Abstract and Applied Analysis

Volume 2012 (2012), Article ID 253890, 14 pages

doi:10.1155/2012/253890

Research Article

Enhanced Multistage Differential Transform Method: Application to the Population Models

Younghae Do¹ and Bongsoo Jang²

¹Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

²Ulsan National Institute of Science and Technology (UNIST), Ulsan Metropolitan City 689-798, Republic of Korea

Received 25 March 2012; Accepted 1 April 2012

Academic Editor: Shaher Momani

Copyright © 2012 Younghae Do and Bongsoo Jang. 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.

We present an efficient computational algorithm, namely, the enhanced multistage differential transform method (E-MsDTM) for solving prey-predator systems. Since the differential transform method (DTM) is based on the Taylor series, it is difficult to obtain accurate approximate solutions in large domain. To overcome this difficulty, the multistage differential transform method (MsDTM) has been introduced and succeeded to have reliable approximate solutions for many problems. In MsDTM, it is the key to update an initial condition in each subdomain. The standard MsDTM utilizes the approximate solution directly to assign the new initial value. Because of local convergence of the Taylor series, the error is accumulated in a large domain. In E-MsDTM, we propose the new technique to update an initial condition by using integral operator. To demonstrate efficiency of the proposed method, several numerical tests are performed and compared with ones obtained by other numerical methods such as MsDTM, multistage variational iteration method (MVIM), and fourth-order Runge-Kutta method (RK4).

1. Introduction

The analytical approaches such as the differential transform method (DTM) and the homotopy perturbation method (HPM) have been succeeded to solve differential equations and integral equations in many fields [1–10]. The basic concept of the DTM is based on the Taylor series. Suppose the solution is represented by the Taylor series. DTM provides a recursive relations for the coefficients of the Taylor series which can be solved easily. However, it is different from the conventional high-order Taylor series method which requires a complex symbolic computation. Even if DTM provides a promising algorithm for solving many linear and nonlinear problems arising in various science and engineering fields, it also has some drawbacks. Since DTM is based on the Taylor series which gives a local convergences and it has only a truncated approximation, it does not exhibits a good approximation in a large domain. Recently, the multistage DTM (MsDTM) [11–16] is proposed to accelerate the convergence of the truncated approximation in a large domain as well as to improve the accuracy of the standard DTM. The MsDTM applies the standard DTM in each subdomain. In MsDTM, it is a key to improve accuracy as well as convergence how to assign a new initial condition at each subdomain. In this paper, we demonstrate that the conventional initial updating in MsDTM accumulates an error for the nonlinear systems. In order to overcome those difficulties, a modified version of MsDTM, which is called a enhanced MsDTM (E-MsDTM), is presented. We apply the E-MsDTM to the two prey-predator models. One is a simple prey-predator model [17] as follows:

where and represent the populations of prey and predator such as rabbits and foxes, respectively, at time and , and are known constants. Another model is a Gauss-type prey-predator model [18] which describes the population dynamics that incorporate functional responses. Specially, we study the following Holling-type II () and Holling-type III ():

where and are the population size for the prey and the predator, respectively; , and are the positive constants representing prey intrinsic growth rate, carrying rate of prey to predator, and death rate of predator, respectively.

2. Description of Enhanced Multistage Differential Transform Method

Let us describe the basic definition and some properties of the standard DTM. Suppose a function is the solution of the following equation:

Let us define the differential transform of at as

The differential inverse transform of is defined by

Suppose that . Using some fundamental properties for the DTM, we have the following recursive relation:

where is the differential transform of . One can obtain the by solving the recursive relation (2.4). Then can be approximated by the finite-term Taylor series. In other words,

It is well known that the approximated error is given by

where . Thus, it is easy to see that, for the fixed term , the error is getting bigger if is increasing. To reduce the error, the more terms are added. But it requires cumbersome computational work. Moreover, it is possible that the error blows up even with many terms if the value is outside of convergence radius of the function .

To overcome difficulties arising in the standard DTM, the multistage differential transform method is introduced. The basic idea of the MsDTM is to apply the standard DTM to each subdomain. To describe the MsDTM we consider the equally spaced nodal points ; and . On th subdomain , we define . The differential transform of at is defined by

The differential inverse transform of is defined by

Similarly, can be approximated by solving the recursive relation (2.4) on . Then we have

It is an issue to give the initial condition in each subdomain . Since in (2.1) is given by the initial condition, we have . For , we approximate by computing and assign the approximate value as the initial condition in . Then . This standard MsDTM has been successful to obtain accurate approximation for many problems in [11–13]. However, it is easy to see that the approximate solution obtained by the standard MsDTM may yield an increasing error due to the nonlinearity because the DTM employs only finite term Taylor series for the given problems. For instance, let us consider

Applying the DTM gives

The basic properties of DTM have the following recursive relation for :

One can solve the above recursive relation easily to obtain the differential transforms . Then we can approximate

Since the recursive relation (2.12) is derived by equating the coefficients for in (2.11), the differential transforms are determined by solving

However, the right-hand side of (2.11) has the coefficients of , so that as many as coefficients of , are not considered to approximate . Thus, it could have a poor approximation . It is supposed to take care of all terms in obtaining the differential transform especially for the nonlinear problem.

In what follows we describe some modifications of MsDTM, which is called the Enhanced MsDTM. Suppose that is approximated by the standard MsDTM, that is, . Then, the solution can be determined by solving the following problem:

Applying the DTM to (2.15) gives the differential transform . Then is approximated by

In the standard MsDTM, which gives . As mentioned before, it could give inaccurate approximate solution . Here, taking the integral operator to (2.15), the solution can be represented by

Thus, we have

Since in (2.16), we can approximate the value of by

Then, . For each subdomain , repeat this process (2.19) to approximate . Thus, the solution can be approximated by

where

3. Application

To apply the MsDTM, we consider a subdomain with uniform length stated in the previous section. For each subdomain , we apply the DTM to the model problem (1.1), then the following recursive relation is obtained:

In order to compare the numerical performance for the enhanced MsDTM, we demonstrate comparative approximations with the standard MsDTM, the multistage variational iteration method (MVIM) [17, 19, 20], and the fourth-order Runge-Kutta method (RK4). In MVIM, for each subdomain , the solutions and can be determined by

where

Here, are general Lagrange multiplier which are determined by solving the stationary problem. Here, the multipliers are given by [17]

In order to apply DTM to the Holling-type prey-predator model (1.2) we let be the functional response of predator to prey as

For each subdomain , using the basic properties of DTM to (1.2) gives the following recursive relation:

where for ,

and for

4. Numerical Experiment and Discussion

In this section, several numerical tests are demonstrated to show the effectiveness of the E-MsDTM. For the simple prey-predator model (1.1), the constants in three different cases are shown in Table 1. Figure 1 shows the population and for the case 1 by using RK4 with time step in . With a dense time step all approximations are fitted well in graph. Thus it is difficult to show the effectiveness for the proposed method, E-MsDTM. In order to show the performance of each method clearly, we fix the time-step at MsDTM, MVIM, and E-MsDTM. Figure 2 shows the absolute error of the approximations obtained by RK4 and other methods with one iteration (a) and two iterations (b). Since the solution is stable for in Figure 1, all methods give reliable approximations for . However, MVIM and MsDTM give a large error compared with the result by E-MsDTM in . Moreover, E-MsDTM gives an accurate approximation in all domain . In case 2, all numerical results present the similar behavior shown in case 1. See Figures 3 and 4. In order to see the effectiveness of the E-MsDTM in a more complex population system, we consider the population model in case 3 in a large domain . Here we used a RK4 at and all other methods at . As seen in Figure 5 this system exhibits a oscillatory dynamics as well as many peaks. In Figure 6 the absolute errors are demonstrated as the number of iteration is increasing. All approximations have large errors at peaks, but still the E-MsDTM yields a better performance than others. In order to see the performance associated to the number of iteration, the maximum absolute errors between RK4 and other methods are presented in Figure 7. As the number of iteration is increasing, it is shown that MVIM and MsDTM give similar performance. However, the accuracy of the E-MsDTM is dramatically increasing when the number of iteration is bigger than 2.

Table 1: Various parameters.

Case

1
1
1
0.1
1
14
18

2
0.1
1
1
1
16
10

3
2
0.1
1
1
0.1
0.2

Figure 1: Population of and by RK4 () in case 1.

(a) Absolute error with one iteration: (left), (right)

(b) Absolute error with two iterations: (left), (right)

Figure 2: Comparison of absolute errors in case 1.

Figure 3: Population of and by RK4 ()   in case 2.

(a) Absolute error with one iteration: (left), right

(b) Absolute error with two iterations: (left), right

Figure 4: Comparison of absolute errors in case 2.

Figure 5: Population of and by RK4 () in case 3.

(a) Absolute error with one iteration: (left), right

(b) Absolute error with two iterations: (left), right

(c) Absolute error with three iterations: (left), right

Figure 6: Comparison of absolute errors in case 3.

Figure 7: Maximum error with the number of iteration.

For the Holling-type prey-predator model (1.2), the parameters are set by and . For the Holling-type II (), it has been known [18] that the model (1.2) has at least one limit cycle if and no limit cycle if and the critical point , where ,   is globally asymptotically stable if and unstable if . The results obtained by the nonstandard numerical scheme [18], MsDTM and E-MsDTM are demonstrated in Figures 8 and 9. Here, the time step is used in nonstandard scheme, in both MsDTM and E-MsDTM. Only one iteration is employed in both MsDTM, and E-MsDTM. As shown in Figures 8 and 9, all results by nonstandard scheme and E-MsDTM are good agreement. However, the numerical results by MsDTM present a wrong dynamical behavior. For Holling-type III (), it is shown in Figures 10 and 11 that all numerical results by nonstandard scheme and E-MsDTM show the accurate dynamical behavior, but the result by MsDTM does not. Moreover, E-MsDTM gives the approximate solutions with a coarse time step , which shows that E-MsDTM is very effective method.

Figure 8: (Top) and (Bottom); nonstandard scheme (Left, ), MsDTM (Middle, ), E-MsDTM (Right, ); .

Figure 9: (Top) and (Bottom); nonstandard scheme (Left, ), MsDTM (Middle, ), E-MsDTM (Right, ); .

Figure 10: (Top) and (Bottom); nonstandard scheme (Left, ), MsDTM (Middle, ), E-MsDTM (Right, ); .

Figure 11: (Top) and (Bottom); nonstandard scheme (Left, ), MsDTM (Middle, ), E-MsDTM (Right, )  .

5. Conclusion

In this paper, we propose an efficient computational algorithm, namely the enhanced multistage differential transform method for solving several prey-predator systems. The basic idea of the MsDTM is to apply the standard DTM to each sub-domain by updating an initial condition. However, it is shown that the standard initial updating gives a numerical inefficiency for the complex nonlinear systems. In E-MsDTM, an initial condition at each sub-domain is updated by using the integral operator which is equivalent to the system. For all illustrative examples the E-MsDTM yields accurate and stable numerical solutions in comparison with the ones obtained by other methods such as the standard MsDTM and MVIM. Also, the E-MsDTM gives almost the same results at low computational cost compared with ones obtained by RK4 and nonstandard numerical scheme. Thus, it can be concluded that the E-MsDTM is an effective numerical tool for solving differential equations. Here, all computations are performed by using Mathematica 8.0.

Acknowledgments

B. Jang is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (no. 2010–0013297), and Y. Do is supported by WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science, and Technology (no. R32-2009-000-20021-0).

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