A reliable iterative method for Cauchy problems

Abstract

In the present paper, 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] is used to solve the Cauchy problems. In this iterative method the solution is obtained in the series form that converge to the exact solution with easily computed components. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficulty arising in calculating calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier in Variational Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations in Homotopy Perturbation Method (HPM) and can be easily comprehended with only a basic knowledge of Calculus. The results show that the present method is very effective, simple and provide the analytic solutions. The software used for the calculations in this study was MATHEMATICA r 8.0.

Keywords: New iterative method; Cauchy problems; Inviscid Burger’s equation; transport equation