Accelerating the Modified Picard Iteration By Using Green's Function Approach

Abstract

We consider the Picard's iteration method as a technique for solving initial value problems of the first and second order linear differential equations. The basic idea is the use of Green's function to collect some of the terms in a perfect differential term, and then use the decomposition techniques. For the second order differential equations, we transform the equation to a system of two first order equations and in addition we use the Gauss Seidel technique. The algorithm of the proposed method is discussed. Comparisons with the classical Picard method and modified Picard have illustrated the rapid convergence of the proposed method. Numerical examples have illustrated that the technique obtains the theoretical fixed point quicker than that obtained with other techniques including the modified Picard.