Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2009, Article ID 710250, 13 pages
http://dx.doi.org/10.1155/2009/710250
Research Article

On a New Reliable Algorithm

1Department of Science, Faculty of Nursing and Sciences, Jerash Private University, 26150 Jerash, Jordan
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

Received 8 August 2009; Revised 30 November 2009; Accepted 22 December 2009

Academic Editor: Dumitru Baleanu

Copyright © 2009 A. K. Alomari 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

An analytical expression for the solution of the prey-predator problem by an adaptation of the homotopy analysis method (HAM) is presented. The HAM is treated as an algorithm for approximating the solution of the problem in a sequence of time intervals; that is, HAM is converted into a hybrid numeric-analytic method called the multistage HAM (MSHAM). Comparisons between the MSHAM solutions and the fourth-order Runge-Kutta (RK4) numerical solutions reveal that the new technique is a promising tool for the nonlinear systems of ordinary differential equations.

1. Introduction

Most modelling of biological problems is characterized by systems of ordinary differential equations (ODEs). For example, the relationship of increasing and decreasing in the population of two kind of animals (such as rabbits and foxes) can be described by the so-called mathematical model of the prey-predator problem which is given by a system of nonlinear equations:

where and are, respectively, the populations of rabbits and foxes at the time and , and are known coefficients (for more details, see [1]). Problems of this nature have been solved using classical numerical techniques such as Runge-Kutta and are now published in most textbooks on differential equations.

Authors in [2, 3] used the Adomian decomposition method (ADM) to handle the systems of prey-predator problem. Yusufoglu and Erbas [4] and Rafei et al. [5] employed the variational iteration method (VIM) to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. Biazar et al. [6] used the power series method (PSM) to handle the systems. All the solutions above are in the form of convergent power series with polynomial base function.

In recent years, a great deal of attention has been devoted to study HAM, which was first envisioned by Liao in his Ph.D. thesis [7] for solving a wide range of problems whose mathematical models yield differential equation or system of differential equations [812]. HAM has successfully been applied to many situations. For example, Hayat and Sajid [13], Hayat and Khan [14], and Hayat et al. [15, 16] used HAM to solve many kinds of modelling in fluid problems. Xu and Liao [17] applied HAM to give the dual solutions of boundary layer flow over an upstream moving plate. Abbasbandy [18] used HAM to present solitary wave solutions to the Kuramoto-Sivashinsky equation. Sami Bataineh et al. [19] modified the HAM (MHAM) to show that Taylor series converge to the exact solution by expanding the coefficients variables using Taylor series. Alomari et al. [20] introduced a new reliable algorithm based on an adaptation of the standard HAM to solve Chen system. In recent years, this method has been successfully employed to solve many types of problems in science and engineering [2128].

In this paper, we are interested to find the approximate analytic solution of the system of coupled nonlinear ODEs (1.1) by treating the HAM as an algorithm for approximating the solution of the problem in a sequence of time intervals. We shall call this technique as the multistage homotopy analysis method (MSHAM). The freedom of choosing the linear operator and the auxiliary parameter is still present in this modification. Different from the series solution in [25], the solution we present here uses a combination of base functions, namely, the polynomial and exponential functions, and it is effective for longer time. Comparison with the classical fourth-order Runge-Kutta (RK4) shall be made.

2. Solution Procedure

Consider the following general system of first-order ordinary differential equations (ODEs):

where are (linear or nonlinear) real-valued functions, is the initial condition, and .

2.1. Solution by HAM

In HAM [8], system (2.1) is first written in the form

where are nonlinear operators, denotes the independent variable, and are the unknown functions. By means of generalizing the traditional homotopy method, Liao [8] constructs the so-called zeroth-order deformation equation:

where is an embedding parameter, are the nonzero auxiliary parameters, is an auxiliary linear operator, are the initial guesses of and are the unknown functions. It is important to note that one has great freedom to choose auxiliary objects such as and in HAM. We note that, in the framework of HAM, the solution can be represented by many different base functions such as the polynomial functions, exponential functions and rational functions, and so forth. Obviously, when and , both and hold. Thus, as increases from to , the solution varies from the initial guess to the solution . Expanding in Taylor series with respect to , one has where

If the auxiliary linear operators, the initial guesses, the auxiliary parameters , and the auxiliary function are so properly chosen, then the series (2.4) converges at and

which must be one of the solutions of the original nonlinear equations, as proved by Liao [8].

Define the vector

Differentiating (2.3) times with respect to the embedding parameter and then setting and finally dividing them by , we have the so-called mth-order deformation equation:

where

It should be emphasized that is governed by the linear (2.8) with the linear boundary conditions that come from the original problem, which can be easily solved by symbolic computation softwares such as Maple and Mathematica. Also,

2.2. Solution by MSHAM

The approximate solutions (2.10) are generally, as shall be shown in the numerical experiments of this paper, not valid for large . A simple way of ensuring validity of the approximations for large is to treat (2.3) and (2.8) as an algorithm for approximating the solutions of (2.1) in a sequence of intervals: the solution from will be derived by subdividing this interval into and applying the HAM solution on each subinterval. The initial approximation in each interval is taken from the solution on the previous interval:

where is the left-end point of each subinterval.

Now we solve (2.8) for unknowns . In order to carry out the iteration in every subinterval of equal length , , , ,, we need to know the values of the following:

But, in general, we do not have these information at our disposal except at the initial point . A simple way for obtaining the necessary values could be by means of the previous -term approximations of the preceding subinterval given by (2.10), that is,

3. Application

In this part, we apply the MSHAM for the prey-predator problem subject to the initial conditions

We note that when we have the initial condition of (1.1). Let the set of the base functions be

So the solutions are

where and are the coefficients. It is straightforward to choose

as our initial approximations of and , and the linear operator should be

with the property

where is the integration constant, which will be determined by the initial condition.

If and indicate the embedding and nonzero auxiliary parameters, respectively, then the zeroth-order deformation problems are of the following form:

subject to the initial conditions

in which we define the nonlinear operators and as

For and , the above zeroth-order equations (3.7) have the solutions

When increases from to , then and vary from and to and . Expanding and in Taylor series with respect to , we have

in which

where is chosen in such a way that these series are convergent at . Therefore, we have through (3.11) that

Define the vectors

Differentiating the zeroth-order equations (3.7) times with respect to , then setting , and finally dividing by , we have the mth-order deformation equations:

with the following boundary conditions:

for all , where

This way, it is easy to solve the linear nonhomogeneous Equations (3.16) at general initial conditions by using Maple, one after the other in the order Thus we successfully have By the same way we can get the first fourth term to be as analytical approximate solution as and terms. Now we divide the interval to subintervals by time step ; Then we start from the initial conditions and we get the solution on the interval . Further, we take and and , so we get the solution on the new interval , and so on. Therefore, by choosing this initial approximation on the starting of each interval, the solution on the whole interval should be continuous. It is worth mentioning that if we take and we fixed and , then the solution will be the standard HAM solution which is not effective at large value of .

4. Analysis of Results

In this section, we compare the fourth term of the MSHAM solution using step size with RK4 using step size for the following cases.

Case 1. , , , , and the initial conditions and .

Case 2. , , , , and the initial conditions and .

Case 3. , , , , and the initial conditions and .

Case 4. , , , , and the initial conditions and .

Case 5. , , , , and the initial conditions and .

Figure 1 presents the -curves for 5th-order of approximations of the prey-predator problem at different cases. It is clear that is in the convergent region which is parallel to the -axis. Figure 2 shows the MSHAM solutions in comparison with the RK4 and HAM solutions for the time span using Digits in Maple software. This 4th-term HAM solution is not accurate enough when compared with the RK4 solution but the 4th-term MSHAM solution works very well. The results obtained in [25] are effective for but the result in MSHAM is effective for , that means the MSHAM is more effective than the methods in [25]. Moreover, the absolute errors between the new algorithm and RK4 using the same benchmark for Cases 2 and 5 are presented in Tables 1 and 2, respectively. These show high accuracy of the new method since the absolute error is up to in Case 2 and in Case 5, which is not possible to get if the linear operator is .

tab1
Table 1: The absolute error from the MSHAM and RK4 for Case 2 of the prey-predator problem.
tab2
Table 2: The absolute error from the MSHAM and RK4 for Case 5 of the prey-predator problem.
fig1
Figure 1: The -curves for 5th-order of approximations. (a) and (b) The curves of for Cases 1 and 5. (c) and (d) The curves of for Cases 1 and 5.
fig2
Figure 2: Comparisons between the populations of rabbits and foxes at time for (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, and (e) Case 5, using the 4th-term classical HAM, 4th-term MSHAM (), and RK4 ().

5. Conclusions

In this paper, a prey-predator problem was simulated accurately by MSHAM. MSHAM has the advantages of giving an analytical form of the solution within each time interval which is not possible in purely numerical techniques like RK4, which provides solution only at the two ends of a given time interval, and provided that the interval is chosen small enough for convergence. The method also gives the freedom to choose the auxiliary linear operator and the auxiliary parameter . The present technique offers an explicit time marching algorithm that works accurately over such a bigger time span. The results presented in this paper suggest that MSHAM is also readily applicable to more complex cases of nonlinear ordinary differential equations.

Acknowledgments

The authors would like to acknowledge the financial supports received from Universiti Kebangsaan Malaysia under the Science Fund UKM-ST-06-FRGS0008-2008, and UKM-GUP-BTT-07-25-174. The authors would also like to thank the reviewers for their valuable comments and suggestions.

References

  1. G. F. Simmons, Differential Equations with Applications and Historical Notes, International Series in Pure and Applied Mathematic, McGraw-Hill, New York, NY, USA, 1972. View at MathSciNet
  2. J. Biazar and R. Montazeri, “A computational method for solution of the prey and predator problem,” Applied Mathematics and Computation, vol. 163, no. 2, pp. 841–847, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. S. H. Chowdhury, I. Hashim, and S. Mawa, “Solution of prey-predator problem by numeric-analytic technique,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1008–1012, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  4. E. Yusufoğlu and B. Erbaş, “He's variational iteration method applied to the solution of the prey and predator problem with variable coefficients,” Physics Letters A, vol. 372, no. 21, pp. 3829–3835, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  5. M. Rafei, H. Daniali, and D. D. Ganji, “Variational interation method for solving the epidemic model and the prey and predator problem,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1701–1709, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  6. J. Biazar, M. Ilie, and A. Khoshkenar, “A new approach to the solution of the prey and predator problem and comparison of the results with the Adomian method,” Applied Mathematics and Computation, vol. 171, no. 1, pp. 486–491, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. J. Liao, Homotopy analysis method and its application, Ph.D. dissertation, Shanghai Jiao Tong University, Shanghai, China, 1992.
  8. S. J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. View at MathSciNet
  9. S. J. Liao, “Comparison between the homotopy analysis method and homotopy perturbation method,” Applied Mathematics and Computation, vol. 169, no. 2, pp. 1186–1194, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. J. Liao, “An explicit, totally analytic approximate solution for Blasius' viscous flow problems,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 759–778, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  11. S. J. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. J. Liao, “Series solutions of unsteady boundary-layer flows over a stretching flat plate,” Studies in Applied Mathematics, vol. 117, no. 3, pp. 239–263, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. T. Hayat and M. Sajid, “On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder,” Physics Letters A, vol. 361, no. 4-5, pp. 316–322, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. T. Hayat and M. Khan, “Homotopy solutions for a generalized second-grade fluid past a porous plate,” Nonlinear Dynamics, vol. 42, no. 4, pp. 395–405, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. T. Hayat, Z. Abbas, and M. Sajid, “Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction,” Physics Letters A, vol. 372, no. 14, pp. 2400–2408, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. T. Hayat, M. Sajid, and M. Ayub, “On explicit analytic solution for MHD pipe flow of a fourth grade fluid,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 4, pp. 745–751, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  17. H. Xu and S. Liao, “Dual solutions of boundary layer flow over an upstream moving plate,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 2, pp. 350–358, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Abbasbandy, “Solitary wave solutions to the Kuramoto-Sivashinsky equation by means of the homotopy analysis method,” Nonlinear Dynamics, vol. 52, no. 1-2, pp. 35–40, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. Sami Bataineh, M. S. M. Noorani, and I. Hashim, “On a new reliable modification of homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 409–423, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  20. A. K. Alomari, M. S. M. Noorani, and R. Nazar, “Adaptation of homotopy analysis method for the numeric-analytic solution of Chen system,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2336–2346, 2009. View at Publisher · View at Google Scholar
  21. S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 360, no. 1, pp. 109–113, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Y. Tan and S. Abbasbandy, “Homotopy analysis method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 3, pp. 539–546, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. A. K. Alomari, M. S. M. Noorani, and R. Nazar, “Solutions of heat-like and wave-like equations with variable coefficients by means of the homotopy analysis method,” Chinese Physics Letters, vol. 25, no. 2, pp. 589–592, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. A. K. Alomari, M. S. M. Noorani, and R. Nazar, “Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1196–1207, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  25. S. Abbasbandy, “The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation,” Physics Letters A, vol. 361, no. 6, pp. 478–483, 2007. View at Publisher · View at Google Scholar · View at Scopus
  26. A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Modified homotopy analysis method for solving systems of second-order BVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 430–442, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  27. A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “The homotopy analysis method for Cauchy reaction-diffusion problems,” Physics Letters A, vol. 372, no. 5, pp. 613–618, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  28. F. M. Allan, “Construction of analytic solution to chaotic dynamical systems using the homotopy analysis method,” Chaos, Solitons and Fractals, vol. 39, no. 4, pp. 1744–1752, 2009. View at Publisher · View at Google Scholar · View at Scopus