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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


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:


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)


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


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


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


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.

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