Numerical Solution of Nonlinear Volterra Integral Equations System Using Simpson’s 3/8 Rule

Adem Kılıçman,L Kargaran Dehkordi,M Tavassoli Kajani

doi:10.1155/2012/603463

Adem Kılıçman, L Kargaran Dehkordi + Show 1 more

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https://doi.org/10.1155/2012/603463

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Abstract

The Simpson’s 3/8 rule is used to solve the nonlinear Volterra integral equations system. Using this rule the system is converted to a nonlinear block system and then by solving this nonlinear system we find approximate solution of nonlinear Volterra integral equations system. One of the advantages of the proposed method is its simplicity in application. Further, we investigate the convergence of the proposed method and it is shown that its convergence is of order O(h4). Numerical examples are given to show abilities of the proposed method for solving linear as well as nonlinear systems. Our results show that the proposed method is simple and effective.

Highlights

IntroductionWe consider the system of second kind Volterra integral equations VIE given by fx gx x
We consider the system of second kind Volterra integral equations VIE given by fx gx x K x, s, fs ds,≤ s ≤ x ≤ X, whereT gx g1 x, g2 x, . . . , gn x, fx f1 x, f2 x, . . . , fn x, ⎤ ⎡k1,1 x, s, f1, . . . , fn · · · k1,n x, s, f1, . . . , fn ⎢ ⎥ K x, s, fs ⎣ ⎦
The Simpson’s 3/8 rule is used to solve the nonlinear Volterra integral equations system. Using this rule the system is converted to a nonlinear block system and by solving this nonlinear system we find approximate solution of nonlinear Volterra integral equations system

Summary

Introduction

We consider the system of second kind Volterra integral equations VIE given by fx gx x. Fn Numerical solution of Volterra integral equations system has been considered by many authors. HPM 8 was proposed by He in 1999 for the first time and recently Yusufoğlu has proposed this method 9 for solving a system of Fredholm-Volterra type integral equations. More details on the modeling of complexity we refer to 11, direct operational method to solve a system of linear in-homogenous couple fractional differential equations, see 12 and for a class of fractional oscillatory system, see 13. We consider block by block method by using Simpson’s 3/8 rule for solving linear and nonlinear systems of Volterra integral equations.

Starting Method

The General Scheme

Convergence Analysis

Numerical Results

Conclusion