A reliable iterative method for solving the epidemic model and the prey and predator problems

Abstract

In the present article, we implement the new iterative method proposed by Daftardar-Gejji and Jafari (NIM) [V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] to solve two problems; the first one is the problem of spread of a non-fatal disease in a population which is assumed to have constant size over the period of the epidemic, and the other one is the problem of the prey and predator. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM), does not require to calculate Lagrange multiplier as in Variational Iteration Method (VIM) and no needs to construct a homotopy as in Homotopy Perturbation Method (HPM). The results obtained are compared with the results by existing methods and prove that the presented method is very effective, simple and does not require any restrictive assumptions for nonlinear terms. The software used for the numerical calculations in this study was MATHEMATICA 8.0.

SOR-Like New Iterative Method for Solving the Epidemic Model and the Prey and Predator Problem

“…A comparison between the results obtained by the SORlike new iterative method and the NIM, ADM, and VIM in [2,3,7] shows that the results of four-term approximations of the of both SOR-like and the NIM and VIM is the same as the results of six-term approximations of the ADM. In fact, the results obtained by the SOR-Like new iterative method are completely the same with the VIM and NIM, but it appears the results obtained by ADM in [2] are not accurate enough.…”

“…Continuing in this manner, we can find the rest of the components which are calculated but not listed. Rafei et al [3], and the results obtained by AL-Jawary [7] using the NIM, the following values, for parameters, are considered: A comparison between the results obtained by the SOR-like method and the NIM, ADM, and VIM in cases 1-2 is completely the same for ω � 0 and will not be plotted (see [2,3,7]).…”

“…To solve this problem, Biazar [2,5] used the Adomian decomposition method (ADM), Rafei et al [3][4][5][6] used the variational iterative method (VIM) and the homotopy perturbation method (HPM), and AL-Jawary [7] used the new iterative method (NIM).…”

“…Foxes eat the rabbits and rabbits eat clover, and there is an increase and decrease in the number of foxes and rabbits. For more details on the mathematical modeling leading to the following system of nonlinear equations governing the problem, we refer the readers to [6,8]: where x(t) and y(t) are, respectively, the populations of the rabbits and the foxes at time t. To solve the prey and predator problem, Biazar et al [2,5] and Chowdhury et al [9] used the Adomian decomposition and the power series methods, Rafei et al [3] used the variational iterative method (VIM), and AL-Jawary [7] used the new iterative method (NIM). Recently, many attempts have been made to develop analytic and approximate methods to solve the epidemic model.…”