Solving second kind integral equations with hybrid Fourier and block–pulse functions

https://doi.org/10.1016/j.amc.2003.11.038Get rights and content

Abstract

In this paper, we use a combination of Fourier and block–pulse functions on the interval [0,1], to solve the linear integral equation of the second kind. We convert the integral equation, to a system of linear equations. We show that our estimates have a good degree of accuracy.

Introduction

In recent years, many different basic functions have used to estimate the solution of system, such as orthonormal bases and wavelets [3]. In this paper we are going to use a simple base, that is a combination of block–pulse functions on [0,1], and Fourier functions, which called the hybrid Fourier block–pulse functions [2], [5].

Definition 1.1

Fourier functions on the interval [0,2π]: are defined byφ0(t)=1,φm(t)=cosmt,m=1,2,3,…φm(t)=sinmt,m=1,2,3,…These functions are orthogonal in Hilbert space L2[0,2π] [1].

Definition 1.2

A set of block–pulse functions bi(λ), i=1,2,…,m on the interval [0,1) are defined as follows:bi(λ)=1i−1m⩽λ<im0otherwiseThe block–pulse functions on [0,1) are disjoint, that is, for i=1,2,…,m, j=1,2,…,m we have: bi(t)bj(t)=δijbi(t), also these functions have the property of orthogonality on [0,1).

Definition 1.3

For m=0,1,2,…,2r and n=1,2,…,N the hybrid Fourier block–pulse functions are defined asb(n,m,t)=1m=0φm(2πNt)m=1,2,…,rφm−r(2πNt)m=r+1,r+2,…,2rn−1N⩽t<nN0otherwise.

Section snippets

Function approximation

A function x(t)∈L2[0,1) may be expanded asx(t)=∑n=1m=0X(n,m)b(n,m,t),whereX(n,m)=(x(t),b(n,m,t))(b(n,m,t),b(n,m,t))in (2.2), (.,.) denotes the inner product.

If the infinite series in (2.1) is truncated, then (2.1) can be written asx(t)≃xNr(t)=∑n=1Nm=02rX(n,m)b(n,m,t)=XTB(t),where B(t) and X given byB(t)=[b(1,0,t),b(1,1,t),…,b(1,2r,t),b(2,0,t),…,b(N,2r,t)]T,X=[X(1,0),X(1,1),…,X(1,2r),X(2,0),…,X(N,2r)]T.We can also approximate the function k(t,s)∈L2([0,1)×[0,1)) as follows:k(t,s)≃kNr(t,s)=BT

Quadrature formulae

We often want to calculate the inner products of functions and hybrid Fourier and block–pulse functions when we use Galerkin methods for linear integral equation. Sweldens and Piessens [4] obtained a quadrature formulae for wavelet. We give a method of construction of quadrature formulae for the calculation of inner products of smooth functions and hybrid Fourier and block–pulse functions. The idea of quadrature formulae is to find weights ω(n,m)i and abscissae t(n,m)i such that01f(t)b(n,m,t)d

Numerical examples

Example 1

Consider the integral equation:y(t)+∫01sin(4πt+2πs)y(s)ds=cos(2πt)+12sin(4πt)with exact solution y(t)=cos(2πt), we choose r=1, N=1, thereforeX=010K=000000000D=10001200012so by solving the linear system of (IKD)Y=X we obtainY=010thereforey11(t)=b(1,1,t)=cos(2πt)and this solution is exact solution.

Example 2

Consider the integral equation:y(t)=∫01t2es(t−1)y(s)ds+(1−t)et+twith exact solution y(t)=et, results are shown in Table 1 (with r=2, N=5).

Example 3

Consider the integral equation:y(t)=∫014ln(2+t−s)y(s)ds+3t+2ln

References (5)

  • E. Kreyzing

    Introduction functional analysis with applications

    (1978)
  • K. Maleknejad, M.Tavassoli Kajani, Solving second kind integral equations by Galerkin methods with hybrid Legandre and...
There are more references available in the full text version of this article.

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