889119905 = int
119887
119886
100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816119889119905
le 119870 (120572 1198911) + 119870 (120572 119891
2)
(26)
Then fromTheorem 2 119896(119905) is fuzzy Riemann integrable on 119868By ldquolinearityrdquo of the fuzzy integral we get
int119868
1198911(119905) 119889119905 = int
119868
1198912(119905) 119889119905 + int
119868
119896 (119905) 119889119905 (27)
Thus int119868
1198911(119905)119889119905⊖int
119868
1198912(119905)119889119905 exists and int
119868
1198911(119905)119889119905⊖int
119868
1198912(119905)119889119905 =
int119868
119896(119905)119889119905
Theorem 11 One considers a fuzzy valued function 119865(119909 119905) 119869 times 119868 rarr 119864 verifying the following assumptions
(1198601) For all 119909 isin 119869 119905 997891rarr 119865(119909 119905) is continuous and fuzzyRiemann integrable on 119868
(1198602) For all 119905 isin 119868 119909 997891rarr 119865(119909 119905) is (i)-differentiable on theinterval 119869
(1198603) For all 119909 isin 119869 119905 997891rarr (120597119865120597119909)(119909 119905) is continuous on 119868
(1198604) For all 119905 isin 119868 119909 997891rarr (120597119865120597119909)(119909 119905) is continuous on 119869
(1198605) For all 120572 isin [0 1] there exist a couple of continuouscrisp functions 120593
120572(119905) and120595
120572(119905) which are integrable on
119868 verifying for all 119909 isin 119869 119905 isin 119868
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(28)
Therefore the fuzzy mapping 120601(119909) = int119868
119865(119909 119905)119889119905 is (i)-differ-entiable on 119869 and
1206011015840
(119909) = int119868
120597119865
120597119909(119909 119905) 119889119905 forall119909 isin 119869 (29)
Moreover if one replaces assumption (1198602) by the alternative
condition
(11986010158402) for all 119905 isin 119868 119909 997891rarr 119865(119909 119905) is (ii)-differentiable on 119869
then the fuzzy function 120601(119909) is (ii)-differentiable on 119869 and (29)remains true
Proof Assume that (1198601)ndash(1198605) hold true Let 119909 isin 119869 120585
0gt 0
being very small and define the auxiliary functions
1198921(120585 119905) =
119865 (119909 + 120585 119905) ⊖ 119865 (119909 119905)
120585 120585 isin ]0 120585
0]
120597119865
120597119909(119909 119905) 120585 = 0
1198922(120585 119905) =
119865 (119909 119905) ⊖ 119865 (119909 minus 120585 119905)
120585 120585 isin ]0 120585
0]
120597119865
120597119909(119909 119905) 120585 = 0
(30)
For fixed 120585 isin]0 1205850] we have
120601 (119909 + 120585) ⊖ 120601 (119909)
120585
=1
120585(int119868
119865 (119909 + 120585 119905) 119889119905 ⊖ int119868
119865 (119909 119905) 119889119905)
= int119868
119865 (119909 + 120585 119905) ⊖ 119865 (119909 119905)
120585119889119905 = int
119868
1198921(120585 119905) 119889119905
(31)
where the existence of theHukuhara differences is ensured bythe (i)-differentiability of 119909 997891rarr 119865(119909 119905) and by Lemma 10
Analogously we get
120601 (119909) ⊖ 120601 (119909 minus 120585)
120585= int119868
1198922(120585 119905) 119889119905 (32)
From assumptions (1198601)ndash(1198604) we deduce that 119892
1and 119892
2
satisfy conditions (1198671)-(1198672) of Theorem 9
International Journal of Differential Equations 5
On the other hand using the finite increments theoremwe obtain
1003816100381610038161003816100381610038161198921
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198921 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(33)
Similarly we have
1003816100381610038161003816100381610038161198922
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198922 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(34)
Inequalities (33) and (34) which are obviously also true for120585 = 0 ensure that 119892
1and 119892
2satisfy condition (119867
3) of
Theorem 9Applying the latter theorem we get
lim120585rarr0+
120601 (119909 + 120585) ⊖ 120601 (119909)
120585= int119868
1198921(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
lim120585rarr0+
120601 (119909) ⊖ 120601 (119909 minus 120585)
120585= int119868
1198922(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
(35)
Therefore 120601 is (i)-differentiable at 119909 and
1206011015840
(119909) = int119868
120597119865
120597119909(119909 119905) 119889119905 (36)
The proof under assumption (11986010158402) instead of (119860
2) is similar
to the first case
Theorem 12 One considers a fuzzy function 119906(120585 120591)
[0infin[times[0infin[rarr 119864 Suppose that the mapping 119865(120585 120591) =
119890minus119904120591
119906(120585 120591) satisfies assumptions (1198601)ndash(1198605) above for all 119904 ge 119904
0
for some 1199040gt 0
Let L120591[119906(120585 120591)] or L[119906(120585 120591)] (for short) denote the fuzzy
Laplace transform of 119906(120585 120591) with respect to the time variable120591 Then
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)]) (37)
Proof For fixed 119904 ge 1199040 then usingTheorem 11 we have
L120591[119906120585(120585 120591)] = int
infin
0
119890minus119904120591
119906120585(120585 120591) 119889120591 = int
infin
0
119865120585(120585 120591) 119889120591
=120597
120597120585(int
infin
0
119865 (120585 120591) 119889120591)
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)])
(38)
Theorem 13 Let 119906(120585 120591) be a fuzzy valued function on[0infin[times[0infin[ into 119864 Suppose that the mappings 120591 997891rarr
119865(120585 120591) = 119890minus119904120591
119906(120585 120591) and 120591 997891rarr 119866(120585 120591) = 119890minus119904120591
119906120591(120585 120591) are fuzzy
Riemann integrable on [0infin[ for all 119904 ge 1199040for some 119904
0gt 0
Consider the following
(a) If 119906(120585 120591) is (i)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = 119904L
120591[119906 (120585 120591)] ⊖ 119906 (120585 0) (39)
(b) If 119906(119909 120591) is (ii)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = (minus119906 (120585 0)) ⊖ (minus119904) L
120591[119906 (120585 120591)] (40)
Proof This is a direct result of Theorem 12 by fixing 120585 ge 0and taking the Laplace transforms with respect to 120591
4 Fuzzy Laplace TransformAlgorithm for First-Order FuzzyPartial Differential Equations
Our aim now is to solve the following first-order FPDEusing the fuzzy Laplace transform method under stronglygeneralized differentiability
119906120585(120585 120591) + 119886119906
120591(120585 120591) = 119891 (120585 120591 119906 (120585 120591))
119906 (120585 0) = 119892 (120585)
119906 (0 120591) = ℎ (120591)
(41)
where 119906(120585 120591) is a fuzzy function of 120585 ge 0 120591 ge 0 119886 is areal constant and 119891(120585 120591 119906) 119892(120585) and ℎ(120591) are fuzzy valuedfunctions such that 119891(120585 120591 119906) is linear with respect to 119906 Forshort assume that 119886 ge 0 (case 119886 lt 0 is similar)
By using fuzzy Laplace transformwith respect to 120591 we get
L120591[119906120585(120585 120591)] + 119886L
120591[119906120591(120585 120591)] = L
120591[119891 (120585 120591 119906 (120585 120591))] (42)
Therefore we have to distinguish the following cases forsolving (42)
6 International Journal of Differential Equations
(a) Case 1 If 119906 is (i)-differentiable with respect to 120585 and120591 then by Laplace transform
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
(43)
where 119891(120585 120591 119906(120585 120591) 120572) = min119891(120585 120591 V)V isin (119906(120585120591 120572) 119906(120585 120591 120572)) and 119891(120585 120591 119906(120585 120591) 120572) = max119891(120585 120591V)V isin (119906(120585 120591 120572) 119906(120585 120591 120572))Using Theorems 12 and 13 we get the followingdifferential system
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(44)
satisfying the following initial conditions
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
(45)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
1(119904 120572)
L [119906 (120585 120591 120572)] = 1198701(119904 120572)
(46)
where (1198671(119904 120572) 119870
1(119904 120572)) is solution of system (44)
under (45)By the inverse Laplace transform we get
119906 (120585 120591 120572) =Lminus1
[1198671(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198701(119904 120572)]
(47)
(b) Case 2 If 119906 is (i)-differentiable with respect to 120585 and(ii)-differentiable with respect to 120591 then byTheorems12 and 13 we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(48)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198672(119904 120572)
L [119906 (120585 120591 120572)] = 1198702(119904 120572)
(49)
where (1198672(119904 120572) 119870
2(119904 120572)) is solution of system (48)
under (45)Thus
119906 (120585 120591 120572) =Lminus1
[1198672(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198702(119904 120572)]
(50)
(c) Case 3 If 119906 is (ii)-differentiable with respect to 120585and (i)-differentiable with respect to 120591 then we getthe following differential system satisfying the initialconditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(51)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198673(119904 120572)
L [119906 (120585 120591 120572)] = 1198703(119904 120572)
(52)
where (1198673(119904 120572) 119870
3(119904 120572)) is solution of system (51)
under (45)Therefore
119906 (120585 120591 120572) =Lminus1
[1198673(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198703(119904 120572)]
(53)
(d) Case 4 If 119906 is (ii)-differentiable with respect to 120585and 120591 then we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(54)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
4(119904 120572)
L [119906 (120585 120591 120572)] = 1198704(119904 120572)
(55)
International Journal of Differential Equations 7
where (1198674(119901 120572) 119870
4(119901 120572)) is solution of system (54)
under (45)
Hence
119906 (120585 120591 120572) =Lminus1
[1198674(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198704(119904 120572)]
(56)
5 Numerical Examples
Example 1 Consider
119906120585(120585 120591) = 3119906
120591(120585 120591) + 120585
119906 (120585 0 120572) = 3120585 sdot (120572 2 minus 120572) +1205852
2
119906 (0 120591 120572) = 120591 sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(57)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby Laplace transform we get
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] minus 9120572120585 minus
31205852
2
+120585
119904
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] + 9 (120572 minus 2) 120585
minus31205852
2+120585
119904
(58)
This differential system satisfies the following initialconditions
L [119906 (0 120591 120572)] =L [120572120591] =120572
1199042
L [119906 (0 120591 120572)] =L [(2 minus 120572) 120591] =(2 minus 120572)
1199042
(59)
Solving (58) under (59) we get
L [119906 (120585 120591 120572)] =
(3120572120585 + 1205852
2)
119904+120572
1199042
L [119906 (120585 120591 120572)] =
((6 minus 3120572) 120585 + 1205852
2)
119904+2 minus 120572
1199042
(60)
By the inverse Laplace transform we deduce
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(61)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 119906 (120585 120591 120572) minus 119906 (120585 120591 120572)
= 2 (1 minus 120572) (3120585 + 120591) ge 0
len (119906120585(120585 120591 120572)) = 119906
120585(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 6 (1 minus 120572) ge 0
len (119906120591(120585 120591 120572)) = 119906
120591(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 2 (1 minus 120572) ge 0
(62)
So this solution is valid for all 120585 ge 0 and 120591 ge 0Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(63)
where119867 is the unit step function or the Heaviside function
119867(120577) =
1 120577 ge 0
0 120577 lt 0
(64)
Therefore this solution 119906 is valid only over Δ = (120585 120591) |120585 ge 0 120591 ge 0 120591 le 3120585
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then similarly
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(65)
As in Case 2 one can verify that this solution is valid onlyover Δ1015840 = (120585 120591) | 120585 ge 0 120591 ge 0 120591 ge 3120585
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 then
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(66)
One can verify that function 119906 is not (ii)-differentiablewith respect to either 120585 or 120591
So no solution exists in this case
8 International Journal of Differential Equations
Example 2 Consider
119906120585(120585 120591) = 119906
120591(120585 120591)
119906 (120585 0 120572) = cos (120585) sdot (120572 2 minus 120572)
119906 (0 120591 120572) = cos (120591) sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(67)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby application of the algorithm above one obtains
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (68)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 2 (1 minus 120572) cos (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
(69)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 therefore
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (70)
Then this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (71)
Hence this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 thensimilarly
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (72)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
6 Conclusion
Theorems of continuity and differentiability for a fuzzyfunction defined via a fuzzy improper Riemann integral areprovedwhich are used to prove some results concerning fuzzyLaplace transform Then using Laplace transform methodthe solutions for some linear fuzzy partial differential equa-tions (FPDEs) of first order are given For future research onecan apply this method to solve FPDEs of high order
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H-C Wu ldquoThe improper fuzzy Riemann integral and itsnumerical integrationrdquo Information Sciences vol 111 no 1ndash4 pp109ndash137 1998
[2] T Allahviranloo and M B Ahmadi ldquoFuzzy Laplace trans-formsrdquo Soft Computing vol 14 no 3 pp 235ndash243 2010
[3] S SalahshourMKhezerloo SHajighasemi andMKhorasanyldquoSolving fuzzy integral equations of the second kind by fuzzylaplace transform methodrdquo International Journal of IndustrialMathematics vol 4 no 1 pp 21ndash29 2012
[4] S Salahshour and T Allahviranloo ldquoApplications of fuzzyLaplace transformsrdquo Soft Computing vol 17 no 1 pp 145ndash1582013
[5] E ElJaoui S Melliani and L S Chadli ldquoSolving second-orderfuzzy differential equations by the fuzzy Laplace transformmethodrdquo Advances in Difference Equations vol 2015 article 6614 pages 2015
[6] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[7] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[8] I J Rudas B Bede and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007
[9] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[10] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons amp Fractals vol 38no 1 pp 112ndash119 2008
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![Page 2: Research Article On Fuzzy Improper Integral and Its ...downloads.hindawi.com/journals/ijde/2016/7246027.pdf · Wu introduced in [] the improper fuzzy Riemann integral and presented](https://reader035.cupdf.com/reader035/viewer/2022071021/5fd5401e6ba5386cb526b789/html5/thumbnails/2.jpg)
2 International Journal of Differential Equations
where 119889 is the Hausdorff distance defined in 119875119888(R)Then it is
clear that (119864119863) is a complete metric space (for more detailsabout the metric119863 see [7])
Definition 1 (see [2]) One defines a fuzzy number V inparametric form as a couple (V V) of mappings V(120572) and V(120572)0 le 120572 le 1 verifying the following properties
(1) V(120572) is bounded increasing left continuous in ]0 1]and right continuous at 0
(2) V(120572) is bounded decreasing left continuous in ]0 1]and right continuous at 0
(3) V(120572) le V(120572) for all 0 le 120572 le 1
The following identity holds true (see [8])
119863(1205831 1205832) = sup0le120572le1
max 100381610038161003816100381610038161003816120583120572
1
minus 120583120572
2
1003816100381610038161003816100381610038161003816100381610038161003816120583120572
1minus 120583120572
2
1003816100381610038161003816 (4)
Theorem 2 (see [1]) One considers a fuzzy valued function119865(119909) = (119865(119909 120572) 119865(119909 120572)) defined on [119886infin[ Suppose that forall fixed 120572 isin [0 1] the crisp functions 119865(119909 120572) 119865(119909 120572) areintegrable on [119886 119887] for every 119887 ge 119886 and that there exist twopositive constants 119870(120572) and 119870(120572) such that int119887
119886
|119865(119909 120572)|119889119909 le
119870(120572) and int119887119886
|119865(119909 120572)|119889119909 le 119870(120572) for every 119887 ge 119886 Then 119865(119909)is fuzzy Riemann integrable (in the sense of Wu) on [119886infin[ itsimproper fuzzy integral intinfin
119886
119865(119909)119889119909 isin 119864 and
int
infin
119886
119865 (119909) 119889119909 = (int
infin
119886
119865 (119909 120572) 119889119909 int
infin
119886
119865 (119909 120572) 119889119909) (5)
For 1205831 1205832isin 119864 if there exists an element 120583
3isin 119864 such that
1205833= 1205831+ 1205832 then 120583
3is called the Hukuhara difference of 120583
1
and 1205832 which we denote by 120583
1⊖ 1205832
Definition 3 (see [2]) A mapping 119865 (119886 119887) rarr 119864 is said to bestrongly generalized differentiable at 119909 isin (119886 119887) if there exists1198651015840
(119909) isin 119864 such that
(i) for all ℎ gt 0 being very small there exist 119865(119909 + ℎ) ⊖119865(119909) 119865(119909) ⊖ 119865(119909 minus ℎ) and the limits
limℎrarr0+
119865 (119909 + ℎ) ⊖ 119865 (119909)
ℎ= limℎrarr0+
119865 (119909) ⊖ 119865 (119909 minus ℎ)
ℎ
= 1198651015840
(119909)
(6)
or
(ii) for all ℎ gt 0 being very small there exist 119865(119909)⊖119865(119909+ℎ) 119865(119909 minus ℎ) ⊖ 119865(119909) and the limits
limℎrarr0+
119865 (119909) ⊖ 119865 (119909 + ℎ)
(minusℎ)= limℎrarr0+
119865 (119909 minus ℎ) ⊖ 119865 (119909)
(minusℎ)
= 1198651015840
(119909)
(7)
or(iii) for all ℎ gt 0 being very small there exist 119865(119909 + ℎ) ⊖
119865(119909) 119865(119909 minus ℎ) ⊖ 119865(119909) and the limits
limℎrarr0+
119865 (119909 + ℎ) ⊖ 119865 (119909)
ℎ= limℎrarr0+
119865 (119909 minus ℎ) ⊖ 119865 (119909)
(minusℎ)
= 1198651015840
(119909)
(8)
or(iv) for all ℎ gt 0 being very small there exist 119865(119909)⊖119865(119909+
ℎ) 119865(119909) ⊖ 119865(119909 minus ℎ) and the limits
limℎrarr0+
119865 (119909) ⊖ 119865 (119909 + ℎ)
(minusℎ)= limℎrarr0+
119865 (119909) ⊖ 119865 (119909 minus ℎ)
ℎ
= 1198651015840
(119909)
(9)
The next theorem permits us to consider only case (i) orcase (ii) of Definition 3 almost everywhere in the domain ofthe mappings studied
Theorem4 (see [9]) If119865 (119886 119887) rarr 119864 is a strongly generalizeddifferentiable function on (119886 119887) in the sense of Definition 3 (iii)or (iv) then 1198651015840(119909) isin R for each 119909 isin (119886 119887)
Theorem 5 (see eg [10]) We consider a fuzzy function 119865 R rarr 119864 which is represented by 119865(119909) = (119865(119909 120572) 119865(119909 120572)) forall 120572 isin [0 1]
(1) If 119865 is (i)-differentiable then the crisp functions119865(119909 120572) and 119865(119909 120572) are differentiable and 1198651015840(119909) =(1198651015840
(119909 120572) 1198651015840
(119909 120572))(2) If 119865 is (ii)-differentiable then the crisp functions119865(119909 120572) and 119865(119909 120572) are differentiable and 1198651015840(119909) =(1198651015840
(119909 120572) 1198651015840
(119909 120572))
Definition 6 (see [2]) If 119865 [0infin[rarr 119864 is a continuousmapping such that 119890minus119904119909119865(119909) is fuzzy Riemann integrableon [0infin[ then intinfin
0
119890minus119904119909
119865(119909)119889119909 is called the fuzzy Laplacetransform of 119865 which one denotes by
L [119865 (119909)] = intinfin
0
119890minus119904119909
119865 (119909) 119889119909 119904 gt 0 (10)
Denote by L(119896(119909)) the classical Laplace transform of acrisp function 119896(119909) and then
L [119865 (119909)] = (L (119865 (119909 120572)) L (119865 (119909 120572))) (11)
Theorem 7 (see [2]) Let 119865 [0infin[rarr 119864 be a fuzzy valuedfunction and 1198651015840 its derivative on [0infin[ Then if 119865 is (i)-differentiable
L [1198651015840 (119909)] = 119904L [119865 (119909)] ⊖ 119865 (0) (12)
or if 119865 is (ii)-differentiable
L [1198651015840 (119909)] = (minus119865 (0)) ⊖ (minus119904) L [119865 (119909)] (13)
provided that the Laplace transforms of 119865 and 1198651015840 exist
International Journal of Differential Equations 3
3 Continuity and Differentiability ofFuzzy Improper Integral
In this section 119868 denotes one of the intervals ] minus infin 119887] or[119887infin[ or ]minusinfininfin[ where 119887 isin R 119869 denotes another intervaland 119860 is a nonempty subset of R
Lemma 8 Let 1198911(119905) 1198912(119905) be two fuzzy valued functions
which are fuzzy Riemann integrable on 119868 in the sense of Wu
(see [1]) such that the real function119863(1198911(119905) 1198912(119905)) is integrable
on 119868 and then
119863(int119868
1198911(119905) 119889119905 int
119868
1198912(119905) 119889119905) le int
119868
119863(1198911(119905) 1198912(119905)) 119889119905 (14)
Proof From identity (4) we have
119863(int119868
1198911(119905) 119889119905 int
119868
1198912(119905) 119889119905) = sup
0le120572le1
max 1003816100381610038161003816100381610038161003816int119868
1198911
(119905 120572) minus 1198912
(119905 120572) 119889119905
1003816100381610038161003816100381610038161003816
1003816100381610038161003816100381610038161003816int119868
1198911(119905 120572) 119889119905 minus 119891
2(119905 120572) 119889119905
1003816100381610038161003816100381610038161003816
le sup0le120572le1
max int119868
1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816119889119905 int119868
100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816119889119905
le sup0le120572le1
int119868
max 1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816 119889119905
le int119868
sup0le120572le1
max 1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816 119889119905 = int
119868
119863(1198911(119905) 1198912(119905)) 119889119905
(15)
Theorem 9 Let 119865(119909 119905) 119860 times 119868 rarr 119864 be a fuzzy functionsatisfying the following conditions
(1198671) For all 119909 isin 119860 119905 997891rarr 119865(119909 119905) is continuous on 119868
(1198672) For each 119905 isin 119868 119909 997891rarr 119865(119909 119905) is continuous on 119860 sub R
(1198673) For all 120572 isin [0 1] there exist a couple of nonnegativecontinuous crisp functions 120593
120572(119905) and 120595
120572(119905) which are
integrable on 119868 verifying for all 119909 isin 119860 119905 isin 1198681003816100381610038161003816119865 (119909 119905 120572)
1003816100381610038161003816 le 120593120572 (119905)
10038161003816100381610038161003816119865 (119909 119905 120572)
10038161003816100381610038161003816le 120595120572(119905)
(16)
Therefore the fuzzy mapping 120601(119909) = int119868
119865(119909 119905)119889119905 is continuouson 119860
Proof Let 119909 isin 119860 and let 119909119896infin
119896=1be a sequence of elements of
119860 which converges to 119909 as 119896 rarr infin For 119896 isin N 119905 isin 119868 and120572 isin [0 1] we have
119865 (119909119896 119905 0) le 119865 (119909
119896 119905 120572) le 119865 (119909
119896 119905 1)
119865 (119909119896 119905 1) le 119865 (119909
119896 119905 120572) le 119865 (119909
119896 119905 0)
(17)
Thus1003816100381610038161003816119865 (119909119896 119905 120572)
1003816100381610038161003816 le max 1003816100381610038161003816119865 (119909119896 119905 1)1003816100381610038161003816 1003816100381610038161003816119865 (119909119896 119905 0)
1003816100381610038161003816
le max 1205930(119905) 1205931(119905) = 119892 (119905)
10038161003816100381610038161003816119865 (119909119896 119905 120572)
10038161003816100381610038161003816le max 10038161003816100381610038161003816119865 (119909119896 119905 1)
1003816100381610038161003816100381610038161003816100381610038161003816119865 (119909119896 119905 0)
10038161003816100381610038161003816
le max 1205950(119905) 1205951(119905) = ℎ (119905)
(18)
By tending 119896 rarr infin and using assumption (1198672) we obtain
1003816100381610038161003816119865 (119909 119905 120572)1003816100381610038161003816 le max 120593
0(119905) 1205931(119905) = 119892 (119905)
10038161003816100381610038161003816119865 (119909 119905 120572)
10038161003816100381610038161003816le max 120595
0(119905) 1205951(119905) = ℎ (119905)
(19)
Therefore
119863(119865 (119909119896 119905) 119865 (119909 119905))
= sup0le120572le1
max 1003816100381610038161003816119865 (119909119896 119905 120572) minus 119865 (119909 119905 120572)1003816100381610038161003816
10038161003816100381610038161003816119865 (119909119896 119905 120572) minus 119865 (119909 119905 120572)
10038161003816100381610038161003816
119863 (119865 (119909119896 119905) 119865 (119909 119905)) le 2 (119892 (119905) + ℎ (119905))
(20)
From (1198671) and (119867
3) we deduce that the mappings 119892(119905) ℎ(119905)
and119863(119865(119909119896 119905) 119865(119909 119905)) are all integrable on 119868
On the other hand we get the following inequality fromLemma 8
119863(int119868
119865 (119909119896 119905) 119889119909 int
119868
119865 (119909 119905) 119889119909)
le int119868
119863(119865 (119909119896 119905) 119865 (119909 119905)) 119889119909
(21)
That is
119863(120601 (119909119896) 120601 (119909)) le int
119868
119863(119865 (119909119896 119905) 119865 (119909 119905)) 119889119909 (22)
By assumption (1198672) we have119863(119865(119909
119896 119905) 119865(119909 119905)) rarr 0 as 119896 rarr
infin
4 International Journal of Differential Equations
So by the dominated convergence theoremint119868
119863(119865(119909119896 119905) 119865(119909 119905))119889119909 rarr 0 as 119896 rarr infin
From inequality (22) we deduce that 120601(119909119896) rarr 120601(119909) as
119896 rarr infinConsequently 120601 is continuous on 119860
Lemma 10 One considers two fuzzy valued functions 1198911(119905)
1198912(119905) 119868 rarr 119864 which are fuzzy Riemann integrable on 119868 (in the
sense of Wu) such that 1198911(119905) ⊖ 119891
2(119905) exists for all 119905 isin 119868 then
1198911(119905) ⊖ 119891
2(119905) is fuzzy Riemann integrable on 119868 the Hukuhara
difference int119868
1198911(119905)119889119905 ⊖ int
119868
1198912(119905)119889119905 is well defined and
int119868
(1198911(119905) ⊖ 119891
2(119905)) 119889119909 = int
119868
1198911(119905) 119889119905 ⊖ int
119868
1198912(119905) 119889119905 (23)
Proof Let 119896(119905) = 1198911(119905) ⊖ 119891
2(119905) that is 119891
1(119905) = 119891
2(119905) + 119896(119905) It
is clear that there exist positive constants 119870(120572 1198911) 119870(120572 119891
1)
119870(120572 1198912) and119870(120572 119891
2) such that for all 119886 le 119887 in 119868 we have
int
119887
119886
1003816100381610038161003816100381610038161198911
(119905 120572)100381610038161003816100381610038161003816119889119905 le 119870 (120572 119891
1)
int
119887
119886
100381610038161003816100381610038161198911(119905 120572)
10038161003816100381610038161003816119889119905 le 119870 (120572 119891
1)
int
119887
119886
1003816100381610038161003816100381610038161198912
(119905 120572)100381610038161003816100381610038161003816119889119905 le 119870 (120572 119891
2)
int
119887
119886
100381610038161003816100381610038161198912(119905 120572)
10038161003816100381610038161003816119889119905 le 119870 (120572 119891
2)
(24)
Hence
int
119887
119886
1003816100381610038161003816119896 (119905 120572)1003816100381610038161003816 119889119905 = int
119887
119886
1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816119889119905
le 119870 (120572 1198911) + 119870 (120572 119891
2)
(25)
and similarly
int
119887
119886
10038161003816100381610038161003816119896 (119905 120572)
10038161003816100381610038161003816119889119905 = int
119887
119886
100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816119889119905
le 119870 (120572 1198911) + 119870 (120572 119891
2)
(26)
Then fromTheorem 2 119896(119905) is fuzzy Riemann integrable on 119868By ldquolinearityrdquo of the fuzzy integral we get
int119868
1198911(119905) 119889119905 = int
119868
1198912(119905) 119889119905 + int
119868
119896 (119905) 119889119905 (27)
Thus int119868
1198911(119905)119889119905⊖int
119868
1198912(119905)119889119905 exists and int
119868
1198911(119905)119889119905⊖int
119868
1198912(119905)119889119905 =
int119868
119896(119905)119889119905
Theorem 11 One considers a fuzzy valued function 119865(119909 119905) 119869 times 119868 rarr 119864 verifying the following assumptions
(1198601) For all 119909 isin 119869 119905 997891rarr 119865(119909 119905) is continuous and fuzzyRiemann integrable on 119868
(1198602) For all 119905 isin 119868 119909 997891rarr 119865(119909 119905) is (i)-differentiable on theinterval 119869
(1198603) For all 119909 isin 119869 119905 997891rarr (120597119865120597119909)(119909 119905) is continuous on 119868
(1198604) For all 119905 isin 119868 119909 997891rarr (120597119865120597119909)(119909 119905) is continuous on 119869
(1198605) For all 120572 isin [0 1] there exist a couple of continuouscrisp functions 120593
120572(119905) and120595
120572(119905) which are integrable on
119868 verifying for all 119909 isin 119869 119905 isin 119868
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(28)
Therefore the fuzzy mapping 120601(119909) = int119868
119865(119909 119905)119889119905 is (i)-differ-entiable on 119869 and
1206011015840
(119909) = int119868
120597119865
120597119909(119909 119905) 119889119905 forall119909 isin 119869 (29)
Moreover if one replaces assumption (1198602) by the alternative
condition
(11986010158402) for all 119905 isin 119868 119909 997891rarr 119865(119909 119905) is (ii)-differentiable on 119869
then the fuzzy function 120601(119909) is (ii)-differentiable on 119869 and (29)remains true
Proof Assume that (1198601)ndash(1198605) hold true Let 119909 isin 119869 120585
0gt 0
being very small and define the auxiliary functions
1198921(120585 119905) =
119865 (119909 + 120585 119905) ⊖ 119865 (119909 119905)
120585 120585 isin ]0 120585
0]
120597119865
120597119909(119909 119905) 120585 = 0
1198922(120585 119905) =
119865 (119909 119905) ⊖ 119865 (119909 minus 120585 119905)
120585 120585 isin ]0 120585
0]
120597119865
120597119909(119909 119905) 120585 = 0
(30)
For fixed 120585 isin]0 1205850] we have
120601 (119909 + 120585) ⊖ 120601 (119909)
120585
=1
120585(int119868
119865 (119909 + 120585 119905) 119889119905 ⊖ int119868
119865 (119909 119905) 119889119905)
= int119868
119865 (119909 + 120585 119905) ⊖ 119865 (119909 119905)
120585119889119905 = int
119868
1198921(120585 119905) 119889119905
(31)
where the existence of theHukuhara differences is ensured bythe (i)-differentiability of 119909 997891rarr 119865(119909 119905) and by Lemma 10
Analogously we get
120601 (119909) ⊖ 120601 (119909 minus 120585)
120585= int119868
1198922(120585 119905) 119889119905 (32)
From assumptions (1198601)ndash(1198604) we deduce that 119892
1and 119892
2
satisfy conditions (1198671)-(1198672) of Theorem 9
International Journal of Differential Equations 5
On the other hand using the finite increments theoremwe obtain
1003816100381610038161003816100381610038161198921
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198921 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(33)
Similarly we have
1003816100381610038161003816100381610038161198922
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198922 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(34)
Inequalities (33) and (34) which are obviously also true for120585 = 0 ensure that 119892
1and 119892
2satisfy condition (119867
3) of
Theorem 9Applying the latter theorem we get
lim120585rarr0+
120601 (119909 + 120585) ⊖ 120601 (119909)
120585= int119868
1198921(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
lim120585rarr0+
120601 (119909) ⊖ 120601 (119909 minus 120585)
120585= int119868
1198922(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
(35)
Therefore 120601 is (i)-differentiable at 119909 and
1206011015840
(119909) = int119868
120597119865
120597119909(119909 119905) 119889119905 (36)
The proof under assumption (11986010158402) instead of (119860
2) is similar
to the first case
Theorem 12 One considers a fuzzy function 119906(120585 120591)
[0infin[times[0infin[rarr 119864 Suppose that the mapping 119865(120585 120591) =
119890minus119904120591
119906(120585 120591) satisfies assumptions (1198601)ndash(1198605) above for all 119904 ge 119904
0
for some 1199040gt 0
Let L120591[119906(120585 120591)] or L[119906(120585 120591)] (for short) denote the fuzzy
Laplace transform of 119906(120585 120591) with respect to the time variable120591 Then
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)]) (37)
Proof For fixed 119904 ge 1199040 then usingTheorem 11 we have
L120591[119906120585(120585 120591)] = int
infin
0
119890minus119904120591
119906120585(120585 120591) 119889120591 = int
infin
0
119865120585(120585 120591) 119889120591
=120597
120597120585(int
infin
0
119865 (120585 120591) 119889120591)
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)])
(38)
Theorem 13 Let 119906(120585 120591) be a fuzzy valued function on[0infin[times[0infin[ into 119864 Suppose that the mappings 120591 997891rarr
119865(120585 120591) = 119890minus119904120591
119906(120585 120591) and 120591 997891rarr 119866(120585 120591) = 119890minus119904120591
119906120591(120585 120591) are fuzzy
Riemann integrable on [0infin[ for all 119904 ge 1199040for some 119904
0gt 0
Consider the following
(a) If 119906(120585 120591) is (i)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = 119904L
120591[119906 (120585 120591)] ⊖ 119906 (120585 0) (39)
(b) If 119906(119909 120591) is (ii)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = (minus119906 (120585 0)) ⊖ (minus119904) L
120591[119906 (120585 120591)] (40)
Proof This is a direct result of Theorem 12 by fixing 120585 ge 0and taking the Laplace transforms with respect to 120591
4 Fuzzy Laplace TransformAlgorithm for First-Order FuzzyPartial Differential Equations
Our aim now is to solve the following first-order FPDEusing the fuzzy Laplace transform method under stronglygeneralized differentiability
119906120585(120585 120591) + 119886119906
120591(120585 120591) = 119891 (120585 120591 119906 (120585 120591))
119906 (120585 0) = 119892 (120585)
119906 (0 120591) = ℎ (120591)
(41)
where 119906(120585 120591) is a fuzzy function of 120585 ge 0 120591 ge 0 119886 is areal constant and 119891(120585 120591 119906) 119892(120585) and ℎ(120591) are fuzzy valuedfunctions such that 119891(120585 120591 119906) is linear with respect to 119906 Forshort assume that 119886 ge 0 (case 119886 lt 0 is similar)
By using fuzzy Laplace transformwith respect to 120591 we get
L120591[119906120585(120585 120591)] + 119886L
120591[119906120591(120585 120591)] = L
120591[119891 (120585 120591 119906 (120585 120591))] (42)
Therefore we have to distinguish the following cases forsolving (42)
6 International Journal of Differential Equations
(a) Case 1 If 119906 is (i)-differentiable with respect to 120585 and120591 then by Laplace transform
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
(43)
where 119891(120585 120591 119906(120585 120591) 120572) = min119891(120585 120591 V)V isin (119906(120585120591 120572) 119906(120585 120591 120572)) and 119891(120585 120591 119906(120585 120591) 120572) = max119891(120585 120591V)V isin (119906(120585 120591 120572) 119906(120585 120591 120572))Using Theorems 12 and 13 we get the followingdifferential system
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(44)
satisfying the following initial conditions
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
(45)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
1(119904 120572)
L [119906 (120585 120591 120572)] = 1198701(119904 120572)
(46)
where (1198671(119904 120572) 119870
1(119904 120572)) is solution of system (44)
under (45)By the inverse Laplace transform we get
119906 (120585 120591 120572) =Lminus1
[1198671(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198701(119904 120572)]
(47)
(b) Case 2 If 119906 is (i)-differentiable with respect to 120585 and(ii)-differentiable with respect to 120591 then byTheorems12 and 13 we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(48)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198672(119904 120572)
L [119906 (120585 120591 120572)] = 1198702(119904 120572)
(49)
where (1198672(119904 120572) 119870
2(119904 120572)) is solution of system (48)
under (45)Thus
119906 (120585 120591 120572) =Lminus1
[1198672(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198702(119904 120572)]
(50)
(c) Case 3 If 119906 is (ii)-differentiable with respect to 120585and (i)-differentiable with respect to 120591 then we getthe following differential system satisfying the initialconditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(51)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198673(119904 120572)
L [119906 (120585 120591 120572)] = 1198703(119904 120572)
(52)
where (1198673(119904 120572) 119870
3(119904 120572)) is solution of system (51)
under (45)Therefore
119906 (120585 120591 120572) =Lminus1
[1198673(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198703(119904 120572)]
(53)
(d) Case 4 If 119906 is (ii)-differentiable with respect to 120585and 120591 then we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(54)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
4(119904 120572)
L [119906 (120585 120591 120572)] = 1198704(119904 120572)
(55)
International Journal of Differential Equations 7
where (1198674(119901 120572) 119870
4(119901 120572)) is solution of system (54)
under (45)
Hence
119906 (120585 120591 120572) =Lminus1
[1198674(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198704(119904 120572)]
(56)
5 Numerical Examples
Example 1 Consider
119906120585(120585 120591) = 3119906
120591(120585 120591) + 120585
119906 (120585 0 120572) = 3120585 sdot (120572 2 minus 120572) +1205852
2
119906 (0 120591 120572) = 120591 sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(57)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby Laplace transform we get
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] minus 9120572120585 minus
31205852
2
+120585
119904
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] + 9 (120572 minus 2) 120585
minus31205852
2+120585
119904
(58)
This differential system satisfies the following initialconditions
L [119906 (0 120591 120572)] =L [120572120591] =120572
1199042
L [119906 (0 120591 120572)] =L [(2 minus 120572) 120591] =(2 minus 120572)
1199042
(59)
Solving (58) under (59) we get
L [119906 (120585 120591 120572)] =
(3120572120585 + 1205852
2)
119904+120572
1199042
L [119906 (120585 120591 120572)] =
((6 minus 3120572) 120585 + 1205852
2)
119904+2 minus 120572
1199042
(60)
By the inverse Laplace transform we deduce
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(61)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 119906 (120585 120591 120572) minus 119906 (120585 120591 120572)
= 2 (1 minus 120572) (3120585 + 120591) ge 0
len (119906120585(120585 120591 120572)) = 119906
120585(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 6 (1 minus 120572) ge 0
len (119906120591(120585 120591 120572)) = 119906
120591(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 2 (1 minus 120572) ge 0
(62)
So this solution is valid for all 120585 ge 0 and 120591 ge 0Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(63)
where119867 is the unit step function or the Heaviside function
119867(120577) =
1 120577 ge 0
0 120577 lt 0
(64)
Therefore this solution 119906 is valid only over Δ = (120585 120591) |120585 ge 0 120591 ge 0 120591 le 3120585
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then similarly
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(65)
As in Case 2 one can verify that this solution is valid onlyover Δ1015840 = (120585 120591) | 120585 ge 0 120591 ge 0 120591 ge 3120585
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 then
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(66)
One can verify that function 119906 is not (ii)-differentiablewith respect to either 120585 or 120591
So no solution exists in this case
8 International Journal of Differential Equations
Example 2 Consider
119906120585(120585 120591) = 119906
120591(120585 120591)
119906 (120585 0 120572) = cos (120585) sdot (120572 2 minus 120572)
119906 (0 120591 120572) = cos (120591) sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(67)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby application of the algorithm above one obtains
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (68)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 2 (1 minus 120572) cos (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
(69)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 therefore
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (70)
Then this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (71)
Hence this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 thensimilarly
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (72)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
6 Conclusion
Theorems of continuity and differentiability for a fuzzyfunction defined via a fuzzy improper Riemann integral areprovedwhich are used to prove some results concerning fuzzyLaplace transform Then using Laplace transform methodthe solutions for some linear fuzzy partial differential equa-tions (FPDEs) of first order are given For future research onecan apply this method to solve FPDEs of high order
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H-C Wu ldquoThe improper fuzzy Riemann integral and itsnumerical integrationrdquo Information Sciences vol 111 no 1ndash4 pp109ndash137 1998
[2] T Allahviranloo and M B Ahmadi ldquoFuzzy Laplace trans-formsrdquo Soft Computing vol 14 no 3 pp 235ndash243 2010
[3] S SalahshourMKhezerloo SHajighasemi andMKhorasanyldquoSolving fuzzy integral equations of the second kind by fuzzylaplace transform methodrdquo International Journal of IndustrialMathematics vol 4 no 1 pp 21ndash29 2012
[4] S Salahshour and T Allahviranloo ldquoApplications of fuzzyLaplace transformsrdquo Soft Computing vol 17 no 1 pp 145ndash1582013
[5] E ElJaoui S Melliani and L S Chadli ldquoSolving second-orderfuzzy differential equations by the fuzzy Laplace transformmethodrdquo Advances in Difference Equations vol 2015 article 6614 pages 2015
[6] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[7] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[8] I J Rudas B Bede and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007
[9] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[10] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons amp Fractals vol 38no 1 pp 112ndash119 2008
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Stochastic AnalysisInternational Journal of
![Page 3: Research Article On Fuzzy Improper Integral and Its ...downloads.hindawi.com/journals/ijde/2016/7246027.pdf · Wu introduced in [] the improper fuzzy Riemann integral and presented](https://reader035.cupdf.com/reader035/viewer/2022071021/5fd5401e6ba5386cb526b789/html5/thumbnails/3.jpg)
International Journal of Differential Equations 3
3 Continuity and Differentiability ofFuzzy Improper Integral
In this section 119868 denotes one of the intervals ] minus infin 119887] or[119887infin[ or ]minusinfininfin[ where 119887 isin R 119869 denotes another intervaland 119860 is a nonempty subset of R
Lemma 8 Let 1198911(119905) 1198912(119905) be two fuzzy valued functions
which are fuzzy Riemann integrable on 119868 in the sense of Wu
(see [1]) such that the real function119863(1198911(119905) 1198912(119905)) is integrable
on 119868 and then
119863(int119868
1198911(119905) 119889119905 int
119868
1198912(119905) 119889119905) le int
119868
119863(1198911(119905) 1198912(119905)) 119889119905 (14)
Proof From identity (4) we have
119863(int119868
1198911(119905) 119889119905 int
119868
1198912(119905) 119889119905) = sup
0le120572le1
max 1003816100381610038161003816100381610038161003816int119868
1198911
(119905 120572) minus 1198912
(119905 120572) 119889119905
1003816100381610038161003816100381610038161003816
1003816100381610038161003816100381610038161003816int119868
1198911(119905 120572) 119889119905 minus 119891
2(119905 120572) 119889119905
1003816100381610038161003816100381610038161003816
le sup0le120572le1
max int119868
1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816119889119905 int119868
100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816119889119905
le sup0le120572le1
int119868
max 1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816 119889119905
le int119868
sup0le120572le1
max 1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816 119889119905 = int
119868
119863(1198911(119905) 1198912(119905)) 119889119905
(15)
Theorem 9 Let 119865(119909 119905) 119860 times 119868 rarr 119864 be a fuzzy functionsatisfying the following conditions
(1198671) For all 119909 isin 119860 119905 997891rarr 119865(119909 119905) is continuous on 119868
(1198672) For each 119905 isin 119868 119909 997891rarr 119865(119909 119905) is continuous on 119860 sub R
(1198673) For all 120572 isin [0 1] there exist a couple of nonnegativecontinuous crisp functions 120593
120572(119905) and 120595
120572(119905) which are
integrable on 119868 verifying for all 119909 isin 119860 119905 isin 1198681003816100381610038161003816119865 (119909 119905 120572)
1003816100381610038161003816 le 120593120572 (119905)
10038161003816100381610038161003816119865 (119909 119905 120572)
10038161003816100381610038161003816le 120595120572(119905)
(16)
Therefore the fuzzy mapping 120601(119909) = int119868
119865(119909 119905)119889119905 is continuouson 119860
Proof Let 119909 isin 119860 and let 119909119896infin
119896=1be a sequence of elements of
119860 which converges to 119909 as 119896 rarr infin For 119896 isin N 119905 isin 119868 and120572 isin [0 1] we have
119865 (119909119896 119905 0) le 119865 (119909
119896 119905 120572) le 119865 (119909
119896 119905 1)
119865 (119909119896 119905 1) le 119865 (119909
119896 119905 120572) le 119865 (119909
119896 119905 0)
(17)
Thus1003816100381610038161003816119865 (119909119896 119905 120572)
1003816100381610038161003816 le max 1003816100381610038161003816119865 (119909119896 119905 1)1003816100381610038161003816 1003816100381610038161003816119865 (119909119896 119905 0)
1003816100381610038161003816
le max 1205930(119905) 1205931(119905) = 119892 (119905)
10038161003816100381610038161003816119865 (119909119896 119905 120572)
10038161003816100381610038161003816le max 10038161003816100381610038161003816119865 (119909119896 119905 1)
1003816100381610038161003816100381610038161003816100381610038161003816119865 (119909119896 119905 0)
10038161003816100381610038161003816
le max 1205950(119905) 1205951(119905) = ℎ (119905)
(18)
By tending 119896 rarr infin and using assumption (1198672) we obtain
1003816100381610038161003816119865 (119909 119905 120572)1003816100381610038161003816 le max 120593
0(119905) 1205931(119905) = 119892 (119905)
10038161003816100381610038161003816119865 (119909 119905 120572)
10038161003816100381610038161003816le max 120595
0(119905) 1205951(119905) = ℎ (119905)
(19)
Therefore
119863(119865 (119909119896 119905) 119865 (119909 119905))
= sup0le120572le1
max 1003816100381610038161003816119865 (119909119896 119905 120572) minus 119865 (119909 119905 120572)1003816100381610038161003816
10038161003816100381610038161003816119865 (119909119896 119905 120572) minus 119865 (119909 119905 120572)
10038161003816100381610038161003816
119863 (119865 (119909119896 119905) 119865 (119909 119905)) le 2 (119892 (119905) + ℎ (119905))
(20)
From (1198671) and (119867
3) we deduce that the mappings 119892(119905) ℎ(119905)
and119863(119865(119909119896 119905) 119865(119909 119905)) are all integrable on 119868
On the other hand we get the following inequality fromLemma 8
119863(int119868
119865 (119909119896 119905) 119889119909 int
119868
119865 (119909 119905) 119889119909)
le int119868
119863(119865 (119909119896 119905) 119865 (119909 119905)) 119889119909
(21)
That is
119863(120601 (119909119896) 120601 (119909)) le int
119868
119863(119865 (119909119896 119905) 119865 (119909 119905)) 119889119909 (22)
By assumption (1198672) we have119863(119865(119909
119896 119905) 119865(119909 119905)) rarr 0 as 119896 rarr
infin
4 International Journal of Differential Equations
So by the dominated convergence theoremint119868
119863(119865(119909119896 119905) 119865(119909 119905))119889119909 rarr 0 as 119896 rarr infin
From inequality (22) we deduce that 120601(119909119896) rarr 120601(119909) as
119896 rarr infinConsequently 120601 is continuous on 119860
Lemma 10 One considers two fuzzy valued functions 1198911(119905)
1198912(119905) 119868 rarr 119864 which are fuzzy Riemann integrable on 119868 (in the
sense of Wu) such that 1198911(119905) ⊖ 119891
2(119905) exists for all 119905 isin 119868 then
1198911(119905) ⊖ 119891
2(119905) is fuzzy Riemann integrable on 119868 the Hukuhara
difference int119868
1198911(119905)119889119905 ⊖ int
119868
1198912(119905)119889119905 is well defined and
int119868
(1198911(119905) ⊖ 119891
2(119905)) 119889119909 = int
119868
1198911(119905) 119889119905 ⊖ int
119868
1198912(119905) 119889119905 (23)
Proof Let 119896(119905) = 1198911(119905) ⊖ 119891
2(119905) that is 119891
1(119905) = 119891
2(119905) + 119896(119905) It
is clear that there exist positive constants 119870(120572 1198911) 119870(120572 119891
1)
119870(120572 1198912) and119870(120572 119891
2) such that for all 119886 le 119887 in 119868 we have
int
119887
119886
1003816100381610038161003816100381610038161198911
(119905 120572)100381610038161003816100381610038161003816119889119905 le 119870 (120572 119891
1)
int
119887
119886
100381610038161003816100381610038161198911(119905 120572)
10038161003816100381610038161003816119889119905 le 119870 (120572 119891
1)
int
119887
119886
1003816100381610038161003816100381610038161198912
(119905 120572)100381610038161003816100381610038161003816119889119905 le 119870 (120572 119891
2)
int
119887
119886
100381610038161003816100381610038161198912(119905 120572)
10038161003816100381610038161003816119889119905 le 119870 (120572 119891
2)
(24)
Hence
int
119887
119886
1003816100381610038161003816119896 (119905 120572)1003816100381610038161003816 119889119905 = int
119887
119886
1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816119889119905
le 119870 (120572 1198911) + 119870 (120572 119891
2)
(25)
and similarly
int
119887
119886
10038161003816100381610038161003816119896 (119905 120572)
10038161003816100381610038161003816119889119905 = int
119887
119886
100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816119889119905
le 119870 (120572 1198911) + 119870 (120572 119891
2)
(26)
Then fromTheorem 2 119896(119905) is fuzzy Riemann integrable on 119868By ldquolinearityrdquo of the fuzzy integral we get
int119868
1198911(119905) 119889119905 = int
119868
1198912(119905) 119889119905 + int
119868
119896 (119905) 119889119905 (27)
Thus int119868
1198911(119905)119889119905⊖int
119868
1198912(119905)119889119905 exists and int
119868
1198911(119905)119889119905⊖int
119868
1198912(119905)119889119905 =
int119868
119896(119905)119889119905
Theorem 11 One considers a fuzzy valued function 119865(119909 119905) 119869 times 119868 rarr 119864 verifying the following assumptions
(1198601) For all 119909 isin 119869 119905 997891rarr 119865(119909 119905) is continuous and fuzzyRiemann integrable on 119868
(1198602) For all 119905 isin 119868 119909 997891rarr 119865(119909 119905) is (i)-differentiable on theinterval 119869
(1198603) For all 119909 isin 119869 119905 997891rarr (120597119865120597119909)(119909 119905) is continuous on 119868
(1198604) For all 119905 isin 119868 119909 997891rarr (120597119865120597119909)(119909 119905) is continuous on 119869
(1198605) For all 120572 isin [0 1] there exist a couple of continuouscrisp functions 120593
120572(119905) and120595
120572(119905) which are integrable on
119868 verifying for all 119909 isin 119869 119905 isin 119868
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(28)
Therefore the fuzzy mapping 120601(119909) = int119868
119865(119909 119905)119889119905 is (i)-differ-entiable on 119869 and
1206011015840
(119909) = int119868
120597119865
120597119909(119909 119905) 119889119905 forall119909 isin 119869 (29)
Moreover if one replaces assumption (1198602) by the alternative
condition
(11986010158402) for all 119905 isin 119868 119909 997891rarr 119865(119909 119905) is (ii)-differentiable on 119869
then the fuzzy function 120601(119909) is (ii)-differentiable on 119869 and (29)remains true
Proof Assume that (1198601)ndash(1198605) hold true Let 119909 isin 119869 120585
0gt 0
being very small and define the auxiliary functions
1198921(120585 119905) =
119865 (119909 + 120585 119905) ⊖ 119865 (119909 119905)
120585 120585 isin ]0 120585
0]
120597119865
120597119909(119909 119905) 120585 = 0
1198922(120585 119905) =
119865 (119909 119905) ⊖ 119865 (119909 minus 120585 119905)
120585 120585 isin ]0 120585
0]
120597119865
120597119909(119909 119905) 120585 = 0
(30)
For fixed 120585 isin]0 1205850] we have
120601 (119909 + 120585) ⊖ 120601 (119909)
120585
=1
120585(int119868
119865 (119909 + 120585 119905) 119889119905 ⊖ int119868
119865 (119909 119905) 119889119905)
= int119868
119865 (119909 + 120585 119905) ⊖ 119865 (119909 119905)
120585119889119905 = int
119868
1198921(120585 119905) 119889119905
(31)
where the existence of theHukuhara differences is ensured bythe (i)-differentiability of 119909 997891rarr 119865(119909 119905) and by Lemma 10
Analogously we get
120601 (119909) ⊖ 120601 (119909 minus 120585)
120585= int119868
1198922(120585 119905) 119889119905 (32)
From assumptions (1198601)ndash(1198604) we deduce that 119892
1and 119892
2
satisfy conditions (1198671)-(1198672) of Theorem 9
International Journal of Differential Equations 5
On the other hand using the finite increments theoremwe obtain
1003816100381610038161003816100381610038161198921
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198921 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(33)
Similarly we have
1003816100381610038161003816100381610038161198922
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198922 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(34)
Inequalities (33) and (34) which are obviously also true for120585 = 0 ensure that 119892
1and 119892
2satisfy condition (119867
3) of
Theorem 9Applying the latter theorem we get
lim120585rarr0+
120601 (119909 + 120585) ⊖ 120601 (119909)
120585= int119868
1198921(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
lim120585rarr0+
120601 (119909) ⊖ 120601 (119909 minus 120585)
120585= int119868
1198922(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
(35)
Therefore 120601 is (i)-differentiable at 119909 and
1206011015840
(119909) = int119868
120597119865
120597119909(119909 119905) 119889119905 (36)
The proof under assumption (11986010158402) instead of (119860
2) is similar
to the first case
Theorem 12 One considers a fuzzy function 119906(120585 120591)
[0infin[times[0infin[rarr 119864 Suppose that the mapping 119865(120585 120591) =
119890minus119904120591
119906(120585 120591) satisfies assumptions (1198601)ndash(1198605) above for all 119904 ge 119904
0
for some 1199040gt 0
Let L120591[119906(120585 120591)] or L[119906(120585 120591)] (for short) denote the fuzzy
Laplace transform of 119906(120585 120591) with respect to the time variable120591 Then
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)]) (37)
Proof For fixed 119904 ge 1199040 then usingTheorem 11 we have
L120591[119906120585(120585 120591)] = int
infin
0
119890minus119904120591
119906120585(120585 120591) 119889120591 = int
infin
0
119865120585(120585 120591) 119889120591
=120597
120597120585(int
infin
0
119865 (120585 120591) 119889120591)
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)])
(38)
Theorem 13 Let 119906(120585 120591) be a fuzzy valued function on[0infin[times[0infin[ into 119864 Suppose that the mappings 120591 997891rarr
119865(120585 120591) = 119890minus119904120591
119906(120585 120591) and 120591 997891rarr 119866(120585 120591) = 119890minus119904120591
119906120591(120585 120591) are fuzzy
Riemann integrable on [0infin[ for all 119904 ge 1199040for some 119904
0gt 0
Consider the following
(a) If 119906(120585 120591) is (i)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = 119904L
120591[119906 (120585 120591)] ⊖ 119906 (120585 0) (39)
(b) If 119906(119909 120591) is (ii)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = (minus119906 (120585 0)) ⊖ (minus119904) L
120591[119906 (120585 120591)] (40)
Proof This is a direct result of Theorem 12 by fixing 120585 ge 0and taking the Laplace transforms with respect to 120591
4 Fuzzy Laplace TransformAlgorithm for First-Order FuzzyPartial Differential Equations
Our aim now is to solve the following first-order FPDEusing the fuzzy Laplace transform method under stronglygeneralized differentiability
119906120585(120585 120591) + 119886119906
120591(120585 120591) = 119891 (120585 120591 119906 (120585 120591))
119906 (120585 0) = 119892 (120585)
119906 (0 120591) = ℎ (120591)
(41)
where 119906(120585 120591) is a fuzzy function of 120585 ge 0 120591 ge 0 119886 is areal constant and 119891(120585 120591 119906) 119892(120585) and ℎ(120591) are fuzzy valuedfunctions such that 119891(120585 120591 119906) is linear with respect to 119906 Forshort assume that 119886 ge 0 (case 119886 lt 0 is similar)
By using fuzzy Laplace transformwith respect to 120591 we get
L120591[119906120585(120585 120591)] + 119886L
120591[119906120591(120585 120591)] = L
120591[119891 (120585 120591 119906 (120585 120591))] (42)
Therefore we have to distinguish the following cases forsolving (42)
6 International Journal of Differential Equations
(a) Case 1 If 119906 is (i)-differentiable with respect to 120585 and120591 then by Laplace transform
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
(43)
where 119891(120585 120591 119906(120585 120591) 120572) = min119891(120585 120591 V)V isin (119906(120585120591 120572) 119906(120585 120591 120572)) and 119891(120585 120591 119906(120585 120591) 120572) = max119891(120585 120591V)V isin (119906(120585 120591 120572) 119906(120585 120591 120572))Using Theorems 12 and 13 we get the followingdifferential system
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(44)
satisfying the following initial conditions
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
(45)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
1(119904 120572)
L [119906 (120585 120591 120572)] = 1198701(119904 120572)
(46)
where (1198671(119904 120572) 119870
1(119904 120572)) is solution of system (44)
under (45)By the inverse Laplace transform we get
119906 (120585 120591 120572) =Lminus1
[1198671(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198701(119904 120572)]
(47)
(b) Case 2 If 119906 is (i)-differentiable with respect to 120585 and(ii)-differentiable with respect to 120591 then byTheorems12 and 13 we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(48)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198672(119904 120572)
L [119906 (120585 120591 120572)] = 1198702(119904 120572)
(49)
where (1198672(119904 120572) 119870
2(119904 120572)) is solution of system (48)
under (45)Thus
119906 (120585 120591 120572) =Lminus1
[1198672(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198702(119904 120572)]
(50)
(c) Case 3 If 119906 is (ii)-differentiable with respect to 120585and (i)-differentiable with respect to 120591 then we getthe following differential system satisfying the initialconditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(51)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198673(119904 120572)
L [119906 (120585 120591 120572)] = 1198703(119904 120572)
(52)
where (1198673(119904 120572) 119870
3(119904 120572)) is solution of system (51)
under (45)Therefore
119906 (120585 120591 120572) =Lminus1
[1198673(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198703(119904 120572)]
(53)
(d) Case 4 If 119906 is (ii)-differentiable with respect to 120585and 120591 then we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(54)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
4(119904 120572)
L [119906 (120585 120591 120572)] = 1198704(119904 120572)
(55)
International Journal of Differential Equations 7
where (1198674(119901 120572) 119870
4(119901 120572)) is solution of system (54)
under (45)
Hence
119906 (120585 120591 120572) =Lminus1
[1198674(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198704(119904 120572)]
(56)
5 Numerical Examples
Example 1 Consider
119906120585(120585 120591) = 3119906
120591(120585 120591) + 120585
119906 (120585 0 120572) = 3120585 sdot (120572 2 minus 120572) +1205852
2
119906 (0 120591 120572) = 120591 sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(57)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby Laplace transform we get
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] minus 9120572120585 minus
31205852
2
+120585
119904
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] + 9 (120572 minus 2) 120585
minus31205852
2+120585
119904
(58)
This differential system satisfies the following initialconditions
L [119906 (0 120591 120572)] =L [120572120591] =120572
1199042
L [119906 (0 120591 120572)] =L [(2 minus 120572) 120591] =(2 minus 120572)
1199042
(59)
Solving (58) under (59) we get
L [119906 (120585 120591 120572)] =
(3120572120585 + 1205852
2)
119904+120572
1199042
L [119906 (120585 120591 120572)] =
((6 minus 3120572) 120585 + 1205852
2)
119904+2 minus 120572
1199042
(60)
By the inverse Laplace transform we deduce
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(61)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 119906 (120585 120591 120572) minus 119906 (120585 120591 120572)
= 2 (1 minus 120572) (3120585 + 120591) ge 0
len (119906120585(120585 120591 120572)) = 119906
120585(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 6 (1 minus 120572) ge 0
len (119906120591(120585 120591 120572)) = 119906
120591(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 2 (1 minus 120572) ge 0
(62)
So this solution is valid for all 120585 ge 0 and 120591 ge 0Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(63)
where119867 is the unit step function or the Heaviside function
119867(120577) =
1 120577 ge 0
0 120577 lt 0
(64)
Therefore this solution 119906 is valid only over Δ = (120585 120591) |120585 ge 0 120591 ge 0 120591 le 3120585
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then similarly
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(65)
As in Case 2 one can verify that this solution is valid onlyover Δ1015840 = (120585 120591) | 120585 ge 0 120591 ge 0 120591 ge 3120585
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 then
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(66)
One can verify that function 119906 is not (ii)-differentiablewith respect to either 120585 or 120591
So no solution exists in this case
8 International Journal of Differential Equations
Example 2 Consider
119906120585(120585 120591) = 119906
120591(120585 120591)
119906 (120585 0 120572) = cos (120585) sdot (120572 2 minus 120572)
119906 (0 120591 120572) = cos (120591) sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(67)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby application of the algorithm above one obtains
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (68)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 2 (1 minus 120572) cos (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
(69)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 therefore
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (70)
Then this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (71)
Hence this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 thensimilarly
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (72)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
6 Conclusion
Theorems of continuity and differentiability for a fuzzyfunction defined via a fuzzy improper Riemann integral areprovedwhich are used to prove some results concerning fuzzyLaplace transform Then using Laplace transform methodthe solutions for some linear fuzzy partial differential equa-tions (FPDEs) of first order are given For future research onecan apply this method to solve FPDEs of high order
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H-C Wu ldquoThe improper fuzzy Riemann integral and itsnumerical integrationrdquo Information Sciences vol 111 no 1ndash4 pp109ndash137 1998
[2] T Allahviranloo and M B Ahmadi ldquoFuzzy Laplace trans-formsrdquo Soft Computing vol 14 no 3 pp 235ndash243 2010
[3] S SalahshourMKhezerloo SHajighasemi andMKhorasanyldquoSolving fuzzy integral equations of the second kind by fuzzylaplace transform methodrdquo International Journal of IndustrialMathematics vol 4 no 1 pp 21ndash29 2012
[4] S Salahshour and T Allahviranloo ldquoApplications of fuzzyLaplace transformsrdquo Soft Computing vol 17 no 1 pp 145ndash1582013
[5] E ElJaoui S Melliani and L S Chadli ldquoSolving second-orderfuzzy differential equations by the fuzzy Laplace transformmethodrdquo Advances in Difference Equations vol 2015 article 6614 pages 2015
[6] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[7] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[8] I J Rudas B Bede and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007
[9] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[10] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons amp Fractals vol 38no 1 pp 112ndash119 2008
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 4: Research Article On Fuzzy Improper Integral and Its ...downloads.hindawi.com/journals/ijde/2016/7246027.pdf · Wu introduced in [] the improper fuzzy Riemann integral and presented](https://reader035.cupdf.com/reader035/viewer/2022071021/5fd5401e6ba5386cb526b789/html5/thumbnails/4.jpg)
4 International Journal of Differential Equations
So by the dominated convergence theoremint119868
119863(119865(119909119896 119905) 119865(119909 119905))119889119909 rarr 0 as 119896 rarr infin
From inequality (22) we deduce that 120601(119909119896) rarr 120601(119909) as
119896 rarr infinConsequently 120601 is continuous on 119860
Lemma 10 One considers two fuzzy valued functions 1198911(119905)
1198912(119905) 119868 rarr 119864 which are fuzzy Riemann integrable on 119868 (in the
sense of Wu) such that 1198911(119905) ⊖ 119891
2(119905) exists for all 119905 isin 119868 then
1198911(119905) ⊖ 119891
2(119905) is fuzzy Riemann integrable on 119868 the Hukuhara
difference int119868
1198911(119905)119889119905 ⊖ int
119868
1198912(119905)119889119905 is well defined and
int119868
(1198911(119905) ⊖ 119891
2(119905)) 119889119909 = int
119868
1198911(119905) 119889119905 ⊖ int
119868
1198912(119905) 119889119905 (23)
Proof Let 119896(119905) = 1198911(119905) ⊖ 119891
2(119905) that is 119891
1(119905) = 119891
2(119905) + 119896(119905) It
is clear that there exist positive constants 119870(120572 1198911) 119870(120572 119891
1)
119870(120572 1198912) and119870(120572 119891
2) such that for all 119886 le 119887 in 119868 we have
int
119887
119886
1003816100381610038161003816100381610038161198911
(119905 120572)100381610038161003816100381610038161003816119889119905 le 119870 (120572 119891
1)
int
119887
119886
100381610038161003816100381610038161198911(119905 120572)
10038161003816100381610038161003816119889119905 le 119870 (120572 119891
1)
int
119887
119886
1003816100381610038161003816100381610038161198912
(119905 120572)100381610038161003816100381610038161003816119889119905 le 119870 (120572 119891
2)
int
119887
119886
100381610038161003816100381610038161198912(119905 120572)
10038161003816100381610038161003816119889119905 le 119870 (120572 119891
2)
(24)
Hence
int
119887
119886
1003816100381610038161003816119896 (119905 120572)1003816100381610038161003816 119889119905 = int
119887
119886
1003816100381610038161003816100381610038161198911
(119905 120572) minus 1198912
(119905 120572)100381610038161003816100381610038161003816119889119905
le 119870 (120572 1198911) + 119870 (120572 119891
2)
(25)
and similarly
int
119887
119886
10038161003816100381610038161003816119896 (119905 120572)
10038161003816100381610038161003816119889119905 = int
119887
119886
100381610038161003816100381610038161198911(119905 120572) minus 119891
2(119905 120572)
10038161003816100381610038161003816119889119905
le 119870 (120572 1198911) + 119870 (120572 119891
2)
(26)
Then fromTheorem 2 119896(119905) is fuzzy Riemann integrable on 119868By ldquolinearityrdquo of the fuzzy integral we get
int119868
1198911(119905) 119889119905 = int
119868
1198912(119905) 119889119905 + int
119868
119896 (119905) 119889119905 (27)
Thus int119868
1198911(119905)119889119905⊖int
119868
1198912(119905)119889119905 exists and int
119868
1198911(119905)119889119905⊖int
119868
1198912(119905)119889119905 =
int119868
119896(119905)119889119905
Theorem 11 One considers a fuzzy valued function 119865(119909 119905) 119869 times 119868 rarr 119864 verifying the following assumptions
(1198601) For all 119909 isin 119869 119905 997891rarr 119865(119909 119905) is continuous and fuzzyRiemann integrable on 119868
(1198602) For all 119905 isin 119868 119909 997891rarr 119865(119909 119905) is (i)-differentiable on theinterval 119869
(1198603) For all 119909 isin 119869 119905 997891rarr (120597119865120597119909)(119909 119905) is continuous on 119868
(1198604) For all 119905 isin 119868 119909 997891rarr (120597119865120597119909)(119909 119905) is continuous on 119869
(1198605) For all 120572 isin [0 1] there exist a couple of continuouscrisp functions 120593
120572(119905) and120595
120572(119905) which are integrable on
119868 verifying for all 119909 isin 119869 119905 isin 119868
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(28)
Therefore the fuzzy mapping 120601(119909) = int119868
119865(119909 119905)119889119905 is (i)-differ-entiable on 119869 and
1206011015840
(119909) = int119868
120597119865
120597119909(119909 119905) 119889119905 forall119909 isin 119869 (29)
Moreover if one replaces assumption (1198602) by the alternative
condition
(11986010158402) for all 119905 isin 119868 119909 997891rarr 119865(119909 119905) is (ii)-differentiable on 119869
then the fuzzy function 120601(119909) is (ii)-differentiable on 119869 and (29)remains true
Proof Assume that (1198601)ndash(1198605) hold true Let 119909 isin 119869 120585
0gt 0
being very small and define the auxiliary functions
1198921(120585 119905) =
119865 (119909 + 120585 119905) ⊖ 119865 (119909 119905)
120585 120585 isin ]0 120585
0]
120597119865
120597119909(119909 119905) 120585 = 0
1198922(120585 119905) =
119865 (119909 119905) ⊖ 119865 (119909 minus 120585 119905)
120585 120585 isin ]0 120585
0]
120597119865
120597119909(119909 119905) 120585 = 0
(30)
For fixed 120585 isin]0 1205850] we have
120601 (119909 + 120585) ⊖ 120601 (119909)
120585
=1
120585(int119868
119865 (119909 + 120585 119905) 119889119905 ⊖ int119868
119865 (119909 119905) 119889119905)
= int119868
119865 (119909 + 120585 119905) ⊖ 119865 (119909 119905)
120585119889119905 = int
119868
1198921(120585 119905) 119889119905
(31)
where the existence of theHukuhara differences is ensured bythe (i)-differentiability of 119909 997891rarr 119865(119909 119905) and by Lemma 10
Analogously we get
120601 (119909) ⊖ 120601 (119909 minus 120585)
120585= int119868
1198922(120585 119905) 119889119905 (32)
From assumptions (1198601)ndash(1198604) we deduce that 119892
1and 119892
2
satisfy conditions (1198671)-(1198672) of Theorem 9
International Journal of Differential Equations 5
On the other hand using the finite increments theoremwe obtain
1003816100381610038161003816100381610038161198921
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198921 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(33)
Similarly we have
1003816100381610038161003816100381610038161198922
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198922 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(34)
Inequalities (33) and (34) which are obviously also true for120585 = 0 ensure that 119892
1and 119892
2satisfy condition (119867
3) of
Theorem 9Applying the latter theorem we get
lim120585rarr0+
120601 (119909 + 120585) ⊖ 120601 (119909)
120585= int119868
1198921(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
lim120585rarr0+
120601 (119909) ⊖ 120601 (119909 minus 120585)
120585= int119868
1198922(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
(35)
Therefore 120601 is (i)-differentiable at 119909 and
1206011015840
(119909) = int119868
120597119865
120597119909(119909 119905) 119889119905 (36)
The proof under assumption (11986010158402) instead of (119860
2) is similar
to the first case
Theorem 12 One considers a fuzzy function 119906(120585 120591)
[0infin[times[0infin[rarr 119864 Suppose that the mapping 119865(120585 120591) =
119890minus119904120591
119906(120585 120591) satisfies assumptions (1198601)ndash(1198605) above for all 119904 ge 119904
0
for some 1199040gt 0
Let L120591[119906(120585 120591)] or L[119906(120585 120591)] (for short) denote the fuzzy
Laplace transform of 119906(120585 120591) with respect to the time variable120591 Then
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)]) (37)
Proof For fixed 119904 ge 1199040 then usingTheorem 11 we have
L120591[119906120585(120585 120591)] = int
infin
0
119890minus119904120591
119906120585(120585 120591) 119889120591 = int
infin
0
119865120585(120585 120591) 119889120591
=120597
120597120585(int
infin
0
119865 (120585 120591) 119889120591)
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)])
(38)
Theorem 13 Let 119906(120585 120591) be a fuzzy valued function on[0infin[times[0infin[ into 119864 Suppose that the mappings 120591 997891rarr
119865(120585 120591) = 119890minus119904120591
119906(120585 120591) and 120591 997891rarr 119866(120585 120591) = 119890minus119904120591
119906120591(120585 120591) are fuzzy
Riemann integrable on [0infin[ for all 119904 ge 1199040for some 119904
0gt 0
Consider the following
(a) If 119906(120585 120591) is (i)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = 119904L
120591[119906 (120585 120591)] ⊖ 119906 (120585 0) (39)
(b) If 119906(119909 120591) is (ii)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = (minus119906 (120585 0)) ⊖ (minus119904) L
120591[119906 (120585 120591)] (40)
Proof This is a direct result of Theorem 12 by fixing 120585 ge 0and taking the Laplace transforms with respect to 120591
4 Fuzzy Laplace TransformAlgorithm for First-Order FuzzyPartial Differential Equations
Our aim now is to solve the following first-order FPDEusing the fuzzy Laplace transform method under stronglygeneralized differentiability
119906120585(120585 120591) + 119886119906
120591(120585 120591) = 119891 (120585 120591 119906 (120585 120591))
119906 (120585 0) = 119892 (120585)
119906 (0 120591) = ℎ (120591)
(41)
where 119906(120585 120591) is a fuzzy function of 120585 ge 0 120591 ge 0 119886 is areal constant and 119891(120585 120591 119906) 119892(120585) and ℎ(120591) are fuzzy valuedfunctions such that 119891(120585 120591 119906) is linear with respect to 119906 Forshort assume that 119886 ge 0 (case 119886 lt 0 is similar)
By using fuzzy Laplace transformwith respect to 120591 we get
L120591[119906120585(120585 120591)] + 119886L
120591[119906120591(120585 120591)] = L
120591[119891 (120585 120591 119906 (120585 120591))] (42)
Therefore we have to distinguish the following cases forsolving (42)
6 International Journal of Differential Equations
(a) Case 1 If 119906 is (i)-differentiable with respect to 120585 and120591 then by Laplace transform
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
(43)
where 119891(120585 120591 119906(120585 120591) 120572) = min119891(120585 120591 V)V isin (119906(120585120591 120572) 119906(120585 120591 120572)) and 119891(120585 120591 119906(120585 120591) 120572) = max119891(120585 120591V)V isin (119906(120585 120591 120572) 119906(120585 120591 120572))Using Theorems 12 and 13 we get the followingdifferential system
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(44)
satisfying the following initial conditions
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
(45)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
1(119904 120572)
L [119906 (120585 120591 120572)] = 1198701(119904 120572)
(46)
where (1198671(119904 120572) 119870
1(119904 120572)) is solution of system (44)
under (45)By the inverse Laplace transform we get
119906 (120585 120591 120572) =Lminus1
[1198671(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198701(119904 120572)]
(47)
(b) Case 2 If 119906 is (i)-differentiable with respect to 120585 and(ii)-differentiable with respect to 120591 then byTheorems12 and 13 we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(48)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198672(119904 120572)
L [119906 (120585 120591 120572)] = 1198702(119904 120572)
(49)
where (1198672(119904 120572) 119870
2(119904 120572)) is solution of system (48)
under (45)Thus
119906 (120585 120591 120572) =Lminus1
[1198672(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198702(119904 120572)]
(50)
(c) Case 3 If 119906 is (ii)-differentiable with respect to 120585and (i)-differentiable with respect to 120591 then we getthe following differential system satisfying the initialconditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(51)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198673(119904 120572)
L [119906 (120585 120591 120572)] = 1198703(119904 120572)
(52)
where (1198673(119904 120572) 119870
3(119904 120572)) is solution of system (51)
under (45)Therefore
119906 (120585 120591 120572) =Lminus1
[1198673(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198703(119904 120572)]
(53)
(d) Case 4 If 119906 is (ii)-differentiable with respect to 120585and 120591 then we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(54)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
4(119904 120572)
L [119906 (120585 120591 120572)] = 1198704(119904 120572)
(55)
International Journal of Differential Equations 7
where (1198674(119901 120572) 119870
4(119901 120572)) is solution of system (54)
under (45)
Hence
119906 (120585 120591 120572) =Lminus1
[1198674(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198704(119904 120572)]
(56)
5 Numerical Examples
Example 1 Consider
119906120585(120585 120591) = 3119906
120591(120585 120591) + 120585
119906 (120585 0 120572) = 3120585 sdot (120572 2 minus 120572) +1205852
2
119906 (0 120591 120572) = 120591 sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(57)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby Laplace transform we get
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] minus 9120572120585 minus
31205852
2
+120585
119904
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] + 9 (120572 minus 2) 120585
minus31205852
2+120585
119904
(58)
This differential system satisfies the following initialconditions
L [119906 (0 120591 120572)] =L [120572120591] =120572
1199042
L [119906 (0 120591 120572)] =L [(2 minus 120572) 120591] =(2 minus 120572)
1199042
(59)
Solving (58) under (59) we get
L [119906 (120585 120591 120572)] =
(3120572120585 + 1205852
2)
119904+120572
1199042
L [119906 (120585 120591 120572)] =
((6 minus 3120572) 120585 + 1205852
2)
119904+2 minus 120572
1199042
(60)
By the inverse Laplace transform we deduce
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(61)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 119906 (120585 120591 120572) minus 119906 (120585 120591 120572)
= 2 (1 minus 120572) (3120585 + 120591) ge 0
len (119906120585(120585 120591 120572)) = 119906
120585(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 6 (1 minus 120572) ge 0
len (119906120591(120585 120591 120572)) = 119906
120591(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 2 (1 minus 120572) ge 0
(62)
So this solution is valid for all 120585 ge 0 and 120591 ge 0Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(63)
where119867 is the unit step function or the Heaviside function
119867(120577) =
1 120577 ge 0
0 120577 lt 0
(64)
Therefore this solution 119906 is valid only over Δ = (120585 120591) |120585 ge 0 120591 ge 0 120591 le 3120585
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then similarly
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(65)
As in Case 2 one can verify that this solution is valid onlyover Δ1015840 = (120585 120591) | 120585 ge 0 120591 ge 0 120591 ge 3120585
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 then
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(66)
One can verify that function 119906 is not (ii)-differentiablewith respect to either 120585 or 120591
So no solution exists in this case
8 International Journal of Differential Equations
Example 2 Consider
119906120585(120585 120591) = 119906
120591(120585 120591)
119906 (120585 0 120572) = cos (120585) sdot (120572 2 minus 120572)
119906 (0 120591 120572) = cos (120591) sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(67)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby application of the algorithm above one obtains
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (68)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 2 (1 minus 120572) cos (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
(69)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 therefore
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (70)
Then this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (71)
Hence this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 thensimilarly
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (72)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
6 Conclusion
Theorems of continuity and differentiability for a fuzzyfunction defined via a fuzzy improper Riemann integral areprovedwhich are used to prove some results concerning fuzzyLaplace transform Then using Laplace transform methodthe solutions for some linear fuzzy partial differential equa-tions (FPDEs) of first order are given For future research onecan apply this method to solve FPDEs of high order
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H-C Wu ldquoThe improper fuzzy Riemann integral and itsnumerical integrationrdquo Information Sciences vol 111 no 1ndash4 pp109ndash137 1998
[2] T Allahviranloo and M B Ahmadi ldquoFuzzy Laplace trans-formsrdquo Soft Computing vol 14 no 3 pp 235ndash243 2010
[3] S SalahshourMKhezerloo SHajighasemi andMKhorasanyldquoSolving fuzzy integral equations of the second kind by fuzzylaplace transform methodrdquo International Journal of IndustrialMathematics vol 4 no 1 pp 21ndash29 2012
[4] S Salahshour and T Allahviranloo ldquoApplications of fuzzyLaplace transformsrdquo Soft Computing vol 17 no 1 pp 145ndash1582013
[5] E ElJaoui S Melliani and L S Chadli ldquoSolving second-orderfuzzy differential equations by the fuzzy Laplace transformmethodrdquo Advances in Difference Equations vol 2015 article 6614 pages 2015
[6] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[7] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[8] I J Rudas B Bede and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007
[9] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[10] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons amp Fractals vol 38no 1 pp 112ndash119 2008
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Stochastic AnalysisInternational Journal of
![Page 5: Research Article On Fuzzy Improper Integral and Its ...downloads.hindawi.com/journals/ijde/2016/7246027.pdf · Wu introduced in [] the improper fuzzy Riemann integral and presented](https://reader035.cupdf.com/reader035/viewer/2022071021/5fd5401e6ba5386cb526b789/html5/thumbnails/5.jpg)
International Journal of Differential Equations 5
On the other hand using the finite increments theoremwe obtain
1003816100381610038161003816100381610038161198921
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198921 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 + 120585 119905 120572) minus 119865 (119909 119905 120572)
120585
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 + V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(33)
Similarly we have
1003816100381610038161003816100381610038161198922
(120585 119905 120572)100381610038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
10038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
10038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
10038161003816100381610038161003816100381610038161003816
le 120593120572(119905)
10038161003816100381610038161198922 (120585 119905 120572)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
119865 (119909 119905 120572) minus 119865 (119909 minus 120585 119905 120572)
ℎ
100381610038161003816100381610038161003816100381610038161003816
le sup0leVle120585
0
100381610038161003816100381610038161003816100381610038161003816
120597119865
120597119909(119909 minus V 119905 120572)
100381610038161003816100381610038161003816100381610038161003816
le 120595120572(119905)
(34)
Inequalities (33) and (34) which are obviously also true for120585 = 0 ensure that 119892
1and 119892
2satisfy condition (119867
3) of
Theorem 9Applying the latter theorem we get
lim120585rarr0+
120601 (119909 + 120585) ⊖ 120601 (119909)
120585= int119868
1198921(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
lim120585rarr0+
120601 (119909) ⊖ 120601 (119909 minus 120585)
120585= int119868
1198922(0 119905) 119889119905
= int119868
120597119865
120597119909(119909 119905) 119889119905
(35)
Therefore 120601 is (i)-differentiable at 119909 and
1206011015840
(119909) = int119868
120597119865
120597119909(119909 119905) 119889119905 (36)
The proof under assumption (11986010158402) instead of (119860
2) is similar
to the first case
Theorem 12 One considers a fuzzy function 119906(120585 120591)
[0infin[times[0infin[rarr 119864 Suppose that the mapping 119865(120585 120591) =
119890minus119904120591
119906(120585 120591) satisfies assumptions (1198601)ndash(1198605) above for all 119904 ge 119904
0
for some 1199040gt 0
Let L120591[119906(120585 120591)] or L[119906(120585 120591)] (for short) denote the fuzzy
Laplace transform of 119906(120585 120591) with respect to the time variable120591 Then
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)]) (37)
Proof For fixed 119904 ge 1199040 then usingTheorem 11 we have
L120591[119906120585(120585 120591)] = int
infin
0
119890minus119904120591
119906120585(120585 120591) 119889120591 = int
infin
0
119865120585(120585 120591) 119889120591
=120597
120597120585(int
infin
0
119865 (120585 120591) 119889120591)
L120591[119906120585(120585 120591)] =
120597
120597120585(L120591[119906 (120585 120591)])
(38)
Theorem 13 Let 119906(120585 120591) be a fuzzy valued function on[0infin[times[0infin[ into 119864 Suppose that the mappings 120591 997891rarr
119865(120585 120591) = 119890minus119904120591
119906(120585 120591) and 120591 997891rarr 119866(120585 120591) = 119890minus119904120591
119906120591(120585 120591) are fuzzy
Riemann integrable on [0infin[ for all 119904 ge 1199040for some 119904
0gt 0
Consider the following
(a) If 119906(120585 120591) is (i)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = 119904L
120591[119906 (120585 120591)] ⊖ 119906 (120585 0) (39)
(b) If 119906(119909 120591) is (ii)-differentiable with respect to 120591 then
L120591[119906120591(120585 120591)] = (minus119906 (120585 0)) ⊖ (minus119904) L
120591[119906 (120585 120591)] (40)
Proof This is a direct result of Theorem 12 by fixing 120585 ge 0and taking the Laplace transforms with respect to 120591
4 Fuzzy Laplace TransformAlgorithm for First-Order FuzzyPartial Differential Equations
Our aim now is to solve the following first-order FPDEusing the fuzzy Laplace transform method under stronglygeneralized differentiability
119906120585(120585 120591) + 119886119906
120591(120585 120591) = 119891 (120585 120591 119906 (120585 120591))
119906 (120585 0) = 119892 (120585)
119906 (0 120591) = ℎ (120591)
(41)
where 119906(120585 120591) is a fuzzy function of 120585 ge 0 120591 ge 0 119886 is areal constant and 119891(120585 120591 119906) 119892(120585) and ℎ(120591) are fuzzy valuedfunctions such that 119891(120585 120591 119906) is linear with respect to 119906 Forshort assume that 119886 ge 0 (case 119886 lt 0 is similar)
By using fuzzy Laplace transformwith respect to 120591 we get
L120591[119906120585(120585 120591)] + 119886L
120591[119906120591(120585 120591)] = L
120591[119891 (120585 120591 119906 (120585 120591))] (42)
Therefore we have to distinguish the following cases forsolving (42)
6 International Journal of Differential Equations
(a) Case 1 If 119906 is (i)-differentiable with respect to 120585 and120591 then by Laplace transform
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
(43)
where 119891(120585 120591 119906(120585 120591) 120572) = min119891(120585 120591 V)V isin (119906(120585120591 120572) 119906(120585 120591 120572)) and 119891(120585 120591 119906(120585 120591) 120572) = max119891(120585 120591V)V isin (119906(120585 120591 120572) 119906(120585 120591 120572))Using Theorems 12 and 13 we get the followingdifferential system
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(44)
satisfying the following initial conditions
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
(45)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
1(119904 120572)
L [119906 (120585 120591 120572)] = 1198701(119904 120572)
(46)
where (1198671(119904 120572) 119870
1(119904 120572)) is solution of system (44)
under (45)By the inverse Laplace transform we get
119906 (120585 120591 120572) =Lminus1
[1198671(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198701(119904 120572)]
(47)
(b) Case 2 If 119906 is (i)-differentiable with respect to 120585 and(ii)-differentiable with respect to 120591 then byTheorems12 and 13 we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(48)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198672(119904 120572)
L [119906 (120585 120591 120572)] = 1198702(119904 120572)
(49)
where (1198672(119904 120572) 119870
2(119904 120572)) is solution of system (48)
under (45)Thus
119906 (120585 120591 120572) =Lminus1
[1198672(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198702(119904 120572)]
(50)
(c) Case 3 If 119906 is (ii)-differentiable with respect to 120585and (i)-differentiable with respect to 120591 then we getthe following differential system satisfying the initialconditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(51)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198673(119904 120572)
L [119906 (120585 120591 120572)] = 1198703(119904 120572)
(52)
where (1198673(119904 120572) 119870
3(119904 120572)) is solution of system (51)
under (45)Therefore
119906 (120585 120591 120572) =Lminus1
[1198673(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198703(119904 120572)]
(53)
(d) Case 4 If 119906 is (ii)-differentiable with respect to 120585and 120591 then we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(54)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
4(119904 120572)
L [119906 (120585 120591 120572)] = 1198704(119904 120572)
(55)
International Journal of Differential Equations 7
where (1198674(119901 120572) 119870
4(119901 120572)) is solution of system (54)
under (45)
Hence
119906 (120585 120591 120572) =Lminus1
[1198674(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198704(119904 120572)]
(56)
5 Numerical Examples
Example 1 Consider
119906120585(120585 120591) = 3119906
120591(120585 120591) + 120585
119906 (120585 0 120572) = 3120585 sdot (120572 2 minus 120572) +1205852
2
119906 (0 120591 120572) = 120591 sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(57)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby Laplace transform we get
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] minus 9120572120585 minus
31205852
2
+120585
119904
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] + 9 (120572 minus 2) 120585
minus31205852
2+120585
119904
(58)
This differential system satisfies the following initialconditions
L [119906 (0 120591 120572)] =L [120572120591] =120572
1199042
L [119906 (0 120591 120572)] =L [(2 minus 120572) 120591] =(2 minus 120572)
1199042
(59)
Solving (58) under (59) we get
L [119906 (120585 120591 120572)] =
(3120572120585 + 1205852
2)
119904+120572
1199042
L [119906 (120585 120591 120572)] =
((6 minus 3120572) 120585 + 1205852
2)
119904+2 minus 120572
1199042
(60)
By the inverse Laplace transform we deduce
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(61)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 119906 (120585 120591 120572) minus 119906 (120585 120591 120572)
= 2 (1 minus 120572) (3120585 + 120591) ge 0
len (119906120585(120585 120591 120572)) = 119906
120585(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 6 (1 minus 120572) ge 0
len (119906120591(120585 120591 120572)) = 119906
120591(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 2 (1 minus 120572) ge 0
(62)
So this solution is valid for all 120585 ge 0 and 120591 ge 0Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(63)
where119867 is the unit step function or the Heaviside function
119867(120577) =
1 120577 ge 0
0 120577 lt 0
(64)
Therefore this solution 119906 is valid only over Δ = (120585 120591) |120585 ge 0 120591 ge 0 120591 le 3120585
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then similarly
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(65)
As in Case 2 one can verify that this solution is valid onlyover Δ1015840 = (120585 120591) | 120585 ge 0 120591 ge 0 120591 ge 3120585
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 then
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(66)
One can verify that function 119906 is not (ii)-differentiablewith respect to either 120585 or 120591
So no solution exists in this case
8 International Journal of Differential Equations
Example 2 Consider
119906120585(120585 120591) = 119906
120591(120585 120591)
119906 (120585 0 120572) = cos (120585) sdot (120572 2 minus 120572)
119906 (0 120591 120572) = cos (120591) sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(67)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby application of the algorithm above one obtains
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (68)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 2 (1 minus 120572) cos (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
(69)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 therefore
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (70)
Then this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (71)
Hence this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 thensimilarly
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (72)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
6 Conclusion
Theorems of continuity and differentiability for a fuzzyfunction defined via a fuzzy improper Riemann integral areprovedwhich are used to prove some results concerning fuzzyLaplace transform Then using Laplace transform methodthe solutions for some linear fuzzy partial differential equa-tions (FPDEs) of first order are given For future research onecan apply this method to solve FPDEs of high order
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H-C Wu ldquoThe improper fuzzy Riemann integral and itsnumerical integrationrdquo Information Sciences vol 111 no 1ndash4 pp109ndash137 1998
[2] T Allahviranloo and M B Ahmadi ldquoFuzzy Laplace trans-formsrdquo Soft Computing vol 14 no 3 pp 235ndash243 2010
[3] S SalahshourMKhezerloo SHajighasemi andMKhorasanyldquoSolving fuzzy integral equations of the second kind by fuzzylaplace transform methodrdquo International Journal of IndustrialMathematics vol 4 no 1 pp 21ndash29 2012
[4] S Salahshour and T Allahviranloo ldquoApplications of fuzzyLaplace transformsrdquo Soft Computing vol 17 no 1 pp 145ndash1582013
[5] E ElJaoui S Melliani and L S Chadli ldquoSolving second-orderfuzzy differential equations by the fuzzy Laplace transformmethodrdquo Advances in Difference Equations vol 2015 article 6614 pages 2015
[6] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[7] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[8] I J Rudas B Bede and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007
[9] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[10] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons amp Fractals vol 38no 1 pp 112ndash119 2008
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Stochastic AnalysisInternational Journal of
![Page 6: Research Article On Fuzzy Improper Integral and Its ...downloads.hindawi.com/journals/ijde/2016/7246027.pdf · Wu introduced in [] the improper fuzzy Riemann integral and presented](https://reader035.cupdf.com/reader035/viewer/2022071021/5fd5401e6ba5386cb526b789/html5/thumbnails/6.jpg)
6 International Journal of Differential Equations
(a) Case 1 If 119906 is (i)-differentiable with respect to 120585 and120591 then by Laplace transform
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
L [119906120585(120585 120591 120572)] + 119886L [119906
120591(120585 120591 120572)]
=L [119891 (120585 120591 119906 (120585 120591))]
(43)
where 119891(120585 120591 119906(120585 120591) 120572) = min119891(120585 120591 V)V isin (119906(120585120591 120572) 119906(120585 120591 120572)) and 119891(120585 120591 119906(120585 120591) 120572) = max119891(120585 120591V)V isin (119906(120585 120591 120572) 119906(120585 120591 120572))Using Theorems 12 and 13 we get the followingdifferential system
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(44)
satisfying the following initial conditions
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
L [119906 (0 120591 120572)] =L [ℎ (120591 120572)]
(45)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
1(119904 120572)
L [119906 (120585 120591 120572)] = 1198701(119904 120572)
(46)
where (1198671(119904 120572) 119870
1(119904 120572)) is solution of system (44)
under (45)By the inverse Laplace transform we get
119906 (120585 120591 120572) =Lminus1
[1198671(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198701(119904 120572)]
(47)
(b) Case 2 If 119906 is (i)-differentiable with respect to 120585 and(ii)-differentiable with respect to 120591 then byTheorems12 and 13 we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(48)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198672(119904 120572)
L [119906 (120585 120591 120572)] = 1198702(119904 120572)
(49)
where (1198672(119904 120572) 119870
2(119904 120572)) is solution of system (48)
under (45)Thus
119906 (120585 120591 120572) =Lminus1
[1198672(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198702(119904 120572)]
(50)
(c) Case 3 If 119906 is (ii)-differentiable with respect to 120585and (i)-differentiable with respect to 120591 then we getthe following differential system satisfying the initialconditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(51)
Assume that this implies
L [119906 (120585 120591 120572)] = 1198673(119904 120572)
L [119906 (120585 120591 120572)] = 1198703(119904 120572)
(52)
where (1198673(119904 120572) 119870
3(119904 120572)) is solution of system (51)
under (45)Therefore
119906 (120585 120591 120572) =Lminus1
[1198673(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198703(119904 120572)]
(53)
(d) Case 4 If 119906 is (ii)-differentiable with respect to 120585and 120591 then we get the following differential systemsatisfying the initial conditions (45)
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
120597
120597120585(L [119906 (120585 120591 120572)]) + 119886119904L [119906 (120585 120591 120572)]
= 119886119892 (120585) +L [119891 (120585 120591 119906 (120585 120591))]
(54)
Assume that this leads toL [119906 (120585 120591 120572)] = 119867
4(119904 120572)
L [119906 (120585 120591 120572)] = 1198704(119904 120572)
(55)
International Journal of Differential Equations 7
where (1198674(119901 120572) 119870
4(119901 120572)) is solution of system (54)
under (45)
Hence
119906 (120585 120591 120572) =Lminus1
[1198674(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198704(119904 120572)]
(56)
5 Numerical Examples
Example 1 Consider
119906120585(120585 120591) = 3119906
120591(120585 120591) + 120585
119906 (120585 0 120572) = 3120585 sdot (120572 2 minus 120572) +1205852
2
119906 (0 120591 120572) = 120591 sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(57)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby Laplace transform we get
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] minus 9120572120585 minus
31205852
2
+120585
119904
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] + 9 (120572 minus 2) 120585
minus31205852
2+120585
119904
(58)
This differential system satisfies the following initialconditions
L [119906 (0 120591 120572)] =L [120572120591] =120572
1199042
L [119906 (0 120591 120572)] =L [(2 minus 120572) 120591] =(2 minus 120572)
1199042
(59)
Solving (58) under (59) we get
L [119906 (120585 120591 120572)] =
(3120572120585 + 1205852
2)
119904+120572
1199042
L [119906 (120585 120591 120572)] =
((6 minus 3120572) 120585 + 1205852
2)
119904+2 minus 120572
1199042
(60)
By the inverse Laplace transform we deduce
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(61)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 119906 (120585 120591 120572) minus 119906 (120585 120591 120572)
= 2 (1 minus 120572) (3120585 + 120591) ge 0
len (119906120585(120585 120591 120572)) = 119906
120585(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 6 (1 minus 120572) ge 0
len (119906120591(120585 120591 120572)) = 119906
120591(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 2 (1 minus 120572) ge 0
(62)
So this solution is valid for all 120585 ge 0 and 120591 ge 0Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(63)
where119867 is the unit step function or the Heaviside function
119867(120577) =
1 120577 ge 0
0 120577 lt 0
(64)
Therefore this solution 119906 is valid only over Δ = (120585 120591) |120585 ge 0 120591 ge 0 120591 le 3120585
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then similarly
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(65)
As in Case 2 one can verify that this solution is valid onlyover Δ1015840 = (120585 120591) | 120585 ge 0 120591 ge 0 120591 ge 3120585
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 then
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(66)
One can verify that function 119906 is not (ii)-differentiablewith respect to either 120585 or 120591
So no solution exists in this case
8 International Journal of Differential Equations
Example 2 Consider
119906120585(120585 120591) = 119906
120591(120585 120591)
119906 (120585 0 120572) = cos (120585) sdot (120572 2 minus 120572)
119906 (0 120591 120572) = cos (120591) sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(67)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby application of the algorithm above one obtains
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (68)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 2 (1 minus 120572) cos (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
(69)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 therefore
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (70)
Then this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (71)
Hence this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 thensimilarly
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (72)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
6 Conclusion
Theorems of continuity and differentiability for a fuzzyfunction defined via a fuzzy improper Riemann integral areprovedwhich are used to prove some results concerning fuzzyLaplace transform Then using Laplace transform methodthe solutions for some linear fuzzy partial differential equa-tions (FPDEs) of first order are given For future research onecan apply this method to solve FPDEs of high order
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H-C Wu ldquoThe improper fuzzy Riemann integral and itsnumerical integrationrdquo Information Sciences vol 111 no 1ndash4 pp109ndash137 1998
[2] T Allahviranloo and M B Ahmadi ldquoFuzzy Laplace trans-formsrdquo Soft Computing vol 14 no 3 pp 235ndash243 2010
[3] S SalahshourMKhezerloo SHajighasemi andMKhorasanyldquoSolving fuzzy integral equations of the second kind by fuzzylaplace transform methodrdquo International Journal of IndustrialMathematics vol 4 no 1 pp 21ndash29 2012
[4] S Salahshour and T Allahviranloo ldquoApplications of fuzzyLaplace transformsrdquo Soft Computing vol 17 no 1 pp 145ndash1582013
[5] E ElJaoui S Melliani and L S Chadli ldquoSolving second-orderfuzzy differential equations by the fuzzy Laplace transformmethodrdquo Advances in Difference Equations vol 2015 article 6614 pages 2015
[6] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[7] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[8] I J Rudas B Bede and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007
[9] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[10] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons amp Fractals vol 38no 1 pp 112ndash119 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 7: Research Article On Fuzzy Improper Integral and Its ...downloads.hindawi.com/journals/ijde/2016/7246027.pdf · Wu introduced in [] the improper fuzzy Riemann integral and presented](https://reader035.cupdf.com/reader035/viewer/2022071021/5fd5401e6ba5386cb526b789/html5/thumbnails/7.jpg)
International Journal of Differential Equations 7
where (1198674(119901 120572) 119870
4(119901 120572)) is solution of system (54)
under (45)
Hence
119906 (120585 120591 120572) =Lminus1
[1198674(119904 120572)]
119906 (120585 120591 120572) =Lminus1
[1198704(119904 120572)]
(56)
5 Numerical Examples
Example 1 Consider
119906120585(120585 120591) = 3119906
120591(120585 120591) + 120585
119906 (120585 0 120572) = 3120585 sdot (120572 2 minus 120572) +1205852
2
119906 (0 120591 120572) = 120591 sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(57)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby Laplace transform we get
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] minus 9120572120585 minus
31205852
2
+120585
119904
120597
120597120585(L [119906 (120585 120591 120572)]) = 3119904L [119906 (120585 120591 120572)] + 9 (120572 minus 2) 120585
minus31205852
2+120585
119904
(58)
This differential system satisfies the following initialconditions
L [119906 (0 120591 120572)] =L [120572120591] =120572
1199042
L [119906 (0 120591 120572)] =L [(2 minus 120572) 120591] =(2 minus 120572)
1199042
(59)
Solving (58) under (59) we get
L [119906 (120585 120591 120572)] =
(3120572120585 + 1205852
2)
119904+120572
1199042
L [119906 (120585 120591 120572)] =
((6 minus 3120572) 120585 + 1205852
2)
119904+2 minus 120572
1199042
(60)
By the inverse Laplace transform we deduce
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(61)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 119906 (120585 120591 120572) minus 119906 (120585 120591 120572)
= 2 (1 minus 120572) (3120585 + 120591) ge 0
len (119906120585(120585 120591 120572)) = 119906
120585(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 6 (1 minus 120572) ge 0
len (119906120591(120585 120591 120572)) = 119906
120591(120585 120591 120572) minus 119906
120591(120585 120591 120572)
= 2 (1 minus 120572) ge 0
(62)
So this solution is valid for all 120585 ge 0 and 120591 ge 0Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(63)
where119867 is the unit step function or the Heaviside function
119867(120577) =
1 120577 ge 0
0 120577 lt 0
(64)
Therefore this solution 119906 is valid only over Δ = (120585 120591) |120585 ge 0 120591 ge 0 120591 le 3120585
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then similarly
119906 (120585 120591 120572) = 2 (120572 minus 1) (120591 minus 3120585)119867 (120591 minus 3120585) + 3120572120585
+ (2 minus 120572) 120591 +1205852
2
119906 (120585 120591 120572) = 2 (1 minus 120572) (120591 minus 3120585)119867 (120591 minus 3120585) + 3 (2 minus 120572) 120585
+ 120572120591 +1205852
2
(65)
As in Case 2 one can verify that this solution is valid onlyover Δ1015840 = (120585 120591) | 120585 ge 0 120591 ge 0 120591 ge 3120585
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 then
119906 (120585 120591 120572) = 120572 (3120585 + 120591) +1205852
2
119906 (120585 120591 120572) = (2 minus 120572) (3120585 + 120591) +1205852
2
(66)
One can verify that function 119906 is not (ii)-differentiablewith respect to either 120585 or 120591
So no solution exists in this case
8 International Journal of Differential Equations
Example 2 Consider
119906120585(120585 120591) = 119906
120591(120585 120591)
119906 (120585 0 120572) = cos (120585) sdot (120572 2 minus 120572)
119906 (0 120591 120572) = cos (120591) sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(67)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby application of the algorithm above one obtains
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (68)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 2 (1 minus 120572) cos (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
(69)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 therefore
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (70)
Then this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (71)
Hence this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 thensimilarly
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (72)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
6 Conclusion
Theorems of continuity and differentiability for a fuzzyfunction defined via a fuzzy improper Riemann integral areprovedwhich are used to prove some results concerning fuzzyLaplace transform Then using Laplace transform methodthe solutions for some linear fuzzy partial differential equa-tions (FPDEs) of first order are given For future research onecan apply this method to solve FPDEs of high order
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H-C Wu ldquoThe improper fuzzy Riemann integral and itsnumerical integrationrdquo Information Sciences vol 111 no 1ndash4 pp109ndash137 1998
[2] T Allahviranloo and M B Ahmadi ldquoFuzzy Laplace trans-formsrdquo Soft Computing vol 14 no 3 pp 235ndash243 2010
[3] S SalahshourMKhezerloo SHajighasemi andMKhorasanyldquoSolving fuzzy integral equations of the second kind by fuzzylaplace transform methodrdquo International Journal of IndustrialMathematics vol 4 no 1 pp 21ndash29 2012
[4] S Salahshour and T Allahviranloo ldquoApplications of fuzzyLaplace transformsrdquo Soft Computing vol 17 no 1 pp 145ndash1582013
[5] E ElJaoui S Melliani and L S Chadli ldquoSolving second-orderfuzzy differential equations by the fuzzy Laplace transformmethodrdquo Advances in Difference Equations vol 2015 article 6614 pages 2015
[6] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[7] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[8] I J Rudas B Bede and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007
[9] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[10] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons amp Fractals vol 38no 1 pp 112ndash119 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 8: Research Article On Fuzzy Improper Integral and Its ...downloads.hindawi.com/journals/ijde/2016/7246027.pdf · Wu introduced in [] the improper fuzzy Riemann integral and presented](https://reader035.cupdf.com/reader035/viewer/2022071021/5fd5401e6ba5386cb526b789/html5/thumbnails/8.jpg)
8 International Journal of Differential Equations
Example 2 Consider
119906120585(120585 120591) = 119906
120591(120585 120591)
119906 (120585 0 120572) = cos (120585) sdot (120572 2 minus 120572)
119906 (0 120591 120572) = cos (120591) sdot (120572 2 minus 120572)
120585 ge 0 120591 ge 0
(67)
Case 1 If 119906 is (i)-differentiable with respect to 120585 and 120591 thenby application of the algorithm above one obtains
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (68)
The lengths of 119906 119906120585 and 119906
120591are respectively given by
len (119906 (120585 120591 120572)) = 2 (1 minus 120572) cos (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
len (119906120585(120585 120591 120572)) = minus2 (1 minus 120572) sin (120585 + 120591)
(69)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 2 If 119906 is (i)-differentiable with respect to 120585 and (ii)-differentiable with respect to 120591 therefore
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (70)
Then this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[31205872 + 2119896120587 2120587 + 2119896120587] 119896 isin Z
Case 3 If 119906 is (ii)-differentiable with respect to 120585 and (i)-differentiable with respect to 120591 then analogously
119906 (120585 120591 120572) = (120572 minus 1) cos (120585 minus 120591) + cos (120585 + 120591)
119906 (120585 120591 120572) = (1 minus 120572) cos (120585 minus 120591) + cos (120585 + 120591) (71)
Hence this solution is valid for all 120585 ge 0 120591 ge 0 120585 minus 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
Case 4 If 119906 is (ii)-differentiable with respect to 120585 and 120591 thensimilarly
119906 (120585 120591 120572) = 120572 cos (120585 + 120591)
119906 (120585 120591 120572) = (2 minus 120572) cos (120585 + 120591) (72)
So this solution is valid for all 120585 ge 0 120591 ge 0 120585 + 120591 isin[2119896120587 1205872 + 2119896120587] 119896 isin Z
6 Conclusion
Theorems of continuity and differentiability for a fuzzyfunction defined via a fuzzy improper Riemann integral areprovedwhich are used to prove some results concerning fuzzyLaplace transform Then using Laplace transform methodthe solutions for some linear fuzzy partial differential equa-tions (FPDEs) of first order are given For future research onecan apply this method to solve FPDEs of high order
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H-C Wu ldquoThe improper fuzzy Riemann integral and itsnumerical integrationrdquo Information Sciences vol 111 no 1ndash4 pp109ndash137 1998
[2] T Allahviranloo and M B Ahmadi ldquoFuzzy Laplace trans-formsrdquo Soft Computing vol 14 no 3 pp 235ndash243 2010
[3] S SalahshourMKhezerloo SHajighasemi andMKhorasanyldquoSolving fuzzy integral equations of the second kind by fuzzylaplace transform methodrdquo International Journal of IndustrialMathematics vol 4 no 1 pp 21ndash29 2012
[4] S Salahshour and T Allahviranloo ldquoApplications of fuzzyLaplace transformsrdquo Soft Computing vol 17 no 1 pp 145ndash1582013
[5] E ElJaoui S Melliani and L S Chadli ldquoSolving second-orderfuzzy differential equations by the fuzzy Laplace transformmethodrdquo Advances in Difference Equations vol 2015 article 6614 pages 2015
[6] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[7] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986
[8] I J Rudas B Bede and A L Bencsik ldquoFirst order linearfuzzy differential equations under generalized differentiabilityrdquoInformation Sciences vol 177 no 7 pp 1648ndash1662 2007
[9] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[10] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons amp Fractals vol 38no 1 pp 112ndash119 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 9: Research Article On Fuzzy Improper Integral and Its ...downloads.hindawi.com/journals/ijde/2016/7246027.pdf · Wu introduced in [] the improper fuzzy Riemann integral and presented](https://reader035.cupdf.com/reader035/viewer/2022071021/5fd5401e6ba5386cb526b789/html5/thumbnails/9.jpg)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
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