Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

S H Behiry

doi:10.1155/2013/416757

Abstract

A numerical method for solving nonlinear Fredholm integrodifferential equations is proposed. The method is based on hybrid functions approximate. The properties of hybrid of block pulse functions and orthonormal Bernstein polynomials are presented and utilized to reduce the problem to the solution of nonlinear algebraic equations. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.

Highlights

Integrodifferential equations are often involved in mathematical formulation of physical phenomena
Numerous works have been focusing on the development of more advanced and efficient methods for solving integrodifferential equations such as wavelets method [4, 5], Walsh functions method [6], sinc-collocation method [7], homotopy analysis method [8], differential transform method [9], the hybrid Legendre polynomials and blockpulse functions [10], Chebyshev polynomials method [11], and Bernoulli matrix method [12]
The purpose of this work is to utilize the hybrid functions consisting of combination of block-pulse functions with normalized Bernstein polynomials for obtaining numerical solution of nonlinear Fredholm integrodifferential equation: s

Summary

Introduction

Integrodifferential equations are often involved in mathematical formulation of physical phenomena. Block-pulse functions have been studied and applied extensively as a basic set of functions for signals and functions approximations All these studies and applications show that block-pulse functions have definite advantages for solving problems involving integrals and derivatives due to their clearness in expressions and their simplicity in formulations; see [13]. The purpose of this work is to utilize the hybrid functions consisting of combination of block-pulse functions with normalized Bernstein polynomials for obtaining numerical solution of nonlinear Fredholm integrodifferential equation: s. We present Bernstein polynomials and hybrid of block-pulse functions Their useful properties such as functions approximation, convergence analysis, operational matrix of product, and operational matrix of differentiation are given.

Properties of Hybrid Functions

Outline of the Solution Method

Applications and Numerical Results

Conclusion