Using Predictor-Corrector Methods for Numerical Solution of System of Non-linear Volterra Integral Equations of Second Kind

Borhan Jumaa,Abbas Taqi

doi:10.33899/csmj.2009.163825

Abstract

The aim of this paper is solving system of non-linear Volterra integral equations of the second kind (NSVIEK2) numerically using Predictor-Corrector methods (P-CM). Two multistep methods (Adams-Bashforth, Adams-Moulton). Convergence and stability of the methods are proved and some examples are presented to illustrate the methods. Programs are written in matlab program version 7.0.

Highlights

A predictor-corrector method (P-CM) is the combination of an explicit and implicit technique. (Delves and Mohamed, [3]), (Delves and Walsh, [4]), (Hall and Watt [5]).(Ahmed, [1]) Solved system of non-linear Volterra integral equations of the second kind using computational methods, (Babolian and Biazar, [2]) used Adomian decomposition method to find the solution of a Borhan F
[6]) find approximate solutions for a system of non-linear Volterra integral equations using B-Spline function, (Linz, [8]) Solve Volterra integral equations of the second kind using two method, (Maleknejad and Shahrezaee, [9]) solve a system of Volterra integral equation numerically using Runge-Kutta method, (Waswas, [10]), used modified decomposition method to treatment non-linear integral equations and system of non-linear integral equations analytically, (Laurene, [7]) in this book derive the formula of Runge-Kutta method of order, Adams method (Moulton, Bashforth), and Adams Predictor-Corrector method and use this method to find numerical solution of ordinary and partial differential equation
The Adams Predictor-Corrector method is applied for the first time to find the numerical solution for a (NSVIEK2), which is defined by Jumaa, [6]: x

Summary

Introduction

A predictor-corrector method (P-CM) is the combination of an explicit and implicit technique. (Delves and Mohamed, [3]), (Delves and Walsh, [4]), (Hall and Watt [5]). A predictor-corrector method (P-CM) is the combination of an explicit and implicit technique. The Adams Predictor-Corrector method is applied for the first time to find the numerical solution for a (NSVIEK2), which is defined by Jumaa, [6]: x. K (x, t, F(t)) = (k1 (x, t, F(t), ...,km (x, t, F(t)))T , In this paper, the method is based on the explicit fourth-order Adams. The (P-CM) combines the fourth-order Adams Bashforth method as Predictor and the implicit fourth-order Adams-Moulton method as Corrector: w0 is given; w1, w2 , w3 are found from a Runge-Kutta method. The form of the explicit fourth-order Adams Bashforth equation (10) can be written as: wi, j+1. The form of the implicit fourth-order Adams Moulton equation (11) can be written as:. − 5ki (x j+1, t j−1, f1, j−1, f 2, j−1, ..., f m, j−1 ) + ki (x j+1, t j−2 , f1, j−2 , f 2, j−2 ,..., f m, j−2 )]

Stability

Illustrative Examples

Exact2

Exact1

Exact1 A-MM

Conclusions