Numerical solution of integral equations system of the second kind by Block–Pulse functions

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Abstract

This paper endeavors to formulate Block–Pulse functions to propose solutions for the Fredholm integral equations system. To begin with we describe the characteristic of Block–Pulse functions and will go on to indicate that through this method a system of Fredholm integral equations can be reduced to an algebraic equation. Numerical examples presented to illustrate the accuracy of the method.

Introduction

In recent years, many different basic functions have been used to estimate the solution of integral equations. In this paper we use BPF (Block–Pulse functions) as a simple base for solving a system of integral equations. This set of functions was first introduced to electrical engineers by Harmuth. Then several researchers (Gopalsami and Deekshatulu, 1997 [8]; Chen and Tsay, 1977 [9]; Sannuti, 1977 [10]) discussed the Block–Pulse functions and their operational matrix [1], [2].

Section snippets

Definition of BPF

An m-set of BPF is defined as follows:Φi(t)=1,(i-1)Tmt<iTm;0,otherwisewith t  [0, T) and i=1,2,,m,Tm=h. Fig. 1, Fig. 2, Fig. 3, Fig. 4 illustrate Φ1(t) to Φ4(t).

Now we explain the properties of BPF.

  • (i)

    Disjointness:We clearly haveΦi(t)Φj(t)=Φi(t),i=j;0,ij,t  [0, T),i, j = 1, 2, …, m. This property is obtained from definition of BPF.

  • (ii)

    Orthogonality:We have0TΦi(t)Φj(t)dt=h,i=j;0,ij,t  [0, T), i, j = 1, 2, …, m. This property is obtained from the disjointness property.

  • (iii)

    Completeness:

    For every f  L2,{Φ} is complete if Φ

Linear integral equations system

Consider following integral equations system (see [5], [6], [7]):j=1nyj(x)=fi(x)+λj=1nαβkij(x,t)yj(t)dt,i=1,2,,n.

Our problem is to determine Black–Pulse coefficient of yj(x), j = 1, 2, …, n in the interval x  [α, β) from the known functions fi(x), i = 1, 2, …, n and the kernels kij(x, t) i, j = 1, 2, …, n. Usually we set α = 0 to facilities the use of Black–Pulse functions. In case α  0 we set X=x-αβ-αT where T = mh.

We approximate fi, yj, kij by its BPF as follows:fi(x)FiTΦ(x),yi(x)YjTΦ(x),kij(x,t)ΦT(x)kijΨ(t),

Numerical examples

Example 1

Consider the integral equations systemy1(t)+01(t+s)y1(s)ds+01(t+2s2)y2(s)ds=116t+116,y2(t)+01ts2y1(s)ds+t2sy2(s)ds=54t2+14twith exact solution y1(t) = t, y2(t) = t2 results with m = 16 and m = 32 shown in Table 1.

Example 2

Consider the linear integral equations systemy1(t)+01et-sy1(s)ds01e(t+2)sy2(s)ds=2et+et+s-1t+1,y2(t)+01etsy1(s)dse(t+s)y2(s)ds=et+e-t+et+s-1t+1with exact solution y1(t) = et and y2(t) = et. The computational results for m = 16 and m = 32 together with the exact solution are given in Table 2.

Conclusion

Block–Pulse functions which had previously been used in control theory have been developed in this paper to solve a linear Fredholm integral equations system of the second kind by means of references [6], [7]. As examples indicate, in order to increase the accuracy of the numerical results, it is necessary to increase m. Other orthogonal functions can also be used for solving integral equations system of the second kind.

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