Approximate solution of a model describing biological species living together using a new iterative method


  • Majeed AL-Jawary Head of Department of Mathematics, College of Education for Pure Sciences / Ibn-AL-Haithem, Baghdad University, Baghdad, Iraq





In the present article, we implement the new iterative method proposed by Daftardar-Gejji and Jafari (NIM or DJM) [V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] to solve a system of two nonlinear integro-differential equations, which describes biological species living together. The results demonstrate that the method has many merits such as being derivative-free, can be easily comprehended with only a basic knowledge of Calculus. Also, the DJM overcoming 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) that implemented to solve the problem. Numerical examples are presented for several problems, to demonstrate the efficiency of the proposed method. A comparison with some existing techniques such as ADM, HPM and VIM also presented, which shows that the DJM is effective and convenient to use and overcomes the difficulties arising in existing techniques. The software used for the calculations in this study was MATHEMATICA®8.0.

Keywords: Biological species living together, Integro-differential equation, Mathematical biology, New iterative method.


A.M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education, Springer, Beijing, Berlin, 2011.

E. Babolian, J. Biazar, Solving the problem of biological species living together by Adomian decomposition method, Applied Mathematics and Computation. 129 (2002) 339-343.

F. Shakeri, M. Dehghan, Solution of a model describing biological species living together using the variational iteration method, Mathematical and Computer Modelling. 48 (2008) 685-699.

P. Roul, P. Meyer, Numerical solutions of systems of nonlinear integro-differential equations by Homotopy-perturbation method, Applied Mathematical Modelling. 35 (2011) 4234-4242.

V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, Journal of Mathematical Analysis and Applications. 316 (2006) 753-763.

S. Bhalekar, V. Daftardar-Gejji. New iterative method: application to partial differntial equations. Applied Mathematics and Computation. 203 (2008) 778-783.

V. Daftardar-Gejji, S. Bhalekar, Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method. Computers & Mathematics with Applications. 59 (2010)1801-1809.

S. Bhalekar, V. Daftardar-Gejji, Solving evolution equations using a new iterative method. Numerical Methods for Partial Differential Equations. 26 (2010) 906-916.

M. Yaseen, M. Samraiz, S. Naheed, Exact solutions of Laplace equation by DJ method, Results in Physics, 3 (2013) 38-40.

V. Daftardar-Gejji, H. Jafari, Convergence of the New Iterative Method. International Journal of Differential Equations, doi:10.1155/2011/989065.

J. Abdul, Introduction to Integral Equations with Applications, Wiley, New York, 1999.

View Full Article: