International Journal of Differential Equations

Volume 2012 (2012), Article ID 975829, 12 pages

doi:10.1155/2012/975829

Research Article

Solving Fractional-Order Logistic Equation Using a New Iterative Method

Sachin Bhalekar¹ and Varsha Daftardar-Gejji²

¹Department of Mathematics, Shivaji University, Kolhapur 416004, India

²Department of Mathematics, University of Pune, Pune 411007, India

Received 6 May 2012; Revised 11 August 2012; Accepted 5 September 2012

Academic Editor: Om Agrawal

Copyright © 2012 Sachin Bhalekar and Varsha Daftardar-Gejji. 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 fractional version of logistic equation is solved using new iterative method proposed by Daftardar-Gejji and Jafari (2006). Convergence of the series solutions obtained is discussed. The solutions obtained are compared with Adomian decomposition method and homotopy perturbation method.

1. Introduction

The following model describing growth of population was first studied by Pierre Verhulst in 1938 [1]

where is population at time , and is Malthusian parameter describing growth rate and is carrying capacity. Defining gives the following differential equation:

which is called as logistic equation.

Logistic equation of fractional order has been discussed in the literature [2, 3]. El-Sayed et al. [2] have investigated the equation , where is Caputo fractional derivative of order . Momani and Qaralleh [3] have employed Adomian decomposition method (ADM) for solving fractional population growth model in a closed system.

In the present paper we use New Iterative Method (NIM) introduced by Daftardar-Gejji and Jafari [4] to solve fractional version of logistic equation.

NIM is useful for solving a general functional equation of the form

where is a given function, and linear and nonlinear operators, respectively. The NIM has fairly simple algorithm and does not require any knowledge of involved concepts such as Adomian polynomials, homotopy, or Lagrange multipliers. Rigorous convergence analysis of NIM has been worked out recently [5]. This method has been applied by present authors successfully for solving partial differential equations [6], evolution equations [7], and fractional diffusion-wave equations [8].

NIM has been further explored by many researchers. Several numerical methods with higher order convergence can be generated using NIM. M. A. Noor and K. I. Noor [9, 10] have developed a three-step predictor-corrector method for solving nonlinear equation . Further, they have shown that this method has fourth-order convergence [11]. Some new methods [12, 13] are proposed by these authors using NIM. Mohyud-Din et al. [14] solved Hirota-Satsuma coupled KdV system using NIM. These authors [15] also have applied NIM in solutions of some fifth order boundary value problems. Noor and Mohyud-Din [16] have used NIM to solve Helmholtz equations. NIM is applied to solve homogeneous and inhomogeneous advection problems [17], diffusion equations [18], Schrödinger equations [19], time fractional partial differential equations [20], and so on. Yaseen and Samraiz [21] proposed modified NIM and used it to solve Klein-Gordon equations. Srivastava and Rai [22] have proposed a new mathematical model for oxygen delivery through a capillary to tissues in terms of multiterm fractional diffusion equation. They have solved the multi-term fractional diffusion equation using NIM and ADM and have shown that the results are in perfect agreement.

Recently Usman et al. [23] have solved a wide range of physical problems using NIM and various other methods and have shown that NIM has better performance as compared to other methods.

In the present paper, we solve fractional-order logistic equation. We compare the results obtained by NIM, ADM [3], and homotopy perturbation method (HPM) with exact solution. Further, we propose sufficient condition for the convergence of NIM solution of fractional order logistic equation.

2. Preliminaries and Notations

In this section, we set up notation and recall some basic definitions from fractional calculus [24].

Definition 2.1. Riemann-Liouville fractional integration of order is defined as

Definition 2.2. Caputo fractional derivative of order is defined as

Note that for and

3. Fractional-Order Logistic Equation

Consider

with initial condition

Operating on both sides of (3.1) and using (3.2), we get

3.1. Adomian Decomposition Method

Adomian decomposition method [25] is one of the most powerful methods used to solve (1.3). In this method, solution is assumed to be of the form , and the nonlinear operator is represented as

where

are the Adomian polynomials. The terms of the series are given as

For convergence of this series, we refer the reader to [26]. Fractional order logistic equation is solved using ADM in [3].

3.2. New Iterative Method

Recently Daftardar-Gejji and Jafari [4, 8] have used a different decomposition of the term . They have decomposed as follows:

The terms of the series are determined by the following recurrence relation:

Discussion regarding convergence of this series and comparison of ADM and NIM can be found in [5].

In view of NIM, the solution of (3.3) is given by where

The term approximate solution is given by

3.3. Homotopy Perturbation Method

Homotopy perturbation method (HPM) proposed by He [27, 28] is a useful technique to solve nonlinear problems. In view of HPM, we construct the following homotopy for solving fractional order logistic equation (3.1):

where is an embedding parameter. For , the homotopy (3.11) gives a linear equation and for , it gives the original equation (3.1). The solution of homotopy equation (3.11) is assumed to be

The solution of (3.1) is obtained from series (3.12) by substituting . Substituting (3.12) in (3.11) and equating like powers of , we get

and so on. The linear equations given in (3.13) give

and so forth. It can be seen from (3.11) that the HPM solutions are the same as ADM solutions given in [3].

4. Convergence of NIM Solutions

In this section, we provide sufficient condition for the convergence of NIM solution series (3.10).

Let be defined on a rectangle

It is clear that is bounded on . We have , for some real number on .

Lemma 4.1. For the initial value problem

the -term approximate NIM solution (3.10) exists on the interval , where , on rectangle . Moreover, the points , for all whenever .

Proof. We prove the lemma by induction on .

Clearly, the result is true for . Now,

Since the points , for all ,

Thus, .

We assume that the result is true for positive integer and proves for . Using NIM solutions, we get

Induction hypothesis implies that the points , for all . Hence

This implies that the points , for all when .

Theorem 4.2. The NIM solution series converges on the interval to a solution of the IVP (4.2).

Proof. Note that is continuous function defined on a rectangle and on . Further

Thus, is Lipschitz in the second variable on with Lipschitz constant . Now

It can be proved by induction that

This shows that the th term of the NIM solution series is bounded by (()times) the th term of the Mittag-Leffler function

Thus, the NIM series is convergent under given conditions.

5. Illustrative Examples

Mathematica 8 has been used for solving the following problems.

Example 5.1. Consider a fractional-order logistic equation

Equivalent integral equation of (5.1) is

In view of NIM,

and so on. Few terms of ADM [3] and HPM solution are

Figure 1 shows the solutions of (5.2) for by NIM (5-term solution), ADM, and HPM (6-term solution) and exact solution . In Figures 2 and 3 we compare ADM, HPM, and NIM solutions of (5.2) for and , respectively. Figure 4 shows NIM solutions for different values of .

Figure 1: (Example 5.1, ).

Figure 2: (, dashed = ADM & HPM, solid = NIM).

Figure 3: (, dashed = ADM & HPM, solid = NIM).

Figure 4: (Example 5.1, NIM solutions).

Example 5.2. Consider the following logistic equation of fractional order

Applying on both sides of (5.5) and using (5.6), we get

In Figure 5, we compare the solution of (5.7) for by NIM (5-term solution), ADM [3], and HPM (6-term solution) and exact solution . Figure 6 shows NIM solutions of (5.7) for different values of .

Figure 5: (Example 5.2, ).

Figure 6: (NIM solutions).

6. Conclusions

Fractional-order logistic equation is solved using a new iterative method (NIM). From Figures 1–6, it is clear that NIM solutions are more stable than ADM and HPM. Further the condition for the convergence of NIM solution series is also provided. It can be concluded that NIM is a useful technique for solving nonlinear problems.

Acknowledgments

V. Daftardar-Gejji acknowledges the Department of Science and Technology, New Delhi, India for the Research Grants [Project no. SR/S2/HEP-024/2009].

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