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Research ArticleNonlocal Telegraph Equation in Frame of the ConformableTime-Fractional Derivative
Mohamed Bouaouid Khalid Hilal and Said Melliani
Department of Mathematics Sultan Moulay Slimane University BP 523 23000 Beni Mellal Morocco
Correspondence should be addressed to Mohamed Bouaouid bouaouidfstgmailcom
Received 15 August 2018 Revised 23 December 2018 Accepted 31 January 2019 Published 3 March 2019
Academic Editor Sergey Shmarev
Copyright copy 2019 Mohamed Bouaouid et al 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 use the cosine family of linear operators to prove the existence uniqueness and stability of the integral solution of a nonlocaltelegraph equation in frame of the conformable time-fractional derivative Moreover we give its implicit fundamental solution interms of the classical trigonometric functions
1 Introduction and Statement of the Problem
The telegraph equation is better than the heat equation inmodeling of physical phenomena which have a parabolicbehavior [1] The one-dimensional telegraph equation can bewritten as follows
1205972119906 (119905 ℓ)1205971199052 = 1
1198711198621205972119906 (119905 ℓ)
120597ℓ2 minus 119877119866119871119862119906 (119905 ℓ)
minus (119877119871 + 119866
119862) 120597119906 (119905 ℓ)120597119905
(1)
where 119877 and 119866 are respectively the resistance and the con-ductance of resistor119862 is the capacitance of capacitor and 119871 isthe inductance of coil Many concrete applications amount toreplacing the time derivative in the telegraph equation witha fractional derivative For example in the works [2ndash8] theauthors have extensively studied the time-fractional telegraphequation with Caputo fractional derivative For more detailsabout the good effect of the fractional derivative we refer tomonographs [9ndash13]
Recently a new definition of fractional derivative namedldquofractional conformable derivativerdquo is introduce by Khalilet al [14] This novel fractional derivative is compatiblewith the classical derivative and it is excellent for studynonregular solutions Since the subject of the fractionalconformable derivative has attracted the attention of manyauthors in domains such as mechanics [15] electronic [16]
and anomalous diffusion [17]We are interested in studying inthis paper the telegraph model (1) in framework of the time-fractional conformable derivative Precisely we will proposethe following transformations
where 120597120572120597119905120572 and 120597120574120597119905120574 are the time-fractional conformablederivative operators [14] Then we get the fractional con-formable telegraph model associated with the transforma-tions (2) and (3) as follows
where the time-parameter 119905 belongs to an interval [0 120591] with120591 is a fixed positive real number The spatial parameter ℓbelongs to the interval [0 120587]
HindawiAdvances in Mathematical PhysicsVolume 2019 Article ID 7528937 7 pageshttpsdoiorg10115520197528937
2 Advances in Mathematical Physics
We associate to (4) the boundary and the nonlocal initialconditions
where 119901 is a nonnull fixed integer and 0 lt 1199051 lt sdot sdot sdot lt 119905119901 lt 120591The quantities 1199090 1199091 and 1205761 120576119901 are physical measuresThe condition appearing in (8) means the nonlocal condition[18] For physical interpretations of this condition we referto works [19 20] For example in [19] the author used anonlocal condition of the form (8) to describe the diffusionphenomenon of a small amount of gas in a transparent tube
Wenote that it is not easy to find the fundamental solutionof (4) by using the Laplace transformmethod if120572 = 120574 For thisreasonwewill propose the integral solution concept based onan operator theory approach When 120572 = 120574 we investigate animplicit fundamental solution
The content of this paper is organized as follows InSection 2 we recall some needed results of the conformablefractional derivative and cosine family of linear operators InSection 3 we prove the existence uniqueness and stabilityof the integral solution of (4) by using of an operator theoryapproach Section 4 is devoted to an implicit fundamentalsolution of (4) in terms of the classical trigonometric func-tions
2 Preliminaries
We start recalling some concepts on the conformable frac-tional calculus [14]
The conformable fractional derivative of a function 119909 oforder 120572 at 119905 gt 0 is defined by the following limit
We remark that the classical Laplace transform is not com-patible with the conformable fractional derivative For thisreason the adapted transform is defined as [21]
Now we introduce some results concerning the cosine familytheory [22]
A one-parameter family (119862(119905))119905isinR of bounded linearoperators on the Banach space 119883 is called a strongly contin-uous cosine family if and only if
(1) 119862(0) = 119868 (I is the identity operator)(2) 119862(119904 + 119905) + 119862(119904 minus 119905) = 2119862(119904)119862(119905) for all 119905 119904 isin R(3) 119905 997891997888rarr 119862(119905)119909 is continuous for each fixed 119909 isin 119883
The infinitesimal generator119860 of a strongly continuous cosinefamily ((119862(119905))119905isinR (119878(119905))119905isinR) on119883 is defined by
119863(119860) = 119909 isin 119883 119905 997891997888rarr 119862 (119905)sdot 119909 is a twice continuously differentiable function
If 119860 is the infinitesimal generator of a strongly cosine family((119862(119905))119905isinR (119878(119905))119905isinR) on119883 then there exists a constant 120596 ge 0such that for all 120582 with 119877119890(120582) gt 120596 we have
1205822 isin 120588 (119860) (120588 (119860) is the resolvent set of 119860)
120582 (1205822119868 minus 119860)minus1 = int+infin0
119890minus120582119905119862 (119905) 119889119905(1205822119868 minus 119860)minus1 = int+infin
According to [23] the operator 119860 generates a cosine family((119862(119905))119905isinR (119878(119905))119905isinR) on 1198712([0 120587] 119889ℓ) Moreover |119862(119905)| le 1and |119878(119905)| le 1 for all 119905 isin [0 120591]
We denote C120572 the Banach space of continuously (120572)-differentiable functions from [0 120591] into1198712([0 120587] 119889ℓ)with thenorm |119909| = sup119905isin[0120591]119909(119905) + sup119905isin[0120591]119889120572119909(119905)119889119905120572 Here is the classical norm in the space 1198712([0 120587] 119889ℓ)
31 Existence and Uniqueness of the Integral Solution Toexplain integral Duhamelrsquos formula we apply the fractionalLaplace transform to (22) obtaining
minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904
(29)
4 Advances in Mathematical Physics
Accordingly we obtain
1003817100381710038171003817Γ (119910) (119905) minus Γ (119909) (119905)1003817100381710038171003817 +10038171003817100381710038171003817100381710038171003817119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)]10038171003817100381710038171003817100381710038171003817
le 2 [120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816
(30)
Then we get
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816le 2 [120576 +max(119886120591120572
120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816 (31)
Finally Γ has an unique fixed point 119909 in (C120572 ||) which is theintegral solution of the equation (22)
Now we give a result that is better than the previous one
Theorem 3 e Cauchy problem (22) has a unique integralsolution provided that
120576 lt 12 (32)
Proof Define the operator Γ C120572 997888rarr C120572 by
120572 )119889119904] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572
(37)
Accordingly we get
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 [120576 + max (119886 119887120591120572minus120574)120579 ] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (38)
Hence we conclude that
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 (120576 + 1)3 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (39)
The fact 2(120576 + 1)3 lt 1 proves that Γ has an unique fixedpoint 119909 in (C120572 ||120572) which is the integral solution of equation(22)
32 Stability of the Integral Solution Here we give a resultconcerning the nonlocal-condition effect on the stability ofthe integral solution
Theorem 4 Let 119909 and 119910 be solutions associated with (1199090 1199091)and (1199090 1199101) respectivelyen we have the following estimate
1003816100381610038161003816119910 minus 1199091003816100381610038161003816le 2
1 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101minus 11990911003817100381710038171003817
(40)
provided that
120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574) lt 1
2 (41)
Advances in Mathematical Physics 5
Proof We have
119910 (119905) minus 119909 (119905) = 119878 (119905120572120572 ) [1199101 minus 1199091 + 119892 (119910) minus 119892 (119909)]
+ int1199050119904120572minus1119878 (119905120572 minus 119904120572
minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904
(42)
Consequently we get
1003816100381610038161003816119910 minus 1199091003816100381610038161003816 le 2(120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)) 1003816100381610038161003816119910 minus 1199091003816100381610038161003816
+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)
Finally we obtain the following estimation1003816100381610038161003816119910 minus 1199091003816100381610038161003816
le 21 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101
minus 11990911003817100381710038171003817 (44)
4 Implicit Fundamental Solution inthe Case When 120574=120572
Here we give the implicit fundamental solution of (4) byusing the separating variables method To do so let 119906(119905 ℓ) =119909(119905)119910(ℓ) Then (4) becomes as follows
2120572 )int1205870cos(radic41198881198992 + 4119886 minus 1198872
2120572 119905120572)
sdot 1199090 sin (119899119904) 119889119904 + 2119887120587 exp(minus119887119905120572
2120572 )
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887119905120572
2120572 )
sdot 119901sum119894=1
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 [1199091119901
+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904
(52)
Finally replacing 119909119899(119905) in (49) we get
By using the classical Laplace transform in (55) we get
119909119899 (119905) = 2120587exp(minus119887
2119905)
sdot int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587
sdot exp(minus1198872119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887
2119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899119904) 119889119904
(57)
Replacing 119909119899(119905) in (54) we find the fundamental solution ofthe classical telegraph equation (1) as follows
119906 (119905 ℓ) = 2120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 2119887120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 4120587exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899ℓ)
sdot sin (119899119904) 119889119904(58)
Remark 6 If we consider the fractional derivative in Caputorsquossense the implicit fundamental solution of (4) can be writtenin terms of the Mittag-Leffler function However in our casewe have found the implicit fundamental solution in terms ofthe classical trigonometric functions
Remark 7 The integral solution does not impose any con-straint concerning the choice of the derivation parametersThis provides more freedom concerning the choice of sensi-tive parameters 120572 and 120574 in practice situations for modelingnaturel phenomena
5 Conclusion
We have studied a time-conformable fractional telegraphequation with nonlocal condition In the case when 120572 = 120574we have given the implicit fundamental solution in terms ofthe classical trigonometric functions In the general case wehave established the existence uniqueness and stability of theintegral solution
As for future work we intend to give the fundamentalsolution for all values of 120572 and 120574Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] E C Eckstein J AGoldstein andM Leggas ldquoThemathematicsof suspensions Kacwalks and asymptotic analyticityrdquoElectronicJournal of Differential Equations vol 3 pp 39ndash50 1999
[2] R C Cascaval E C Eckstein C L Frota and J A GoldsteinldquoFractional telegraph equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 276 no 1 pp 145ndash159 2002
[3] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[4] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[5] V RHosseiniWChen andZAvazzadeh ldquoNumerical solutionof fractional telegraph equation by using radial basis functionsrdquo
Advances in Mathematical Physics 7
Engineering Analysis with Boundary Elements vol 38 no 12 pp31ndash39 2014
[6] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[7] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[8] V K Srivastava M K Awasthi and S Kumar ldquoAnalyticalapproximations of two and three dimensional time-fractionaltelegraphic equation by reduced differential transformmethodrdquoEgyptian Journal of Basic and Applied Sciences vol 1 no 1 pp60ndash66 2014
[9] A A Kilbas H M Srivastava and J J Trujillo eory andApplications of Fractional Differential Equations North HollandMathematics Studies 204 Elsevier New York NY USA 2006
[10] K S Miller An Introduction to the Fractional Calculus andFractional Differential Equations John Wiley and Sons 1993
[11] K B Oldham and J Spaniere Fractional Calculus AcademicPress New York NY USA 1974
[12] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives eory and Applications Gordon andBreach Yverdon Switzerland 1993
[14] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
[15] W S Chung ldquoFractional Newton mechanics with conformablefractional derivativerdquo Journal of Computational and AppliedMathematics vol 290 pp 150ndash158 2015
[16] L Martınez J Rosales C Carreno and J Lozano ldquoElectricalcircuits described by fractional conformable derivativerdquo Inter-national Journal of Circuit eory and Applications vol 46 no5 pp 1091ndash1100 2018
[17] H W Zhou S Yang and S Q Zhang ldquoConformable deriva-tive approach to anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 491 pp 1001ndash1013 2018
[18] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
[19] K Deng ldquoExponential decay of solutions of semilinearparabolic equations with nonlocal initial conditionsrdquo Journal ofMathematical Analysis and Applications vol 179 no 2 pp 630ndash637 1993
[20] W E Olmstead and C A Roberts ldquoThe one-dimensional heatequation with a nonlocal initial conditionrdquo Applied Mathemat-ics Letters vol 10 no 3 pp 89ndash94 1997
[21] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[22] C C Travis and G F Webb ldquoCosine families and abstract non-linear second order differential equationsrdquo Acta MathematicaHungarica vol 32 no 1-2 pp 75ndash96 1978
[23] M E Hernandez ldquoExistence of solutions to a second order par-tial differential equation with nonlocal conditionsrdquo ElectronicJournal of Differential Equations vol 2003 pp 1ndash10 2003
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International Journal of Mathematics and Mathematical Sciences
where 119901 is a nonnull fixed integer and 0 lt 1199051 lt sdot sdot sdot lt 119905119901 lt 120591The quantities 1199090 1199091 and 1205761 120576119901 are physical measuresThe condition appearing in (8) means the nonlocal condition[18] For physical interpretations of this condition we referto works [19 20] For example in [19] the author used anonlocal condition of the form (8) to describe the diffusionphenomenon of a small amount of gas in a transparent tube
Wenote that it is not easy to find the fundamental solutionof (4) by using the Laplace transformmethod if120572 = 120574 For thisreasonwewill propose the integral solution concept based onan operator theory approach When 120572 = 120574 we investigate animplicit fundamental solution
The content of this paper is organized as follows InSection 2 we recall some needed results of the conformablefractional derivative and cosine family of linear operators InSection 3 we prove the existence uniqueness and stabilityof the integral solution of (4) by using of an operator theoryapproach Section 4 is devoted to an implicit fundamentalsolution of (4) in terms of the classical trigonometric func-tions
2 Preliminaries
We start recalling some concepts on the conformable frac-tional calculus [14]
The conformable fractional derivative of a function 119909 oforder 120572 at 119905 gt 0 is defined by the following limit
We remark that the classical Laplace transform is not com-patible with the conformable fractional derivative For thisreason the adapted transform is defined as [21]
Now we introduce some results concerning the cosine familytheory [22]
A one-parameter family (119862(119905))119905isinR of bounded linearoperators on the Banach space 119883 is called a strongly contin-uous cosine family if and only if
(1) 119862(0) = 119868 (I is the identity operator)(2) 119862(119904 + 119905) + 119862(119904 minus 119905) = 2119862(119904)119862(119905) for all 119905 119904 isin R(3) 119905 997891997888rarr 119862(119905)119909 is continuous for each fixed 119909 isin 119883
The infinitesimal generator119860 of a strongly continuous cosinefamily ((119862(119905))119905isinR (119878(119905))119905isinR) on119883 is defined by
119863(119860) = 119909 isin 119883 119905 997891997888rarr 119862 (119905)sdot 119909 is a twice continuously differentiable function
If 119860 is the infinitesimal generator of a strongly cosine family((119862(119905))119905isinR (119878(119905))119905isinR) on119883 then there exists a constant 120596 ge 0such that for all 120582 with 119877119890(120582) gt 120596 we have
1205822 isin 120588 (119860) (120588 (119860) is the resolvent set of 119860)
120582 (1205822119868 minus 119860)minus1 = int+infin0
119890minus120582119905119862 (119905) 119889119905(1205822119868 minus 119860)minus1 = int+infin
According to [23] the operator 119860 generates a cosine family((119862(119905))119905isinR (119878(119905))119905isinR) on 1198712([0 120587] 119889ℓ) Moreover |119862(119905)| le 1and |119878(119905)| le 1 for all 119905 isin [0 120591]
We denote C120572 the Banach space of continuously (120572)-differentiable functions from [0 120591] into1198712([0 120587] 119889ℓ)with thenorm |119909| = sup119905isin[0120591]119909(119905) + sup119905isin[0120591]119889120572119909(119905)119889119905120572 Here is the classical norm in the space 1198712([0 120587] 119889ℓ)
31 Existence and Uniqueness of the Integral Solution Toexplain integral Duhamelrsquos formula we apply the fractionalLaplace transform to (22) obtaining
minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904
(29)
4 Advances in Mathematical Physics
Accordingly we obtain
1003817100381710038171003817Γ (119910) (119905) minus Γ (119909) (119905)1003817100381710038171003817 +10038171003817100381710038171003817100381710038171003817119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)]10038171003817100381710038171003817100381710038171003817
le 2 [120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816
(30)
Then we get
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816le 2 [120576 +max(119886120591120572
120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816 (31)
Finally Γ has an unique fixed point 119909 in (C120572 ||) which is theintegral solution of the equation (22)
Now we give a result that is better than the previous one
Theorem 3 e Cauchy problem (22) has a unique integralsolution provided that
120576 lt 12 (32)
Proof Define the operator Γ C120572 997888rarr C120572 by
120572 )119889119904] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572
(37)
Accordingly we get
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 [120576 + max (119886 119887120591120572minus120574)120579 ] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (38)
Hence we conclude that
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 (120576 + 1)3 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (39)
The fact 2(120576 + 1)3 lt 1 proves that Γ has an unique fixedpoint 119909 in (C120572 ||120572) which is the integral solution of equation(22)
32 Stability of the Integral Solution Here we give a resultconcerning the nonlocal-condition effect on the stability ofthe integral solution
Theorem 4 Let 119909 and 119910 be solutions associated with (1199090 1199091)and (1199090 1199101) respectivelyen we have the following estimate
1003816100381610038161003816119910 minus 1199091003816100381610038161003816le 2
1 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101minus 11990911003817100381710038171003817
(40)
provided that
120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574) lt 1
2 (41)
Advances in Mathematical Physics 5
Proof We have
119910 (119905) minus 119909 (119905) = 119878 (119905120572120572 ) [1199101 minus 1199091 + 119892 (119910) minus 119892 (119909)]
+ int1199050119904120572minus1119878 (119905120572 minus 119904120572
minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904
(42)
Consequently we get
1003816100381610038161003816119910 minus 1199091003816100381610038161003816 le 2(120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)) 1003816100381610038161003816119910 minus 1199091003816100381610038161003816
+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)
Finally we obtain the following estimation1003816100381610038161003816119910 minus 1199091003816100381610038161003816
le 21 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101
minus 11990911003817100381710038171003817 (44)
4 Implicit Fundamental Solution inthe Case When 120574=120572
Here we give the implicit fundamental solution of (4) byusing the separating variables method To do so let 119906(119905 ℓ) =119909(119905)119910(ℓ) Then (4) becomes as follows
2120572 )int1205870cos(radic41198881198992 + 4119886 minus 1198872
2120572 119905120572)
sdot 1199090 sin (119899119904) 119889119904 + 2119887120587 exp(minus119887119905120572
2120572 )
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887119905120572
2120572 )
sdot 119901sum119894=1
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 [1199091119901
+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904
(52)
Finally replacing 119909119899(119905) in (49) we get
By using the classical Laplace transform in (55) we get
119909119899 (119905) = 2120587exp(minus119887
2119905)
sdot int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587
sdot exp(minus1198872119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887
2119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899119904) 119889119904
(57)
Replacing 119909119899(119905) in (54) we find the fundamental solution ofthe classical telegraph equation (1) as follows
119906 (119905 ℓ) = 2120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 2119887120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 4120587exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899ℓ)
sdot sin (119899119904) 119889119904(58)
Remark 6 If we consider the fractional derivative in Caputorsquossense the implicit fundamental solution of (4) can be writtenin terms of the Mittag-Leffler function However in our casewe have found the implicit fundamental solution in terms ofthe classical trigonometric functions
Remark 7 The integral solution does not impose any con-straint concerning the choice of the derivation parametersThis provides more freedom concerning the choice of sensi-tive parameters 120572 and 120574 in practice situations for modelingnaturel phenomena
5 Conclusion
We have studied a time-conformable fractional telegraphequation with nonlocal condition In the case when 120572 = 120574we have given the implicit fundamental solution in terms ofthe classical trigonometric functions In the general case wehave established the existence uniqueness and stability of theintegral solution
As for future work we intend to give the fundamentalsolution for all values of 120572 and 120574Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] E C Eckstein J AGoldstein andM Leggas ldquoThemathematicsof suspensions Kacwalks and asymptotic analyticityrdquoElectronicJournal of Differential Equations vol 3 pp 39ndash50 1999
[2] R C Cascaval E C Eckstein C L Frota and J A GoldsteinldquoFractional telegraph equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 276 no 1 pp 145ndash159 2002
[3] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[4] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[5] V RHosseiniWChen andZAvazzadeh ldquoNumerical solutionof fractional telegraph equation by using radial basis functionsrdquo
Advances in Mathematical Physics 7
Engineering Analysis with Boundary Elements vol 38 no 12 pp31ndash39 2014
[6] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[7] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[8] V K Srivastava M K Awasthi and S Kumar ldquoAnalyticalapproximations of two and three dimensional time-fractionaltelegraphic equation by reduced differential transformmethodrdquoEgyptian Journal of Basic and Applied Sciences vol 1 no 1 pp60ndash66 2014
[9] A A Kilbas H M Srivastava and J J Trujillo eory andApplications of Fractional Differential Equations North HollandMathematics Studies 204 Elsevier New York NY USA 2006
[10] K S Miller An Introduction to the Fractional Calculus andFractional Differential Equations John Wiley and Sons 1993
[11] K B Oldham and J Spaniere Fractional Calculus AcademicPress New York NY USA 1974
[12] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives eory and Applications Gordon andBreach Yverdon Switzerland 1993
[14] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
[15] W S Chung ldquoFractional Newton mechanics with conformablefractional derivativerdquo Journal of Computational and AppliedMathematics vol 290 pp 150ndash158 2015
[16] L Martınez J Rosales C Carreno and J Lozano ldquoElectricalcircuits described by fractional conformable derivativerdquo Inter-national Journal of Circuit eory and Applications vol 46 no5 pp 1091ndash1100 2018
[17] H W Zhou S Yang and S Q Zhang ldquoConformable deriva-tive approach to anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 491 pp 1001ndash1013 2018
[18] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
[19] K Deng ldquoExponential decay of solutions of semilinearparabolic equations with nonlocal initial conditionsrdquo Journal ofMathematical Analysis and Applications vol 179 no 2 pp 630ndash637 1993
[20] W E Olmstead and C A Roberts ldquoThe one-dimensional heatequation with a nonlocal initial conditionrdquo Applied Mathemat-ics Letters vol 10 no 3 pp 89ndash94 1997
[21] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[22] C C Travis and G F Webb ldquoCosine families and abstract non-linear second order differential equationsrdquo Acta MathematicaHungarica vol 32 no 1-2 pp 75ndash96 1978
[23] M E Hernandez ldquoExistence of solutions to a second order par-tial differential equation with nonlocal conditionsrdquo ElectronicJournal of Differential Equations vol 2003 pp 1ndash10 2003
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
According to [23] the operator 119860 generates a cosine family((119862(119905))119905isinR (119878(119905))119905isinR) on 1198712([0 120587] 119889ℓ) Moreover |119862(119905)| le 1and |119878(119905)| le 1 for all 119905 isin [0 120591]
We denote C120572 the Banach space of continuously (120572)-differentiable functions from [0 120591] into1198712([0 120587] 119889ℓ)with thenorm |119909| = sup119905isin[0120591]119909(119905) + sup119905isin[0120591]119889120572119909(119905)119889119905120572 Here is the classical norm in the space 1198712([0 120587] 119889ℓ)
31 Existence and Uniqueness of the Integral Solution Toexplain integral Duhamelrsquos formula we apply the fractionalLaplace transform to (22) obtaining
minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904
(29)
4 Advances in Mathematical Physics
Accordingly we obtain
1003817100381710038171003817Γ (119910) (119905) minus Γ (119909) (119905)1003817100381710038171003817 +10038171003817100381710038171003817100381710038171003817119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)]10038171003817100381710038171003817100381710038171003817
le 2 [120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816
(30)
Then we get
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816le 2 [120576 +max(119886120591120572
120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816 (31)
Finally Γ has an unique fixed point 119909 in (C120572 ||) which is theintegral solution of the equation (22)
Now we give a result that is better than the previous one
Theorem 3 e Cauchy problem (22) has a unique integralsolution provided that
120576 lt 12 (32)
Proof Define the operator Γ C120572 997888rarr C120572 by
120572 )119889119904] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572
(37)
Accordingly we get
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 [120576 + max (119886 119887120591120572minus120574)120579 ] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (38)
Hence we conclude that
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 (120576 + 1)3 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (39)
The fact 2(120576 + 1)3 lt 1 proves that Γ has an unique fixedpoint 119909 in (C120572 ||120572) which is the integral solution of equation(22)
32 Stability of the Integral Solution Here we give a resultconcerning the nonlocal-condition effect on the stability ofthe integral solution
Theorem 4 Let 119909 and 119910 be solutions associated with (1199090 1199091)and (1199090 1199101) respectivelyen we have the following estimate
1003816100381610038161003816119910 minus 1199091003816100381610038161003816le 2
1 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101minus 11990911003817100381710038171003817
(40)
provided that
120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574) lt 1
2 (41)
Advances in Mathematical Physics 5
Proof We have
119910 (119905) minus 119909 (119905) = 119878 (119905120572120572 ) [1199101 minus 1199091 + 119892 (119910) minus 119892 (119909)]
+ int1199050119904120572minus1119878 (119905120572 minus 119904120572
minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904
(42)
Consequently we get
1003816100381610038161003816119910 minus 1199091003816100381610038161003816 le 2(120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)) 1003816100381610038161003816119910 minus 1199091003816100381610038161003816
+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)
Finally we obtain the following estimation1003816100381610038161003816119910 minus 1199091003816100381610038161003816
le 21 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101
minus 11990911003817100381710038171003817 (44)
4 Implicit Fundamental Solution inthe Case When 120574=120572
Here we give the implicit fundamental solution of (4) byusing the separating variables method To do so let 119906(119905 ℓ) =119909(119905)119910(ℓ) Then (4) becomes as follows
2120572 )int1205870cos(radic41198881198992 + 4119886 minus 1198872
2120572 119905120572)
sdot 1199090 sin (119899119904) 119889119904 + 2119887120587 exp(minus119887119905120572
2120572 )
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887119905120572
2120572 )
sdot 119901sum119894=1
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 [1199091119901
+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904
(52)
Finally replacing 119909119899(119905) in (49) we get
By using the classical Laplace transform in (55) we get
119909119899 (119905) = 2120587exp(minus119887
2119905)
sdot int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587
sdot exp(minus1198872119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887
2119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899119904) 119889119904
(57)
Replacing 119909119899(119905) in (54) we find the fundamental solution ofthe classical telegraph equation (1) as follows
119906 (119905 ℓ) = 2120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 2119887120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 4120587exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899ℓ)
sdot sin (119899119904) 119889119904(58)
Remark 6 If we consider the fractional derivative in Caputorsquossense the implicit fundamental solution of (4) can be writtenin terms of the Mittag-Leffler function However in our casewe have found the implicit fundamental solution in terms ofthe classical trigonometric functions
Remark 7 The integral solution does not impose any con-straint concerning the choice of the derivation parametersThis provides more freedom concerning the choice of sensi-tive parameters 120572 and 120574 in practice situations for modelingnaturel phenomena
5 Conclusion
We have studied a time-conformable fractional telegraphequation with nonlocal condition In the case when 120572 = 120574we have given the implicit fundamental solution in terms ofthe classical trigonometric functions In the general case wehave established the existence uniqueness and stability of theintegral solution
As for future work we intend to give the fundamentalsolution for all values of 120572 and 120574Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] E C Eckstein J AGoldstein andM Leggas ldquoThemathematicsof suspensions Kacwalks and asymptotic analyticityrdquoElectronicJournal of Differential Equations vol 3 pp 39ndash50 1999
[2] R C Cascaval E C Eckstein C L Frota and J A GoldsteinldquoFractional telegraph equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 276 no 1 pp 145ndash159 2002
[3] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[4] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[5] V RHosseiniWChen andZAvazzadeh ldquoNumerical solutionof fractional telegraph equation by using radial basis functionsrdquo
Advances in Mathematical Physics 7
Engineering Analysis with Boundary Elements vol 38 no 12 pp31ndash39 2014
[6] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[7] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[8] V K Srivastava M K Awasthi and S Kumar ldquoAnalyticalapproximations of two and three dimensional time-fractionaltelegraphic equation by reduced differential transformmethodrdquoEgyptian Journal of Basic and Applied Sciences vol 1 no 1 pp60ndash66 2014
[9] A A Kilbas H M Srivastava and J J Trujillo eory andApplications of Fractional Differential Equations North HollandMathematics Studies 204 Elsevier New York NY USA 2006
[10] K S Miller An Introduction to the Fractional Calculus andFractional Differential Equations John Wiley and Sons 1993
[11] K B Oldham and J Spaniere Fractional Calculus AcademicPress New York NY USA 1974
[12] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives eory and Applications Gordon andBreach Yverdon Switzerland 1993
[14] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
[15] W S Chung ldquoFractional Newton mechanics with conformablefractional derivativerdquo Journal of Computational and AppliedMathematics vol 290 pp 150ndash158 2015
[16] L Martınez J Rosales C Carreno and J Lozano ldquoElectricalcircuits described by fractional conformable derivativerdquo Inter-national Journal of Circuit eory and Applications vol 46 no5 pp 1091ndash1100 2018
[17] H W Zhou S Yang and S Q Zhang ldquoConformable deriva-tive approach to anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 491 pp 1001ndash1013 2018
[18] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
[19] K Deng ldquoExponential decay of solutions of semilinearparabolic equations with nonlocal initial conditionsrdquo Journal ofMathematical Analysis and Applications vol 179 no 2 pp 630ndash637 1993
[20] W E Olmstead and C A Roberts ldquoThe one-dimensional heatequation with a nonlocal initial conditionrdquo Applied Mathemat-ics Letters vol 10 no 3 pp 89ndash94 1997
[21] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[22] C C Travis and G F Webb ldquoCosine families and abstract non-linear second order differential equationsrdquo Acta MathematicaHungarica vol 32 no 1-2 pp 75ndash96 1978
[23] M E Hernandez ldquoExistence of solutions to a second order par-tial differential equation with nonlocal conditionsrdquo ElectronicJournal of Differential Equations vol 2003 pp 1ndash10 2003
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
4 Advances in Mathematical Physics
Accordingly we obtain
1003817100381710038171003817Γ (119910) (119905) minus Γ (119909) (119905)1003817100381710038171003817 +10038171003817100381710038171003817100381710038171003817119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)]10038171003817100381710038171003817100381710038171003817
le 2 [120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816
(30)
Then we get
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816le 2 [120576 +max(119886120591120572
120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816 (31)
Finally Γ has an unique fixed point 119909 in (C120572 ||) which is theintegral solution of the equation (22)
Now we give a result that is better than the previous one
Theorem 3 e Cauchy problem (22) has a unique integralsolution provided that
120576 lt 12 (32)
Proof Define the operator Γ C120572 997888rarr C120572 by
120572 )119889119904] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572
(37)
Accordingly we get
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 [120576 + max (119886 119887120591120572minus120574)120579 ] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (38)
Hence we conclude that
1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 (120576 + 1)3 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (39)
The fact 2(120576 + 1)3 lt 1 proves that Γ has an unique fixedpoint 119909 in (C120572 ||120572) which is the integral solution of equation(22)
32 Stability of the Integral Solution Here we give a resultconcerning the nonlocal-condition effect on the stability ofthe integral solution
Theorem 4 Let 119909 and 119910 be solutions associated with (1199090 1199091)and (1199090 1199101) respectivelyen we have the following estimate
1003816100381610038161003816119910 minus 1199091003816100381610038161003816le 2
1 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101minus 11990911003817100381710038171003817
(40)
provided that
120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574) lt 1
2 (41)
Advances in Mathematical Physics 5
Proof We have
119910 (119905) minus 119909 (119905) = 119878 (119905120572120572 ) [1199101 minus 1199091 + 119892 (119910) minus 119892 (119909)]
+ int1199050119904120572minus1119878 (119905120572 minus 119904120572
minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904
(42)
Consequently we get
1003816100381610038161003816119910 minus 1199091003816100381610038161003816 le 2(120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)) 1003816100381610038161003816119910 minus 1199091003816100381610038161003816
+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)
Finally we obtain the following estimation1003816100381610038161003816119910 minus 1199091003816100381610038161003816
le 21 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101
minus 11990911003817100381710038171003817 (44)
4 Implicit Fundamental Solution inthe Case When 120574=120572
Here we give the implicit fundamental solution of (4) byusing the separating variables method To do so let 119906(119905 ℓ) =119909(119905)119910(ℓ) Then (4) becomes as follows
2120572 )int1205870cos(radic41198881198992 + 4119886 minus 1198872
2120572 119905120572)
sdot 1199090 sin (119899119904) 119889119904 + 2119887120587 exp(minus119887119905120572
2120572 )
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887119905120572
2120572 )
sdot 119901sum119894=1
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 [1199091119901
+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904
(52)
Finally replacing 119909119899(119905) in (49) we get
By using the classical Laplace transform in (55) we get
119909119899 (119905) = 2120587exp(minus119887
2119905)
sdot int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587
sdot exp(minus1198872119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887
2119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899119904) 119889119904
(57)
Replacing 119909119899(119905) in (54) we find the fundamental solution ofthe classical telegraph equation (1) as follows
119906 (119905 ℓ) = 2120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 2119887120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 4120587exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899ℓ)
sdot sin (119899119904) 119889119904(58)
Remark 6 If we consider the fractional derivative in Caputorsquossense the implicit fundamental solution of (4) can be writtenin terms of the Mittag-Leffler function However in our casewe have found the implicit fundamental solution in terms ofthe classical trigonometric functions
Remark 7 The integral solution does not impose any con-straint concerning the choice of the derivation parametersThis provides more freedom concerning the choice of sensi-tive parameters 120572 and 120574 in practice situations for modelingnaturel phenomena
5 Conclusion
We have studied a time-conformable fractional telegraphequation with nonlocal condition In the case when 120572 = 120574we have given the implicit fundamental solution in terms ofthe classical trigonometric functions In the general case wehave established the existence uniqueness and stability of theintegral solution
As for future work we intend to give the fundamentalsolution for all values of 120572 and 120574Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] E C Eckstein J AGoldstein andM Leggas ldquoThemathematicsof suspensions Kacwalks and asymptotic analyticityrdquoElectronicJournal of Differential Equations vol 3 pp 39ndash50 1999
[2] R C Cascaval E C Eckstein C L Frota and J A GoldsteinldquoFractional telegraph equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 276 no 1 pp 145ndash159 2002
[3] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[4] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[5] V RHosseiniWChen andZAvazzadeh ldquoNumerical solutionof fractional telegraph equation by using radial basis functionsrdquo
Advances in Mathematical Physics 7
Engineering Analysis with Boundary Elements vol 38 no 12 pp31ndash39 2014
[6] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[7] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[8] V K Srivastava M K Awasthi and S Kumar ldquoAnalyticalapproximations of two and three dimensional time-fractionaltelegraphic equation by reduced differential transformmethodrdquoEgyptian Journal of Basic and Applied Sciences vol 1 no 1 pp60ndash66 2014
[9] A A Kilbas H M Srivastava and J J Trujillo eory andApplications of Fractional Differential Equations North HollandMathematics Studies 204 Elsevier New York NY USA 2006
[10] K S Miller An Introduction to the Fractional Calculus andFractional Differential Equations John Wiley and Sons 1993
[11] K B Oldham and J Spaniere Fractional Calculus AcademicPress New York NY USA 1974
[12] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives eory and Applications Gordon andBreach Yverdon Switzerland 1993
[14] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
[15] W S Chung ldquoFractional Newton mechanics with conformablefractional derivativerdquo Journal of Computational and AppliedMathematics vol 290 pp 150ndash158 2015
[16] L Martınez J Rosales C Carreno and J Lozano ldquoElectricalcircuits described by fractional conformable derivativerdquo Inter-national Journal of Circuit eory and Applications vol 46 no5 pp 1091ndash1100 2018
[17] H W Zhou S Yang and S Q Zhang ldquoConformable deriva-tive approach to anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 491 pp 1001ndash1013 2018
[18] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
[19] K Deng ldquoExponential decay of solutions of semilinearparabolic equations with nonlocal initial conditionsrdquo Journal ofMathematical Analysis and Applications vol 179 no 2 pp 630ndash637 1993
[20] W E Olmstead and C A Roberts ldquoThe one-dimensional heatequation with a nonlocal initial conditionrdquo Applied Mathemat-ics Letters vol 10 no 3 pp 89ndash94 1997
[21] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[22] C C Travis and G F Webb ldquoCosine families and abstract non-linear second order differential equationsrdquo Acta MathematicaHungarica vol 32 no 1-2 pp 75ndash96 1978
[23] M E Hernandez ldquoExistence of solutions to a second order par-tial differential equation with nonlocal conditionsrdquo ElectronicJournal of Differential Equations vol 2003 pp 1ndash10 2003
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904
(42)
Consequently we get
1003816100381610038161003816119910 minus 1199091003816100381610038161003816 le 2(120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)) 1003816100381610038161003816119910 minus 1199091003816100381610038161003816
+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)
Finally we obtain the following estimation1003816100381610038161003816119910 minus 1199091003816100381610038161003816
le 21 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101
minus 11990911003817100381710038171003817 (44)
4 Implicit Fundamental Solution inthe Case When 120574=120572
Here we give the implicit fundamental solution of (4) byusing the separating variables method To do so let 119906(119905 ℓ) =119909(119905)119910(ℓ) Then (4) becomes as follows
2120572 )int1205870cos(radic41198881198992 + 4119886 minus 1198872
2120572 119905120572)
sdot 1199090 sin (119899119904) 119889119904 + 2119887120587 exp(minus119887119905120572
2120572 )
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887119905120572
2120572 )
sdot 119901sum119894=1
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 [1199091119901
+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904
(52)
Finally replacing 119909119899(119905) in (49) we get
By using the classical Laplace transform in (55) we get
119909119899 (119905) = 2120587exp(minus119887
2119905)
sdot int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587
sdot exp(minus1198872119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887
2119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899119904) 119889119904
(57)
Replacing 119909119899(119905) in (54) we find the fundamental solution ofthe classical telegraph equation (1) as follows
119906 (119905 ℓ) = 2120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 2119887120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 4120587exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899ℓ)
sdot sin (119899119904) 119889119904(58)
Remark 6 If we consider the fractional derivative in Caputorsquossense the implicit fundamental solution of (4) can be writtenin terms of the Mittag-Leffler function However in our casewe have found the implicit fundamental solution in terms ofthe classical trigonometric functions
Remark 7 The integral solution does not impose any con-straint concerning the choice of the derivation parametersThis provides more freedom concerning the choice of sensi-tive parameters 120572 and 120574 in practice situations for modelingnaturel phenomena
5 Conclusion
We have studied a time-conformable fractional telegraphequation with nonlocal condition In the case when 120572 = 120574we have given the implicit fundamental solution in terms ofthe classical trigonometric functions In the general case wehave established the existence uniqueness and stability of theintegral solution
As for future work we intend to give the fundamentalsolution for all values of 120572 and 120574Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] E C Eckstein J AGoldstein andM Leggas ldquoThemathematicsof suspensions Kacwalks and asymptotic analyticityrdquoElectronicJournal of Differential Equations vol 3 pp 39ndash50 1999
[2] R C Cascaval E C Eckstein C L Frota and J A GoldsteinldquoFractional telegraph equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 276 no 1 pp 145ndash159 2002
[3] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[4] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[5] V RHosseiniWChen andZAvazzadeh ldquoNumerical solutionof fractional telegraph equation by using radial basis functionsrdquo
Advances in Mathematical Physics 7
Engineering Analysis with Boundary Elements vol 38 no 12 pp31ndash39 2014
[6] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[7] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[8] V K Srivastava M K Awasthi and S Kumar ldquoAnalyticalapproximations of two and three dimensional time-fractionaltelegraphic equation by reduced differential transformmethodrdquoEgyptian Journal of Basic and Applied Sciences vol 1 no 1 pp60ndash66 2014
[9] A A Kilbas H M Srivastava and J J Trujillo eory andApplications of Fractional Differential Equations North HollandMathematics Studies 204 Elsevier New York NY USA 2006
[10] K S Miller An Introduction to the Fractional Calculus andFractional Differential Equations John Wiley and Sons 1993
[11] K B Oldham and J Spaniere Fractional Calculus AcademicPress New York NY USA 1974
[12] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives eory and Applications Gordon andBreach Yverdon Switzerland 1993
[14] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
[15] W S Chung ldquoFractional Newton mechanics with conformablefractional derivativerdquo Journal of Computational and AppliedMathematics vol 290 pp 150ndash158 2015
[16] L Martınez J Rosales C Carreno and J Lozano ldquoElectricalcircuits described by fractional conformable derivativerdquo Inter-national Journal of Circuit eory and Applications vol 46 no5 pp 1091ndash1100 2018
[17] H W Zhou S Yang and S Q Zhang ldquoConformable deriva-tive approach to anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 491 pp 1001ndash1013 2018
[18] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
[19] K Deng ldquoExponential decay of solutions of semilinearparabolic equations with nonlocal initial conditionsrdquo Journal ofMathematical Analysis and Applications vol 179 no 2 pp 630ndash637 1993
[20] W E Olmstead and C A Roberts ldquoThe one-dimensional heatequation with a nonlocal initial conditionrdquo Applied Mathemat-ics Letters vol 10 no 3 pp 89ndash94 1997
[21] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[22] C C Travis and G F Webb ldquoCosine families and abstract non-linear second order differential equationsrdquo Acta MathematicaHungarica vol 32 no 1-2 pp 75ndash96 1978
[23] M E Hernandez ldquoExistence of solutions to a second order par-tial differential equation with nonlocal conditionsrdquo ElectronicJournal of Differential Equations vol 2003 pp 1ndash10 2003
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
By using the classical Laplace transform in (55) we get
119909119899 (119905) = 2120587exp(minus119887
2119905)
sdot int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587
sdot exp(minus1198872119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904
+ 4120587 exp(minus119887
2119905)
sdot int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899119904) 119889119904
(57)
Replacing 119909119899(119905) in (54) we find the fundamental solution ofthe classical telegraph equation (1) as follows
119906 (119905 ℓ) = 2120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870cos(radic41198881198992 + 4119886 minus 1198872
2 119905) 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 2119887120587 exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899ℓ)
sdot sin (119899119904) 119889119904 + 4120587exp(minus119887
2119905)
sdot +infinsum119899=0
int1205870
sin ((radic41198881198992 + 4119886 minus 11988722) 119905)radic41198881198992 + 4119886 minus 1198872 1199091 sin (119899ℓ)
sdot sin (119899119904) 119889119904(58)
Remark 6 If we consider the fractional derivative in Caputorsquossense the implicit fundamental solution of (4) can be writtenin terms of the Mittag-Leffler function However in our casewe have found the implicit fundamental solution in terms ofthe classical trigonometric functions
Remark 7 The integral solution does not impose any con-straint concerning the choice of the derivation parametersThis provides more freedom concerning the choice of sensi-tive parameters 120572 and 120574 in practice situations for modelingnaturel phenomena
5 Conclusion
We have studied a time-conformable fractional telegraphequation with nonlocal condition In the case when 120572 = 120574we have given the implicit fundamental solution in terms ofthe classical trigonometric functions In the general case wehave established the existence uniqueness and stability of theintegral solution
As for future work we intend to give the fundamentalsolution for all values of 120572 and 120574Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] E C Eckstein J AGoldstein andM Leggas ldquoThemathematicsof suspensions Kacwalks and asymptotic analyticityrdquoElectronicJournal of Differential Equations vol 3 pp 39ndash50 1999
[2] R C Cascaval E C Eckstein C L Frota and J A GoldsteinldquoFractional telegraph equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 276 no 1 pp 145ndash159 2002
[3] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[4] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[5] V RHosseiniWChen andZAvazzadeh ldquoNumerical solutionof fractional telegraph equation by using radial basis functionsrdquo
Advances in Mathematical Physics 7
Engineering Analysis with Boundary Elements vol 38 no 12 pp31ndash39 2014
[6] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[7] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[8] V K Srivastava M K Awasthi and S Kumar ldquoAnalyticalapproximations of two and three dimensional time-fractionaltelegraphic equation by reduced differential transformmethodrdquoEgyptian Journal of Basic and Applied Sciences vol 1 no 1 pp60ndash66 2014
[9] A A Kilbas H M Srivastava and J J Trujillo eory andApplications of Fractional Differential Equations North HollandMathematics Studies 204 Elsevier New York NY USA 2006
[10] K S Miller An Introduction to the Fractional Calculus andFractional Differential Equations John Wiley and Sons 1993
[11] K B Oldham and J Spaniere Fractional Calculus AcademicPress New York NY USA 1974
[12] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives eory and Applications Gordon andBreach Yverdon Switzerland 1993
[14] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
[15] W S Chung ldquoFractional Newton mechanics with conformablefractional derivativerdquo Journal of Computational and AppliedMathematics vol 290 pp 150ndash158 2015
[16] L Martınez J Rosales C Carreno and J Lozano ldquoElectricalcircuits described by fractional conformable derivativerdquo Inter-national Journal of Circuit eory and Applications vol 46 no5 pp 1091ndash1100 2018
[17] H W Zhou S Yang and S Q Zhang ldquoConformable deriva-tive approach to anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 491 pp 1001ndash1013 2018
[18] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
[19] K Deng ldquoExponential decay of solutions of semilinearparabolic equations with nonlocal initial conditionsrdquo Journal ofMathematical Analysis and Applications vol 179 no 2 pp 630ndash637 1993
[20] W E Olmstead and C A Roberts ldquoThe one-dimensional heatequation with a nonlocal initial conditionrdquo Applied Mathemat-ics Letters vol 10 no 3 pp 89ndash94 1997
[21] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[22] C C Travis and G F Webb ldquoCosine families and abstract non-linear second order differential equationsrdquo Acta MathematicaHungarica vol 32 no 1-2 pp 75ndash96 1978
[23] M E Hernandez ldquoExistence of solutions to a second order par-tial differential equation with nonlocal conditionsrdquo ElectronicJournal of Differential Equations vol 2003 pp 1ndash10 2003
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Advances in Mathematical Physics 7
Engineering Analysis with Boundary Elements vol 38 no 12 pp31ndash39 2014
[6] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[7] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[8] V K Srivastava M K Awasthi and S Kumar ldquoAnalyticalapproximations of two and three dimensional time-fractionaltelegraphic equation by reduced differential transformmethodrdquoEgyptian Journal of Basic and Applied Sciences vol 1 no 1 pp60ndash66 2014
[9] A A Kilbas H M Srivastava and J J Trujillo eory andApplications of Fractional Differential Equations North HollandMathematics Studies 204 Elsevier New York NY USA 2006
[10] K S Miller An Introduction to the Fractional Calculus andFractional Differential Equations John Wiley and Sons 1993
[11] K B Oldham and J Spaniere Fractional Calculus AcademicPress New York NY USA 1974
[12] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives eory and Applications Gordon andBreach Yverdon Switzerland 1993
[14] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
[15] W S Chung ldquoFractional Newton mechanics with conformablefractional derivativerdquo Journal of Computational and AppliedMathematics vol 290 pp 150ndash158 2015
[16] L Martınez J Rosales C Carreno and J Lozano ldquoElectricalcircuits described by fractional conformable derivativerdquo Inter-national Journal of Circuit eory and Applications vol 46 no5 pp 1091ndash1100 2018
[17] H W Zhou S Yang and S Q Zhang ldquoConformable deriva-tive approach to anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 491 pp 1001ndash1013 2018
[18] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
[19] K Deng ldquoExponential decay of solutions of semilinearparabolic equations with nonlocal initial conditionsrdquo Journal ofMathematical Analysis and Applications vol 179 no 2 pp 630ndash637 1993
[20] W E Olmstead and C A Roberts ldquoThe one-dimensional heatequation with a nonlocal initial conditionrdquo Applied Mathemat-ics Letters vol 10 no 3 pp 89ndash94 1997
[21] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[22] C C Travis and G F Webb ldquoCosine families and abstract non-linear second order differential equationsrdquo Acta MathematicaHungarica vol 32 no 1-2 pp 75ndash96 1978
[23] M E Hernandez ldquoExistence of solutions to a second order par-tial differential equation with nonlocal conditionsrdquo ElectronicJournal of Differential Equations vol 2003 pp 1ndash10 2003
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