Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

S. Mashayekhi,M. Razzaghi,O. Tripak

doi:10.1155/2014/413623

Abstract

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Highlights

There is considerable literature that discusses approximating the solution of linear and nonlinear Hammerstein integral equations [1,2,3,4,5,6]
In the present paper we introduce a new direct computational method to solve nonlinear mixed Volterra-Fredholm integral equations in (1)
The functions z1(t) and z2(t) are approximated by hybrid functions with unknown coefficients. These hybrid functions, which consist of block-pulse functions and Bernoulli polynomials together with their operational matrices of integration and product, are given

Summary

Introduction

There is considerable literature that discusses approximating the solution of linear and nonlinear Hammerstein integral equations [1,2,3,4,5,6]. Reihani and Abadi [7] and Hsiao [8] applied rationalized Haar functions and hybrid of block-pulse functions and Legendre polynomials, respectively, for solving Fredholm and Volterra integral equations of the second kind. In the present paper we introduce a new direct computational method to solve nonlinear mixed Volterra-Fredholm integral equations in (1). The functions z1(t) and z2(t) are approximated by hybrid functions with unknown coefficients These hybrid functions, which consist of block-pulse functions and Bernoulli polynomials together with their operational matrices of integration and product, are given. These matrices are used to evaluate the coefficients of the hybrid functions for solution of nonlinear mixed Volterra-Fredholm integral equations.

Hybrid Functions

Hybrid of Block-Pulse Functions and Bernoulli Polynomials

The Numerical Method

Illustrative Example

Methods

Conclusion