-
Hindawi Publishing CorporationMathematical Problems in
EngineeringVolume 2013, Article ID 161030, 8
pageshttp://dx.doi.org/10.1155/2013/161030
Research ArticleLegendre Wavelets Method for Solving Fractional
PopulationGrowth Model in a Closed System
M. H. Heydari,1 M. R. Hooshmandasl,1 C. Cattani,2 and Ming
Li3
1 Faculty of Mathematics, Yazd University, Yazd 89195741,
Iran2Department of Mathematics, University of Salerno, Via Ponte
Don Melillo, 84084 Fisciano, Italy3 School of Information Science
& Technology, East China Normal University, Shanghai 200241,
China
Correspondence should be addressed to M. R. Hooshmandasl;
[email protected]
Received 7 August 2013; Accepted 17 August 2013
Academic Editor: Cristian Toma
Copyright © 2013 M. H. Heydari et al. This 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.
A new operational matrix of fractional order integration for
Legendre wavelets is derived. Block pulse functions and
collocationmethod are employed to derive a general procedure for
forming this matrix. Moreover, a computational method based on
waveletexpansion together with this operational matrix is proposed
to obtain approximate solution of the fractional population
growthmodel of a species within a closed system. The main
characteristic of the new approach is to convert the problem under
study to anonlinear algebraic equation.
1. Introduction
In recent years, fractional calculus and differential
equationshave found enormous applications in mathematics,
physics,chemistry, and engineering because of the fact that a
realisticmodeling of a physical phenomenon having dependencenot
only at the time instant but also on the previous timehistory can
be successfully achieved by using fractionalcalculus.The
applications of the fractional calculus have beendemonstrated by
many authors. For examples, it has beenapplied to model the
nonlinear oscillation of earthquakes,fluid-dynamic traffic,
frequency dependent damping behav-ior of many viscoelastic
materials, continuum and statisticalmechanics, colored noise,
solidmechanics, economics, signalprocessing, and control theory
[1–5].However, during the lastdecade fractional calculus has
attracted much more attentionof physicists and mathematicians. Due
to the increasingapplications, some schemes have been proposed to
solvefractional differential equations. The most frequently
usedmethods are Adomian decomposition method (ADM) [6,7], homotopy
perturbation method [8], homotopy analysismethod [9], variational
iteration method (VIM) [10], frac-tional differential transform
method (FDTM) [11, 12], frac-tional difference method (FDM) [13],
power series method[14], generalized block pulse operational matrix
method [15],
and Laplace transform method [16]. Also, recently the
Haarwavelets [17], Legendre wavelets [18, 19], and the
Chebyshevwavelets of first kind [20–23] and second kind [24] have
beendeveloped to solve the fractional differential equations. It
isworth noting that wavelets are localized functions, whichare the
basis for energy-bounded functions and in particularfor 𝐿2(𝑅), so
that localized pulse problems can be easilyapproached and analyzed
[25–28].
Approximation by orthogonal family of basis functionshas found
wide applications in science and engineering. Themost commonly used
orthogonal families of functions inrecent years are sine-cosine
functions, block pulse functions,Legendre, Chebyshev, and Laguerre
polynomials and alsoorthogonal wavelets, for example Haar,
Legendre, Cheby-shev, and CAS wavelets. The main advantages of
using anorthogonal basis is that the problem under
considerationreduces to a system of linear or nonlinear algebraic
systemequations [18]; thus this act not only simplifies the
problemenormously but also speeds up the computation work duringthe
implementation. This work can be done by truncatingthe series
expansion in orthogonal basis function for theunknown solution of
the problem and using the operationalmatrices [29]. There are two
main approaches for numericalsolution of fractional differential
equations.
-
2 Mathematical Problems in Engineering
One approach is based on using the operational matrix
offractional derivative to reduce the problem under considera-tion
into a system of algebraic equations and solving this sys-tem to
obtain the numerical solution of the problem. Anotheruseful
approach is based on converting the underlying frac-tional
differential equations into fractional integral equations,and using
the operational matrix of fractional integration, toeliminate the
integral operations and reducing the probleminto solving a system
of algebraic equations. The operationalmatrix of fractional
Riemann-Liouville integration is given by
𝐼𝛼Ψ (𝑥) ≃ 𝑃
𝛼Ψ (𝑥) , (1)
where Ψ(𝑥) = [𝜓1(𝑥), 𝜓
2(𝑥), . . . , 𝜓
�̂�]𝑇, in which 𝜓
𝑖(𝑥) (𝑖 =
1, 2, . . . , �̂�) are orthogonal basis functions which are
orthog-onal with respect to a specific weight function on a
certaininterval [𝑎, 𝑏] and 𝑃𝛼 is the operational matrix of
fractionalintegration ofΨ(𝑥). Notice that𝑃𝛼 is a constant �̂� ×
�̂�matrixand 𝛼 is an arbitrary positive constant.
In view of successful application of wavelet operationalmatrices
in numerical solution of integral and differentialequations,
together with the characteristics of wavelet func-tions, we believe
that they can be applicable in solvingfractional population growth
model. In this paper, the oper-ational matrix of fractional order
integrations for Legendrewavelets is derived, and a general
procedure based oncollocation method and block Pulse functions
(BPFs) forforming this matrix is presented. Then, by using this
matrixa computational method for solving fractional
populationgrowth model in a closed system is proposed. This paper
isorganized as follows. In Section 2, some necessary definitionsof
the fractional calculus are reviewed. In Section 3, the Leg-endre
wavelets with some of their properties are presented.In Section 4,
the proposed method for solving fractionalpopulation growth model
in a closed system is described.Finally a conclusion is drawn in
Section 5.
2. Preliminaries
In this section, we present some notations, definitions,
andpreliminary facts that will be used further in this paper.
The Riemann-Liouville fractional integral operator 𝐼𝛼 oforder 𝛼
≥ 0 on the usual Lebesgue space 𝐿1[0, 𝑏] is given by[30]
(𝐼𝛼𝑢) (𝑥) =
{
{
{
1
Γ (𝛼)∫𝑥
0(𝑥 − 𝑠)
𝛼−1𝑢 (𝑠) 𝑑𝑠, 𝛼 > 0,
𝑢 (𝑥) , 𝛼 = 0.
(2)
The Riemann-Liouville fractional derivative of order 𝛼 > 0
isnormally defined as
𝐷𝛼𝑢 (𝑥) = (
𝑑
𝑑𝑥)
𝑚
𝐼𝑚−𝛼
𝑢 (𝑥) , (𝑚 − 1 < 𝛼 ≤ 𝑚) , (3)
where𝑚 is an integer.
The fractional derivative of order 𝛼 > 0 in the Caputosense
is given by [30]
𝐷𝛼
∗𝑢 (𝑥) =
1
Γ (𝑚 − 𝛼)∫
𝑥
0
(𝑥 − 𝑠)𝑚−𝛼−1
𝑢(𝑚)
(𝑠) 𝑑𝑠,
(𝑚 − 1 < 𝛼 ≤ 𝑚) ,
(4)
where𝑚 is an integer, 𝑥 > 0, and 𝑢(𝑚) ∈ 𝐿1[0, 𝑏].The useful
relation between the Riemann-Liouville oper-
ator andCaputo operator is given by the following
expression:
𝐼𝛼𝐷𝛼
∗𝑢 (𝑥) = 𝑢 (𝑥) −
𝑚−1
∑
𝑘=0
𝑢(𝑘)
(0+)𝑥𝑘
𝑘!, (𝑚 − 1 < 𝛼 ≤ 𝑚) ,
(5)
where𝑚 is an integer, 𝑥 > 0, and 𝑢(𝑚) ∈ 𝐿1[0, 𝑏].
3. The Legendre Wavelets
In this section,we briefly present someproperties of
Legendrewavelets.
3.1. Constructing the Legendre Wavelets. Here we introducea
process to construct the Legendre wavelets on the unitinterval [0,
1], using recursive wavelet construction whichhas been proposed in
[31, 32] for piecewise polynomials on[0, 1]. For this purpose, we
first introduce some notations.Throughout this work, N denotes the
set of all naturalnumbers,N
0= N∪{0} andZ
𝜇= {0, 1, . . . , 𝜇−1}, for a positive
integer 𝜇.For an integer 𝜇 > 1, we consider the following
contractive mappings on the interval 𝐼 = [0, 1]:
𝜓𝜖(𝑡) =
𝑡 + 𝜖
𝜇, 𝑡 ∈ [0, 1] , 𝜖 ∈ Z𝜇. (6)
It is obvious that the mappings {𝜓𝜖} satisfy the following
properties:
𝜓𝜖(𝐼) ⊂ 𝐼, ∀𝜖 ∈ Z
𝜇,
⋃
𝜖∈Z𝜇
𝜓𝜖(𝐼) = 𝐼.
(7)
Now, let 𝐹0denote the finite dimensional linear space on
[0, 1] that is spanned by the Legendre polynomials 𝑃0(2𝑥 −
1), 𝑃1(2𝑥 − 1), . . . , and 𝑃
𝑀−1(2𝑥 − 1), where𝑀 ∈ N and 𝑃
𝑚
are the Legendre polynomials of degree𝑚, namely,
𝐹0= span {𝑃
𝑚(2𝑥 − 1) | 𝑥 ∈ [0, 1] , 𝑚 ∈ Z
𝜇} . (8)
It is well known that the Legendre polynomials 𝑃𝑚
areorthogonal with respect to the weight function 𝑤(𝑥) = 1 onthe
interval [−1, 1].
In order to construct an orthonormal basis for 𝐿2[0, 1],for each
𝜖 ∈ Z
𝜇we define an isometry 𝑇
𝜖on 𝐿2[0, 1] as
follows:
(𝑇𝜖𝑓) (𝑥) = {
√𝜇𝑓 (𝜓−1
𝜖(𝑥)) , 𝑥 ∈ 𝜓
𝜖(𝐼) ,
0, 𝑥 ∉ 𝜓𝜖(𝐼) .
(9)
-
Mathematical Problems in Engineering 3
Starting from the space 𝐹0, we define a sequence of spaces
{𝐹𝑘| 𝑘 ∈ N
0} using the recurrence formula
𝐹𝑘+1
= ⨁
𝜖∈Z𝜇
𝑇𝜖𝐹𝑘, 𝑘 ∈ N
0, (10)
where ⊕ denotes the direct sum; that is, if 𝐴 and 𝐵 are
twosubspaces of 𝐿2[0, 1] with 𝐴 ∩ 𝐵 = {0}, then
𝐴 ⊕ 𝐵 = {𝑓 + 𝑔 : 𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵} . (11)
The sequence of spaces {𝐹𝑘| 𝑘 ∈ N
0} is nested, that is, [32]:
𝐹0⊂ 𝐹1⊂ ⋅ ⋅ ⋅ ⊂ 𝐹
𝑘⊂ 𝐹𝑘+1
⊂ ⋅ ⋅ ⋅ ,
dim𝐹𝑘= 𝑀𝜇
𝑘, 𝑘 ∈ N
0.
(12)
Moreover, similar toTheorem2.4 in [33], it can be proved
that
∞
⋃
𝑘=0
𝐹𝑘= 𝐿2[0, 1] . (13)
Now, we construct an orthonormal basis for each of thespaces
𝐹
𝑘. We first notice that
𝐺0= {√2𝑚 + 1𝑃
𝑚(2𝑥 − 1) | 𝑥 ∈ [0, 1] , 𝑚 ∈ Z𝜇} (14)
is an orthonormal basis for 𝐹0, and moreover for 𝑓(𝑥) ∈
𝐿2[0, 1] with compact support and for 𝜖 ̸= 𝜖 we have
supp {𝑇𝜖𝑓} ∩ supp {𝑇
𝜖𝑓} = 0, 𝜖 ̸= 𝜖
, (15)
where supp(𝑓) denotes the support of the function 𝑓. It canbe
simply seen that [31]
𝐺𝑘= {𝑇𝜖0∘ ⋅ ⋅ ⋅ ∘ 𝑇
𝜖𝑘−1(√2𝑚 + 1𝑃
𝑚(2𝑥 − 1)) |
𝑚 ∈ Z𝑀, 𝜖ℓ∈ Z𝜇, ℓ ∈ Z
𝑘}
(16)
is an orthonormal basis for𝐹𝑘, where “∘” denotes composition
of functions. In other words, if for 𝑛 = 1, 2, . . . , 𝜇𝑘, 𝑘 ∈
N, weset
𝜓𝑛𝑚
(𝑥) = 𝜓 (𝑘,𝑚, 𝑛, 𝑥)
=
{
{
{
√2𝑚+1𝜇𝑘/2𝑃𝑚(2𝜇𝑘𝑥 − 2𝑛+1) , 𝑥∈[
𝑛 − 1
𝜇𝑘,𝑛
𝜇𝑘) ,
0, otherwise,(17)
then {𝜓𝑛𝑚(𝑥) | 𝑛 = 1, 2, . . . , 𝜇
𝑘, 𝑚 ∈ 𝑍
𝑀} forms an ortho-
normal basis for 𝐹𝑘.
3.2. Function Approximation. A function 𝑓(𝑥) defined over[0,
1)may be expanded by the Legendre wavelets as
𝑢 (𝑥) =
∞
∑
𝑛=1
∞
∑
𝑚=0
𝑐𝑛𝑚𝜓𝑛𝑚
(𝑥) , (18)
where 𝑐𝑛𝑚
= (𝑢(𝑥), 𝜓𝑛𝑚(𝑥)), and (⋅, ⋅) denotes the inner
product. If the infinite series in (18) is truncated, then it
canbe written as
𝑢 (𝑥) ≃
𝜇𝑘
∑
𝑛=1
𝑀−1
∑
𝑚=0
𝑐𝑛𝑚𝜓𝑛𝑚
(𝑥) = 𝐶𝑇Ψ (𝑥) , (19)
where 𝑇 indicates transposition, 𝐶 and Ψ(𝑥) are �̂� =𝜇𝑘𝑀 column
vectors which are given by
𝐶 = [𝑐10, . . . , 𝑐
1𝑀−1| 𝑐20, . . . , 𝑐
2𝑀−1| ⋅ ⋅ ⋅ | 𝑐
𝜇𝑘0, . . . , 𝑐
𝜇𝑘𝑀−1
]𝑇
,
Ψ (𝑥) = [𝜓10(𝑥) , . . . , 𝜓
1𝑀−1(𝑥) | 𝜓
20(𝑥) , . . . ,
𝜓2𝑀−1
(𝑥) | ⋅ ⋅ ⋅ | 𝜓𝜇𝑘0(𝑥) , . . . , 𝜓
𝜇𝑘𝑀−1
(𝑥)]𝑇
.
(20)
Taking the collocation points
𝑡𝑖=(2𝑖 − 1)
2�̂�, 𝑖 = 1, 2, . . . , �̂�, (21)
we define the wavelet matrixΦ�̂�×�̂�
as
Φ�̂�×�̂�
= [Ψ(1
2�̂�) , Ψ (
3
2�̂�) , . . . , Ψ (
2�̂� − 1
2�̂�)] . (22)
Indeed Φ�̂�×�̂�
has the following form:
Φ�̂�×�̂�
= (
𝐴 0 0 . . . 0
0 𝐴 0 . . . 0
0 0 𝐴 . . . 0
...... d d
...0 0 . . . 0 𝐴
), (23)
where 𝐴 is an𝑀×𝑀matrix given by
-
4 Mathematical Problems in Engineering
𝐴 =
((((((((
(
𝜓10(
1
2�̂�) 𝜓
10(
3
2�̂�) . . . 𝜓
10(2�̂� − 1
2�̂�)
𝜓11(
1
2�̂�) 𝜓
11(
3
2�̂�) . . . 𝜓
11(2�̂� − 1
2�̂�)
......
......
𝜓𝜇𝑘𝑀−1
(1
2�̂�) 𝜓𝜇𝑘𝑀−1
(3
2�̂�) . . . 𝜓
𝜇𝑘𝑀−1
(2�̂� − 1
2�̂�)
))))))))
)
. (24)
For example, for 𝜇 = 3, 𝑘 = 1, 𝑀 = 2, the Legendre matrixcan be
expressed as:
Φ6×6
= (
(
1.7321 1.7321 0.0 0.0 0.0 0.0
−1.5000 1.5000 0.0 0.0 0.0 0.0
0.0 0.0 1.7321 1.7321 0.0 0.0
0.0 0.0 −1.5000 1.5000 0.0 0.0
0.0 0.0 0.0 0.0 1.7321 1.7321
0.0 0.0 0.0 0.0 −1.5000 1.5000
)
)
. (25)
3.3. Operational Matrix of Fractional Order Integration.
Thefractional integration of order 𝛼 of the vector function Ψ(𝑥)can
be expressed as
(𝐼𝛼Ψ) (𝑥) ≃ 𝑃
𝛼Ψ (𝑥) , (26)
where 𝑃𝛼 is the �̂� × �̂� operational matrix of
fractionalintegration of order 𝛼. In the following we obtain an
explicitform of the matrix 𝑃. For this purpose, we need to
introducea new family of basis functions, namely, block pulse
functions(BPFs).
We define a �̂�-set of BPFs as [34, 35]
𝑏𝑖(𝑥) =
{
{
{
1,𝑖
�̂�≤ 𝑥 <
(𝑖 + 1)
�̂�,
0, otherwise,(27)
where 𝑖 = 0, 1, 2, . . . , (�̂� − 1).The functions 𝑏
𝑖(𝑥) are disjoint and orthogonal.
The Legendre wavelets may be expanded into a �̂�-set ofBPFs
as
Ψ (𝑥) ≃ Φ�̂�×�̂�
𝐵�̂�(𝑥) , (28)
where 𝐵�̂�(𝑥) = [𝑏
0(𝑥), 𝑏1(𝑥), . . . , 𝑏
𝑖(𝑥), . . . , 𝑏
�̂�−1(𝑥)]𝑇.
In [34], Kilicman et al. have given the block pulse opera-tional
matrix of fractional integration 𝑃𝛼
𝐵as
(𝐼𝛼𝐵�̂�) (𝑥) ≃ 𝑃
𝛼
𝐵𝐵�̂�(𝑥) , (29)
where
𝑃𝛼
𝐵=
1
�̂�𝛼
1
Γ (𝛼 + 2)
(
(
1 𝜉1𝜉2. . . 𝜉�̂�−1
0 1 𝜉1. . . 𝜉�̂�−2
0 0 1 . . . 𝜉�̂�−3
0 0 0 d...
0 0 0 0 1
)
)
, (30)
and 𝜉𝑖= (𝑖 + 1)
𝛼+1− 2𝑖𝛼+1
+ (𝑖 − 1)𝛼+1.
Next, we derive the Legendre wavelets operational matrixof
fractional integration. By considering (26) and using (28),and (29)
we have
(𝐼𝛼Ψ) (𝑥) ≃ (𝐼
𝛼Φ�̂�×�̂�
𝐵�̂�) (𝑥) = Φ
�̂�×�̂�(𝐼𝛼𝐵�̂�) (𝑡)
≃ Φ�̂�×�̂�
𝑃𝛼
𝐵𝐵�̂�(𝑥) .
(31)
Thus, by considering (28) and (31), we obtain the
Legendrewavelets operational matrix of fractional integration
as
(𝐼𝛼Ψ) (𝑥) ≃ Φ
�̂�×�̂�𝑃𝛼
𝐵Φ−1
�̂�×�̂�. (32)
To illustrate the calculation procedure we choose 𝜇 = 3, 𝑘 =1, 𝑀
= 2, and 𝛼 = 1/2; thus we have:
-
Mathematical Problems in Engineering 5
𝑃(1/2)
= (
(
0.43433 0.14689 0.35988 −0.069510 0.23430 −0.017626
−0.11016 0.17991 0.052129 −0.028562 0.013219 −0.0032248
0.0 0.0 0.43433 0.14689 0.35988 −0.069510
0.0 0.0 −0.11016 0.17991 0.052129 −0.028562
0.0 0.0 0.0 0.0 0.43433 0.14689
0.0 0.0 0.0 0.0 −0.11016 0.17991
)
)
. (33)
4. Application for Fractional PopulationGrowth Model
As we have already mentioned, the fractional order modelsare
more accurate than integer order models; that is, thereare more
degrees of freedom in the fractional order models.In this section,
we will apply Legendre wavelets for solving afractional population
growth model. The model is character-ized by the nonlinear
fractional Volterra integrodifferentialequation [36] as
follows:
𝐷𝛼
∗𝑝 (𝑡) − 𝑎𝑝 (𝑡) + 𝑏[𝑝 (𝑡)]
2
+ 𝑐𝑝 (𝑡) ∫
𝑡
0
𝑝 (𝜏) 𝑑𝜏 = 0,
𝑝 (0) = 𝑝0, 0 < 𝛼 ≤ 1,
(34)
where 𝛼 is a constant parameter describing the order ofthe time
fractional derivative, 𝑎 > 0 is the birth ratecoefficient, 𝑏
> 0 is the crowding coefficient, 𝑐 > 0 is thetoxicity
coefficient, 𝑝
0is the initial population, and 𝑝(𝑡) is the
population of identical individuals at time 𝑡 which
exhibitscrowding and sensitivity to the amount of toxins
produced[37]. The coefficient 𝑐 indicates the essential behavior of
thepopulation evolution before its level falls to zero in the
longrun. It is worth mentioning that when the toxicity
coefficientis zero, (34) reduces to the well-known logistic
equation[37, 38]. The last term contains the integral which
indicatesthe totalmetabolismor total amount of toxins produced
sincetime zero. The individual death rate is proportional to
thisintegral, and also the population death rate due to
toxicitymust include a factor 𝑝. Due to the fact that the systemis
closed, the presence of the toxic term always causes thepopulation
level falling to zero in the long run, as it willbe seen later. The
relative size of the sensitivity to toxins,𝑐, determines the manner
in which the population evolvesbefore its extinction. It is worth
noting that in case 𝛼 = 1,the fractional equation reduces to a
classical logistic growthmodel, so the proposed method can be also
applied in thissituation. Here we apply the scale time and
population byintroducing the non-dimensional variables 𝑡 = 𝑐𝑡/𝑏 and
𝑢 =𝑏𝑝/𝑎, to obtain the following non-dimensional problem:
𝜅𝐷𝛼
∗𝑢 (𝑡) − 𝑢 (𝑡) + [𝑢 (𝑡)]
2+ 𝑢 (𝑡) ∫
𝑡
0
𝑢 (𝜏) 𝑑𝜏 = 0,
𝑢 (0) = 𝑢0, 0 < 𝛼 ≤ 1,
(35)
where 𝑢(𝑡) is the scaled population of identical individualsat
time 𝑡 and 𝜅 = 𝑐/𝑎𝑏 is a prescribed non-dimensional
parameter.The only equilibrium solution of (35) is the
trivialsolution 𝑢(𝑡) = 0, and the analytical solution for 𝛼 = 1 is
[39]
𝑢 (𝑡) = 𝑢0exp(1
𝜅∫
𝑡
0
(1 − 𝑢 (𝜏) − ∫
𝜏
0
𝑢 (𝑠) 𝑑𝑠) 𝑑𝜏) . (36)
In recent years, several numerical methods have been pro-posed
to solve the classical and fractional population growthmodel, for
instance, the reader is advised to see [36–43]and references
therein. Here we use the operational matrixof fractional
integration for solving nonlinear fractionalintegrodifferential
population model (35). For this purpose,we first approximate𝐷𝛼
∗𝑢(𝑡) as
𝐷𝛼
∗𝑢 (𝑡) ≃ 𝑈
𝑇Ψ (𝑡) , (37)
where 𝑈 is an unknown vector which should be found andΨ(𝑡) is
the vector which is defined in (20).
By using initial condition and (5), we have
𝑢 (𝑡) ≃ 𝑈𝑇𝑃𝛼Ψ (𝑡) + 𝑢
0. (38)
Since Ψ(𝑡) ≃ Φ�̂�×�̂�
𝐵�̂�(𝑡), from (38), we have:
𝑢 (𝑡) ≃ 𝑈𝑇𝑃𝛼Φ�̂�×�̂�
𝐵�̂�(𝑡) + 𝑢
0 [1, 1, . . . , 1] 𝐵�̂� (𝑡) . (39)
Define
𝐴𝑇= [𝑎1, 𝑎2, . . . , 𝑎
�̂�] = 𝑈
𝑇𝑃𝛼Φ�̂�×�̂�
+ 𝑢0 [1, 1, . . . , 1] . (40)
By using (38) and (39), we have 𝑢(𝑡) ≃ 𝐴𝑇𝐵�̂�(𝑡). From (27),
we have
[𝑢 (𝑡)]2≃ [𝑎2
1, 𝑎2
2, . . . , 𝑎
2
�̂�] 𝐵�̂�(𝑡) = 𝐴
𝑇𝐵�̂�(𝑡) . (41)
Also, we have
∫
𝑡
0
𝑢 (𝜏) 𝑑𝜏 ≃ 𝐴𝑇𝑃𝐵𝐵�̂�(𝑡) = 𝐶
𝑇𝐵�̂�(𝑡) , (42)
where 𝐶𝑇 = 𝐴𝑇𝑃𝐵. Now using (27), (39), and (42), we have
𝑢 (𝑡) ∫
𝑡
0
𝑢 (𝜏) 𝑑𝜏 ≃ �̃�𝑇𝐵�̂�(𝑡) , (43)
where
�̃�𝑇= [𝑎1𝑐1, 𝑎2𝑐2, . . . , 𝑎
�̂�𝑐�̂�] . (44)
Now by substituting (37), (39), (41) and (43), into (35),
weobtain
(𝑘𝑈𝑇Φ�̂�×�̂�
− 𝐴𝑇+ 𝐴𝑇+ �̂�𝑇) 𝐵�̂�(𝑡) ≃ 0, (45)
-
6 Mathematical Problems in Engineering
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t
u(t)
𝜅 = 0.2
𝜅 = 0.1
𝜅 = 0.3
𝜅 = 0.4
𝜅 = 0.5
𝜅 = 0.1, 0.2, 0.3, 0.4, 0.5
Figure 1: Numerical solutions of the classical population
growthmodel for different values of 𝜅.
and by replacing ≃ by =, we obtain the following system
ofnonlinear algebraic equations:
𝜅𝑈𝑇Φ�̂�×�̂�
− 𝐴𝑇+ 𝐴𝑇+ �̂�𝑇= 0. (46)
Finally by solving this system and determining 𝐴, we obtainthe
approximate solution of the problem as 𝑢(𝑡) = 𝐴𝑇Ψ(𝑡).
As a numerical example, we consider the nonlinearfractional
integrodifferential equation (35) with the initialcondition 𝑢(0) =
0.1, which is investigated in severalpapers, for instance see
[36–43]. Here our purpose is to studythe mathematical behavior of
the solution of this fractionalpopulation growth model as the order
of the fractionalderivative changes. In particular, we seek to
study the rapidgrowth along the logistic curve that will reach a
peak thenslow exponential decayed for different values of 𝛼. To see
thebehavior solution of this problem for different values of 𝛼,
wewill take advantage of the proposed method and consider
thefollowing two special cases.
Case 1. We investigate the classical population growth model(𝛼 =
1) for some different small values 𝜅. The behavior of thenumerical
solutions for �̂� = 162 (𝜇 = 3, 𝑘 = 3, and 𝑀 =6) is shown in Figure
1. From Figure 1 it can be seen thatas 𝜅 increases, the amplitude
of 𝑢(𝑡) decreases, whereas theexponential decay increases.
Case 2. In this case we investigate the fractional
populationgrowth model (35) for different values of 𝛼 and 𝜅.
From Figures 2, 3, and 4 it can be simply seen that as theorder
of the fractional derivative decreases, the amplitude of𝑢(𝑡)
decreases, whereas the exponential decay increases andalso it can
be concluded that as 𝜅 increases, the maximum of𝑢(𝑡∗) of 𝑢(𝑡)
decreases. This tendency is similar to the case
𝛼 = 1, which we have already mentioned.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t
u(t)
𝛼 = 0.1
𝛼 = 0.75
𝛼 = 0.5
𝛼 = 0.5, 0.75, 1.0
Figure 2: Numerical solutions of the fractional population
growthmodel for 𝜅 = 0.1.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
t
u(t)
𝛼 = 0.1
𝛼 = 0.75
𝛼 = 0.5
𝛼 = 0.5, 0.75, 1.0
Figure 3: Numerical solutions of the fractional population
growthmodel for 𝜅 = 0.3.
5. Conclusion
In this paper, the operational matrix of fractional
orderintegration for Legendre wavelets was derived. Block
pulsefunctions and collocation method were employed to derivea
general procedure for forming this matrix. Moreover, awavelet
expansion together with this operational matrixwas used to obtain
approximate solution of the fractionalpopulation growthmodel of a
species within a closed system.The main characteristic of the new
approach is to convertthe problem under study to a system of
nonlinear algebraicequations by introducing the operational matrix
of fractionalintegration for these basis functions. Analysis of the
behaviorof the model showed that it increases rapidly along
thelogistic curve followed by a slow exponential decay
afterreaching a maximum point, and also when the order of
-
Mathematical Problems in Engineering 7
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
t
u(t)
𝛼 = 0.1
𝛼 = 0.75
𝛼 = 0.5
𝛼 = 0.5, 0.75, 1.0
Figure 4: Numerical solutions of the fractional population
growthmodel for 𝜅 = 0.5.
the fractional derivative 𝛼 decreases, the amplitude of
thesolution decreases, whereas the exponential decay increases.
Acknowledgment
This work was supported in part by the National NaturalScience
Foundation of China under the project Grant nos.61272402, 61070214,
and 60873264.
References
[1] J. H.He, “Nonlinear oscillationwith fractional derivative
and itsapplications,” in Proceedings of the International
Conference onVibrating Engineering, vol. 98, pp. 288–291, Dalian,
China, 1998.
[2] R. L. Bagley and P. J. Torvik, “A theoretical basis for
theapplication of fractional calculus to viscoelasticity,” Journal
ofRheology, vol. 27, no. 3, pp. 201–210, 1983.
[3] F. Mainardi, “Fractional calculus: some basic problems
incontinuum and statistical mechanics,” in Fractals and Frac-tional
Calculus in Continuum Mechanics, A. Carpinteri and F.Mainardi,
Eds., vol. 378, pp. 291–348, Springer, New York, NY,USA, 1997.
[4] Y. A. Rossikhin and M. V. Shitikova, “Applications of
fractionalcalculus to dynamic problems of linear and nonlinear
heredi-tary mechanics of solids,” Applied Mechanics Reviews, vol.
50,pp. 15–67, 1997.
[5] R. T. Baillie, “Longmemory processes and fractional
integrationin econometrics,” Journal of Econometrics, vol. 73, no.
1, pp. 5–59, 1996.
[6] S. Momani and Z. Odibat, “Numerical approach to
differentialequations of fractional order,” Journal of
Computational andApplied Mathematics, vol. 207, no. 1, pp. 96–110,
2007.
[7] S. A. El-Wakil, A. Elhanbaly, and M. A. Abdou,
“Adomiandecomposition method for solving fractional nonlinear
differ-ential equations,” Applied Mathematics and Computation,
vol.182, no. 1, pp. 313–324, 2006.
[8] N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numeri-cal
studies for a multi-order fractional differential equation,”Physics
Letters A, vol. 371, no. 1-2, pp. 26–33, 2007.
[9] I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy
analysismethod for fractional IVPs,” Communications in
NonlinearScience and Numerical Simulation, vol. 14, no. 3, pp.
674–684,2009.
[10] S. Das, “Analytical solution of a fractional diffusion
equation byvariational iteration method,” Computers &
Mathematics withApplications, vol. 57, no. 3, pp. 483–487,
2009.
[11] A. Arikoglu and I. Ozkol, “Solution of fractional
integro-differential equations by using fractional differential
transformmethod,”Chaos, Solitons and Fractals, vol. 40, no. 2, pp.
521–529,2009.
[12] V. S. Ertürk and S. Momani, “Solving systems of
fractionaldifferential equations using differential transform
method,”Journal of Computational and AppliedMathematics, vol. 215,
no.1, pp. 142–151, 2008.
[13] M. M. Meerschaert and C. Tadjeran, “Finite difference
approxi-mations for two-sided space-fractional partial differential
equa-tions,”Applied Numerical Mathematics, vol. 56, no. 1, pp.
80–90,2006.
[14] Z. M. Odibat and N. T. Shawagfeh, “Generalized
Taylor’sformula,” Applied Mathematics and Computation, vol. 186,
no.1, pp. 286–293, 2007.
[15] Y. Li and N. Sun, “Numerical solution of fractional
differentialequations using the generalized block pulse
operationalmatrix,”Computers & Mathematics with Applications,
vol. 62, no. 3, pp.1046–1054, 2011.
[16] I. Podlubny, “The Laplace transform method for linear
differ-ential equations of the fractional order,” UEF-02-94,
Instituteof Experimental Physics, Slovak Academy of Sciences,
Kosice,Slovakia, 1994.
[17] Y. Li and W. Zhao, “Haar wavelet operational matrix of
frac-tional order integration and its applications in solving
thefractional order differential equations,” Applied Mathematicsand
Computation, vol. 216, no. 8, pp. 2276–2285, 2010.
[18] M. Rehman and R. Ali Khan, “The Legendre wavelet methodfor
solving fractional differential equations,”Communications
inNonlinear Science and Numerical Simulation, vol. 16, no. 11,
pp.4163–4173, 2011.
[19] M. H. Heydari, M. R. Hooshmandasl, F. M. M. Ghaini, andF.
Fereidouni, “Two-dimensional legendre wavelets forsolvingfractional
poisson equation with dirichlet boundary condi-tions,”Engineering
AnalysisWith Boundary Elements, vol. 37, pp.1331–1338, 2013.
[20] Y. Li, “Solving a nonlinear fractional differential
equation usingChebyshev wavelets,”Communications in Nonlinear
Science andNumerical Simulation, vol. 15, no. 9, pp. 2284–2292,
2010.
[21] M. H. Heydari, M. R. Hooshmandasl, F. M. M. Ghaini, and
F.Mohammadi, “Wavelet collocation method for solving multiorder
fractional differential equations,” Journal of AppliedMath-ematics,
vol. 2012, Article ID 542401, 19 pages, 2012.
[22] M. H. Heydari, M. R. Hooshmandasl, F. Mohammadi, and
C.Cattani, “Wavelets method for solving systems of nonlinear
sin-gular fractional volterra integro-differential equations,”
Com-munications inNonlinear Science andNumerical Simulation,
vol.19, no. 1, pp. 37–48, 2014.
[23] M. R. Hooshmandasl, M. H. Heydari, and F. M. M.
Ghaini,“Numerical solution of the one-dimensional heat equation
byusing chebyshev wavelets method,” Applied and
ComputationalMathematics, vol. 1, no. 6, Article ID 42401, 19
pages, 2012.
[24] Y. Wang and Q. Fan, “The second kind Chebyshev
waveletmethod for solving fractional differential equations,”
Applied
-
8 Mathematical Problems in Engineering
Mathematics and Computation, vol. 218, no. 17, pp.
8592–8601,2012.
[25] C. Cattani, “Fractional calculus and Shannon wavelet,”
Mathe-matical Problems in Engineering, vol. 2012, Article ID
502812, 26pages, 2012.
[26] C. Cattani, “Shannon wavelets for the solution of
integrodif-ferential equations,”Mathematical Problems in
Engineering, vol.2010, Article ID 408418, 22 pages, 2010.
[27] C. Cattani and A. Kudreyko, “Harmonic wavelet methodtowards
solution of the Fredholm type integral equations of thesecond
kind,” Applied Mathematics and Computation, vol. 215,no. 12, pp.
4164–4171, 2010.
[28] C. Cattani, “Shannon wavelets theory,” Mathematical
Problemsin Engineering, vol. 2008, Article ID 164808, 24 pages,
2008.
[29] J. Biazar and H. Ebrahimi, “Chebyshev wavelets approach
fornonlinear systems of Volterra integral equations,” Computers
&Mathematics with Applications, vol. 63, no. 3, pp. 608–616,
2012.
[30] I. Podlubny, Fractional Differential Equations, vol. 198,
Aca-demic Press, San Diego, Calif, USA, 1999.
[31] Y. Shen and W. Lin, “Collocation method for the
naturalboundary integral equation,” Applied Mathematics Letters,
vol.19, no. 11, pp. 1278–1285, 2006.
[32] C. A. Micchelli and Y. Xu, “Reconstruction and
decompositionalgorithms for biorthogonal multiwavelets,”
MultidimensionalSystems and Signal Processing, vol. 8, no. 1-2, pp.
31–69, 1997.
[33] C. A. Micchelli and Y. Xu, “Using the matrix
refinementequation for the construction of wavelets on invariant
sets,”Applied and Computational Harmonic Analysis, vol. 1, no. 4,
pp.391–401, 1994.
[34] A. Kilicman, A. Zhour, and Z. A. Aziz, “Kronecker
operationalmatrices for fractional calculus and some applications,”
AppliedMathematics and Computation, vol. 187, no. 1, pp. 250–265,
2007.
[35] M. H. Heydari, M. R. Hooshmandasl, and F. M. M. Ghaini,“A
good approximate solution for lienard equation in a largeinterval
using block pulse functions,” Journal of MathematicalExtension,
vol. 7, no. 1, pp. 17–32, 2013.
[36] H. Xu, “Analytical approximations for a population
growthmodel with fractional order,” Communications in
NonlinearScience and Numerical Simulation, vol. 14, no. 5, pp.
1978–1983,2009.
[37] K. G. TeBeest, “Numerical and analytical solutions of
Volterra’spopulation model,” SIAM Review, vol. 39, no. 3, pp.
484–493,1997.
[38] F. M. Scudo, “Vito Volterra and theoretical
ecology,”TheoreticalPopulation Biology, vol. 2, pp. 1–23, 1971.
[39] K. Parand, A. R. Rezaei, and A. Taghavi, “Numerical
approx-imations for population growth model by rational
chebyshevand hermite functions collocation approach: a
comparison,”Mathematical Methods in the Applied Sciences, vol. 33,
no. 17, pp.2076–2086, 2010.
[40] A.-M. Wazwaz, “Analytical approximations and Padé
approx-imants for Volterra’s population model,” Applied
Mathematicsand Computation, vol. 100, no. 1, pp. 13–25, 1999.
[41] K. Al-Khaled, “Numerical approximations for
populationgrowth models,” Applied Mathematics and Computation,
vol.160, no. 3, pp. 865–873, 2005.
[42] K. Al-Khaled, “Analytical approximations for a
populationgrowth model with fractional order,” Communications in
Non-linear Science and Numerical Simulation, vol. 14, pp.
1978–1983.
[43] K. Krishnaveni and S. B. K. Kannan, “Approximate
analyticalsolution for fractional population growth model,”
InternationalJournal of Engineering and Technology, vol. 5, no. 3,
pp. 2832–2836, 2013.
-
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Probability and StatisticsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
CombinatoricsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical
Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
The Scientific World JournalHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014
Stochastic AnalysisInternational Journal of