A reliable method for obtaining approximate solutions of linear and nonlinear Volterra‐Fredholm integro‐differential equations

Abstract

Purpose -Based on the original methods of Adomian a decomposition method has been developed to find the analytic approximation of the linear and nonlinear Volterra-Fredholm (V-F) integro-differential equations under the initial or boundary conditions. Design/methodology/approach -Designed around the methods of Adomian and later researchers. The methodology to obtain numerical solutions of the V-F integro-differential equations is one whose essential features is its rapid convergence and high degree of accuracy which it approximates. This is achieved in only a few terms of its iterative scheme which is devised to avoid linearization, perturbation and any transformation in order to find solutions to given problems. Findings -The scheme was shown to have many advantages over the traditional methods. In particular it provided discretization and provided an efficient numerical solution with high accuracy, minimal calculations as well as an avoidance of physical unrealistic assumptions. Research limitations/implications -A reliable method for obtaining approximate solutions of linear and nonlinear V-F integro-differential using the decomposition method which avoids the tedious work needed by traditional techniques has been developed. Exact solutions were easily obtained. Practical implications -The new method had most of its symbolic and numerical computations performed using the Computer Algebra Systems-Mathematica. Numerical results from selected examples were presented. Originality/value -A new effective and accurate methodology has been developed and demonstrated.

A numerical-analytic method for solving the Cauchy problem for ordinary differential equations

“…In Hashim, Noorani, Ahmad, Bakar, Ismail and Zakaria [2006] authors apply ADM (in a rather modified form) to the Lorenz system. The paper Inc and Cherruault [2005] is devoted to the application of ADM to the solution of the nonlinear Volterra-Fredholm integro-differential equation.…”

“…In [9] authors apply ADM (in a rather modified form) to the Lorenz system. The convergence of ADM applying to the nonlinear Volterra-Fredholm integro-differential equation was studied in [10].…”