Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 161030, 8 pages

http://dx.doi.org/10.1155/2013/161030

Research Article

Legendre Wavelets Method for Solving Fractional Population Growth Model in a Closed System

M. H. Heydari,¹ M. R. Hooshmandasl,¹ C. Cattani,² and Ming Li³

¹Faculty of Mathematics, Yazd University, Yazd 89195741, Iran

²Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

³School of Information Science & Technology, East China Normal University, Shanghai 200241, China

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.

Abstract.

A new operational matrix of fractional order integration for Legendre wavelets is derived. Block pulse functions and collocation method are employed to derive a general procedure for forming this matrix. Moreover, a computational method based on wavelet expansion together with this operational matrix is proposed to obtain approximate solution of the fractional population growth model of a species within a closed system. The main characteristic of the new approach is to convert the problem under study to a nonlinear algebraic equation.

1. Introduction

In recent years, fractional calculus and differential equations have found enormous applications in mathematics, physics, chemistry, and engineering because of the fact that a realistic modeling of a physical phenomenon having dependence not only at the time instant but also on the previous time history can be successfully achieved by using fractional calculus. The applications of the fractional calculus have been demonstrated by many authors. For examples, it has been applied to model the nonlinear oscillation of earthquakes, fluid-dynamic traffic, frequency dependent damping behavior of many viscoelastic materials, continuum and statistical mechanics, colored noise, solid mechanics, economics, signal processing, and control theory [1–5]. However, during the last decade fractional calculus has attracted much more attention of physicists and mathematicians. Due to the increasing applications, some schemes have been proposed to solve fractional differential equations. The most frequently used methods are Adomian decomposition method (ADM) [6, 7], homotopy perturbation method [8], homotopy analysis method [9], variational iteration method (VIM) [10], fractional differential transform method (FDTM) [11, 12], fractional difference method (FDM) [13], power series method [14], generalized block pulse operational matrix method [15], and Laplace transform method [16]. Also, recently the Haar wavelets [17], Legendre wavelets [18, 19], and the Chebyshev wavelets of first kind [20–23] and second kind [24] have been developed to solve the fractional differential equations. It is worth noting that wavelets are localized functions, which are the basis for energy-bounded functions and in particular for , so that localized pulse problems can be easily approached and analyzed [25–28].

Approximation by orthogonal family of basis functions has found wide applications in science and engineering. The most commonly used orthogonal families of functions in recent years are sine-cosine functions, block pulse functions, Legendre, Chebyshev, and Laguerre polynomials and also orthogonal wavelets, for example Haar, Legendre, Chebyshev, and CAS wavelets. The main advantages of using an orthogonal basis is that the problem under consideration reduces to a system of linear or nonlinear algebraic system equations [18]; thus this act not only simplifies the problem enormously but also speeds up the computation work during the implementation. This work can be done by truncating the series expansion in orthogonal basis function for the unknown solution of the problem and using the operational matrices [29]. There are two main approaches for numerical solution of fractional differential equations.

One approach is based on using the operational matrix of fractional derivative to reduce the problem under consideration into a system of algebraic equations and solving this system to obtain the numerical solution of the problem. Another useful approach is based on converting the underlying fractional differential equations into fractional integral equations, and using the operational matrix of fractional integration, to eliminate the integral operations and reducing the problem into solving a system of algebraic equations. The operational matrix of fractional Riemann-Liouville integration is given by

where , in which are orthogonal basis functions which are orthogonal with respect to a specific weight function on a certain interval and is the operational matrix of fractional integration of . Notice that is a constant matrix and is an arbitrary positive constant.

In view of successful application of wavelet operational matrices in numerical solution of integral and differential equations, together with the characteristics of wavelet functions, we believe that they can be applicable in solving fractional population growth model. In this paper, the operational matrix of fractional order integrations for Legendre wavelets is derived, and a general procedure based on collocation method and block Pulse functions (BPFs) for forming this matrix is presented. Then, by using this matrix a computational method for solving fractional population growth model in a closed system is proposed. This paper is organized as follows. In Section 2, some necessary definitions of the fractional calculus are reviewed. In Section 3, the Legendre wavelets with some of their properties are presented. In Section 4, the proposed method for solving fractional population 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, and preliminary facts that will be used further in this paper.

The Riemann-Liouville fractional integral operator of order on the usual Lebesgue space is given by [30]

The Riemann-Liouville fractional derivative of order is normally defined as

where is an integer.

The fractional derivative of order in the Caputo sense is given by [30]

where is an integer, , and .

The useful relation between the Riemann-Liouville operator and Caputo operator is given by the following expression:

where is an integer, , and .

3. The Legendre Wavelets

In this section, we briefly present some properties of Legendre wavelets.

3.1. Constructing the Legendre Wavelets

Here we introduce a process to construct the Legendre wavelets on the unit interval , using recursive wavelet construction which has been proposed in [31, 32] for piecewise polynomials on . For this purpose, we first introduce some notations. Throughout this work, denotes the set of all natural numbers, and , for a positive integer .

For an integer , we consider the following contractive mappings on the interval :

It is obvious that the mappings satisfy the following properties:

Now, let denote the finite dimensional linear space on that is spanned by the Legendre polynomials , and, where and are the Legendre polynomials of degree , namely,

It is well known that the Legendre polynomials are orthogonal with respect to the weight function on the interval .

In order to construct an orthonormal basis for , for each we define an isometry on as follows:

Starting from the space , we define a sequence of spaces using the recurrence formula

where denotes the direct sum; that is, if and are two subspaces of with , then

The sequence of spaces is nested, that is, [32]:

Moreover, similar to Theorem 2.4 in [33], it can be proved that

Now, we construct an orthonormal basis for each of the spaces . We first notice that

is an orthonormal basis for , and moreover for with compact support and for we have

where denotes the support of the function . It can be simply seen that [31]

is an orthonormal basis for , where “” denotes composition of functions. In other words, if for , we set

then forms an orthonormal basis for .

3.2. Function Approximation

A function defined over may be expanded by the Legendre wavelets as

where , and denotes the inner product. If the infinite series in (18) is truncated, then it can be written as

where indicates transposition, and are column vectors which are given by

Taking the collocation points

we define the wavelet matrix as

Indeed has the following form:

where is an matrix given by

For example, for , the Legendre matrix can be expressed as:

3.3. Operational Matrix of Fractional Order Integration

The fractional integration of order of the vector function can be expressed as

where is the operational matrix of fractional integration of order . In the following we obtain an explicit form of the matrix . For this purpose, we need to introduce a new family of basis functions, namely, block pulse functions (BPFs).

We define a -set of BPFs as [34, 35]

where .

The functions are disjoint and orthogonal.

The Legendre wavelets may be expanded into a -set of BPFs as

where .

In [34], Kilicman et al. have given the block pulse operational matrix of fractional integration as

where

and .

Next, we derive the Legendre wavelets operational matrix of fractional integration. By considering (26) and using (28), and (29) we have

Thus, by considering (28) and (31), we obtain the Legendre wavelets operational matrix of fractional integration as

To illustrate the calculation procedure we choose , and ; thus we have:

4. Application for Fractional Population Growth Model

As we have already mentioned, the fractional order models are more accurate than integer order models; that is, there are more degrees of freedom in the fractional order models. In this section, we will apply Legendre wavelets for solving a fractional population growth model. The model is characterized by the nonlinear fractional Volterra integrodifferential equation [36] as follows:

where is a constant parameter describing the order of the time fractional derivative, is the birth rate coefficient, is the crowding coefficient, is the toxicity coefficient, is the initial population, and is the population of identical individuals at time which exhibits crowding and sensitivity to the amount of toxins produced [37]. The coefficient indicates the essential behavior of the population evolution before its level falls to zero in the long run. It is worth mentioning that when the toxicity coefficient is zero, (34) reduces to the well-known logistic equation [37, 38]. The last term contains the integral which indicates the total metabolism or total amount of toxins produced since time zero. The individual death rate is proportional to this integral, and also the population death rate due to toxicity must include a factor . Due to the fact that the system is closed, the presence of the toxic term always causes the population level falling to zero in the long run, as it will be seen later. The relative size of the sensitivity to toxins, , determines the manner in which the population evolves before its extinction. It is worth noting that in case , the fractional equation reduces to a classical logistic growth model, so the proposed method can be also applied in this situation. Here we apply the scale time and population by introducing the non-dimensional variables and , to obtain the following non-dimensional problem:

where is the scaled population of identical individuals at time and is a prescribed non-dimensional parameter. The only equilibrium solution of (35) is the trivial solution , and the analytical solution for is [39]

In recent years, several numerical methods have been proposed to solve the classical and fractional population growth model, for instance, the reader is advised to see [36–43] and references therein. Here we use the operational matrix of fractional integration for solving nonlinear fractional integrodifferential population model (35). For this purpose, we first approximate as

where is an unknown vector which should be found and is the vector which is defined in (20).

By using initial condition and (5), we have

Since , from (38), we have:

Define

By using (38) and (39), we have . From (27), we have

Also, we have

where . Now using (27), (39), and (42), we have

where

Now by substituting (37), (39), (41) and (43), into (35), we obtain

and by replacing by , we obtain the following system of nonlinear algebraic equations:

Finally by solving this system and determining , we obtain the approximate solution of the problem as .

As a numerical example, we consider the nonlinear fractional integrodifferential equation (35) with the initial condition , which is investigated in several papers, for instance see [36–43]. Here our purpose is to study the mathematical behavior of the solution of this fractional population growth model as the order of the fractional derivative changes. In particular, we seek to study the rapid growth along the logistic curve that will reach a peak then slow exponential decayed for different values of . To see the behavior solution of this problem for different values of , we will take advantage of the proposed method and consider the following two special cases.

Case 1. We investigate the classical population growth model () for some different small values . The behavior of the numerical solutions for and is shown in Figure 1. From Figure 1 it can be seen that as increases, the amplitude of decreases, whereas the exponential decay increases.

Figure 1: Numerical solutions of the classical population growth model for different values of .

Case 2. In this case we investigate the fractional population growth model (35) for different values of and .

From Figures 2, 3, and 4 it can be simply seen that as the order of the fractional derivative decreases, the amplitude of decreases, whereas the exponential decay increases and also it can be concluded that as increases, the maximum of of decreases. This tendency is similar to the case , which we have already mentioned.

Figure 2: Numerical solutions of the fractional population growth model for .

Figure 3: Numerical solutions of the fractional population growth model for .

Figure 4: Numerical solutions of the fractional population growth model for .

5. Conclusion

In this paper, the operational matrix of fractional order integration for Legendre wavelets was derived. Block pulse functions and collocation method were employed to derive a general procedure for forming this matrix. Moreover, a wavelet expansion together with this operational matrix was used to obtain approximate solution of the fractional population growth model of a species within a closed system. The main characteristic of the new approach is to convert the problem under study to a system of nonlinear algebraic equations by introducing the operational matrix of fractional integration for these basis functions. Analysis of the behavior of the model showed that it increases rapidly along the logistic curve followed by a slow exponential decay after reaching a maximum point, and also when the order of the fractional derivative decreases, the amplitude of the solution decreases, whereas the exponential decay increases.

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China under the project Grant nos. 61272402, 61070214, and 60873264.

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