Solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via rationalized Haar functions

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Abstract

Rationalized Haar functions are developed to approximate of the nonlinear Volterra–Fredholm–Hammerstein integral equations. The properties of rationalized Haar functions are first presented, and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.

Introduction

In this paper we present a rationalized Haar functions method for solving nonlinear Volterra–Fredholm–Hammerstein integral equations. Several numerical methods for approximating the solution of Hammerstein integral equations are known. For Fredholm–Hammerstein integral equations, the classical method of successive approximations was introduced in [1]. A variation of the Nystrom method was presented in [2]. A collocation type method was developed in [3]. In [4], Brunner applied a collocation-type method to nonlinear Volterra–Hammerstein integral equations and integro-differential equations, and discussed its connection with the iterated collocation method. Guoqiang [5] introduced and discussed the asymptotic error expansion of a collocation-type method for Volterra–Hammerstein integral equations.

Orthogonal functions, often used to represent an arbitrary time functions, have received considerable 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 systems of algebraic equations, thus greatly simplifying the problem. Orthogonal functions have also been proposed to solve linear integral equations. Special attention has been given to applications of Walsh functions [6], block-pulse functions [7], Laguerre series [8], Legendre polynomials [9], Chebyshev polynomials [10] and Fourier series [11]. The orthogonal set of Haar functions is a group of square waves with magnitude of +2i/2, −2i/2 and 0, i = 0, 1, 2, … [12]. The use of the Haar functions comes from the rapid convergence feature of Haar series in expansion of function compared with that of Walsh series [13]. Lynch and Reis [14] 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 rationalized Haar (RH) transform. The RH transform preserves all the properties of the original Haar transform and can be efficiently implemented using digital pipeline architecture [15]. The corresponding functions are known as RH functions. The RH functions are composed of only three amplitudes +1, −1 and 0. Further, Ohkita and Kobayashi [16], [17] applied RH functions to solve linear ordinary differential equation [16] and linear first and second order partial differential equations [17].

Very few references have been found in technical literature dealing with Volterra–Fredholm integral equations. Yalcinbas [18] applied Taylor series to the following nonlinear Volterra–Fredholm integral equationy(t)=f(t)+λ10tκ1(t,s)[y(s)]pds+λ201κ2(t,s)[y(s)]qds,0t,s1,where p and q are nonnegative integers and λ1 and λ2 are constants. Moreover, f(t), the kernels κ1(t, s) and κ2(t, s) are assumed to have nth derivatives on the interval 0  t, s  1. Also Yousefi and Razzaghi [19] applied Legendre wavelet for solvey(t)=f(t)+λ10tκ1(t,s)F1(y(s))ds+λ201κ2(t,s)F2(y(s))ds.In the present article, we are concerned with the application of rationalized Haar to the numerical solution of a nonlinear Volterra–Fredholm–Hammerstein integral equation of the form:y(t)=f(t)+λ10tκ1(t,s)g1(s,y(s))ds+λ201κ2(t,s)g2(s,y(s))ds,0t,s1,where f(t), the kernels κ1(t, s) and κ2(t, s) are assumed to be in L2(R) on the interval 0  t, s  1. We assume that Eq. (3) has a unique solution y(t) to be determined. The method consists of expanding the solution by rationalized Haar with unknown coefficients. The properties of rationalized Haar together with the Newton–Cotes nodes [20] are then utilized to evaluate the unknown coefficients and find an approximate solution to Eq. (3). In this method time and computations are small.

The article is organized as follows: In Section 2, we describe the basic formulation of rationalized Haar required for our subsequent development. Section 3 is devoted to the solution of Eq. (3) by using rationalized Haar. In Section 4, we report our numerical finding and demonstrate the accuracy of the proposed scheme by considering numerical examples.

Section snippets

Rationalized Haar functions

The RH function RH(r, t), r = 1, 2, 3, … are composed of three values +1, −1 and 0 and can be defined on the interval [0, 1) as [17]:RH(r,t)=1,J1t<J1/2,-1,J1/2t<J0,0,otherwise,whereJu=j-u2i,u=0,12,1.The value of r is defined by two parameters i and j asr=2i+j-1,i=0,1,2,3,j=1,2,3,,2i,RH(0, t) is defined for i = j = 0 and is given byRH(0,t)=1,0t<1.The orthogonality property is given by01RH(r,t)RH(v,t)dt=2-i,r=v,0,rv,wherev=2n+m-1,n=0,1,2,3,m=1,2,3,,2n.

Function approximation

A function f(t) defined over the interval [0, 1)

Nonlinear Volterra–Fredholm–Hammerstein integral equations

Consider Volterra–Fredholm–Hammerstein integral equations given in Eq. (3). To solve for y(t), we first approximate the solution not to Eq. (3), but rather to an equivalent equation:z1(t)=g1(t,y(t)),z2(t)=g2(t,y(t)),0t<1.From Eq. (3) we getz1(t)=g1(t,f(t)+λ10tκ1(t,s)z1(s)ds+λ201κ2(t,s)z2(s)ds),z2(t)=g2(t,f(t)+λ10tκ1(t,s)z1(s)ds+λ201κ2(t,s)z2(s)ds).Suppose z1(t), z2(t) and κ1, κ2 can be expressed approximately asz1(t)=A1TΦ(t),z2(t)=A2TΦ(t),κ1(t,s)=ΦT(t)H1Φ(s),κ2(t,s)=ΦT(t)H2Φ(s),where A1, A2

Illustrative examples

Example 1

Consider the nonlinear Volterra–Fredholm–Hammerstein integral equation given in [18] byy(t)=-130t6+13t4-t2+53t-54+0t(t-s)[y(s)]2ds+01(t+s)[y(s)]ds,0t,s1.

We applied the method presented in this paper and solved Eq. (30) with k = 8 and k = 16. The computational result together with the exact solution y(t) = t2  2 are given in Table 1.

Example 2

Considery(t)=2cost-2+30tsin(t-s)cos2sds+67-6cos101(1-s)(cos2t)(s+y(s))ds,0t<l.

We solved Eq. (31) using the method in Section 3. The computational result for k = 16

Conclusion

In the present work the rationalized Haar functions are used to solve the solution of nonlinear Volterra–Fredholm–Hammerstein integral equations. The matrices Φ^k×k and Φ^k×k-1 introduces in Eqs. (15)and (17) contain many zeros, and these zeros make the Haar transform faster than other square such as Walsh and block-pulse functions, hence making rationalized Haar functions computationally very attractive and time is small. The problem has been reduced to a problem of solving a system of

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