Numerical solution of integro-differential equations by using rationalized Haar functions method

1 Introduction

In this paper, we present a rationalized Haar (RH) functions method for solving integro-differential equations. Several numerical methods for approximating the solution of integro-differential equations are known. For example, [1] Brunner (1982) applied a collocation-type method to integro-differential equations, and discussed its connection with the iterated collocation method.

Orthogonal functions, often used to represent an arbitrary time functions, have received considerable attention in dealing with various problems of dynamic systems. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations thus greatly simplifying the problem. The approach is based on converting the underlying differential equations into an integral equations through integration, approximating a various signal involved in the equation by truncated orthogonal series [straight phi] (t )=[[straight phi]₀ ,[straight phi]₁ , ... ,[straight phi]_{k -1} ]^T and using the operational matrix of integration P , to eliminate the integral operations. The elements [straight phi]₀ ,[straight phi]₁ , ... ,[straight phi]_{k -1} are the basic functions, orthogonal on certain interval [a ,b ], and the matrix P can be uniquely determined based on particular orthogonal functions.

The orthogonal set of Haar functions is a group of square waves with magnitude of 2^1/2 , -2^1/2 , and 0, i =0, 1, 2,... See [6] Razzaghi and Nazarzadeh (1999). Just these zeros make the Haar transform faster than other square functions such as Walsh function. [3] Lynch and Reis (1976) have rationalized the Haar transform by deleting the irrational numbers and introducing the integral powers of two. This modification results in what is called the RH transform. The RH transform preserves all the properties of the original Haar transform and can be efficiently implemented using digital pipeline architecture. See [7] Reis et al. (1976). The corresponding functions are known as RH functions. The RH functions are composed of only three amplitudes 1, -1 and 0. Further [4], [5] Ohkita and Kobayashi (1986, 1988) applied RH functions to solve linear ordinary differential equations and linear first and second order partial differential equations.

In the present paper, we apply RH functions to solve integro-differential equations. The method consists of reducing...