The Combined RKM and ADM for Solving Nonlinear Weakly Singular Volterra Integrodifferential Equations

Sixing Shi,Xueqin Lv

doi:10.1155/2012/258067

Abstract

The reproducing kernel method (RKM) and the Adomian decomposition method (ADM) are applied to solventh-order nonlinear weakly singular Volterra integrodifferential equations. The numerical solutions of this class of equations have been a difficult topic to analyze. The aim of this paper is to use Taylor’s approximation and then transform the givennth-order nonlinear Volterra integrodifferential equation into an ordinary nonlinear differential equation. Using the RKM and ADM to solve ordinary nonlinear differential equation is an accurate and efficient method. Some examples indicate that this method is an efficient method to solventh-order nonlinear Volterra integro-differential equations.

Highlights

In this paper, we consider the following nth-order nonlinear weakly singular Volterra integrodifferential equation of the following form 1–4 : n x ai xuixfxK t, x um t dt, a x b, i0 a where m > 1, f x is a given function, u x is the unknown function, and K t, x is the kernel of the integro equation
We have by setting yt Fut, 2.2 n ai xuixfx i0 x yt a x − t α dt, a x b, 0 < α < 1
We introduce how to determine the inverse operator L−1 of L

Summary

Introduction

We consider the following nth-order nonlinear weakly singular Volterra integrodifferential equation of the following form 1–4 :. To our knowledge there is still no viable analytic approach for solving weakly singular Volterra integro-differential equations. The present work is motivated by the desire to obtain approximate solution to nth-order nonlinear weakly singular Volterra integro-differential equation, where the integrand is weakly singular in the sense that its integral is continuous at the singular point, that is, its kernel K t, x 1/ x − t α is singular as t → x. The RKM has been applied successfully to solving linear and nonlinear problems 18–20.

Taylor’s Approximation

Analysis of Reproducing Kernel Hilbert Space

Representation of the Inverse Operator

Numerical Examples

Conclusion