Solving second kind integral equations with hybrid Fourier and block–pulse functions
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 byThese 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:The 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 as
Section snippets
Function approximation
A function x(t)∈L2[0,1) may be expanded aswherein (2.2), (.,.) denotes the inner product.
If the infinite series in (2.1) is truncated, then (2.1) can be written aswhere B(t) and X given byWe can also approximate the function k(t,s)∈L2([0,1)×[0,1)) as follows:
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 that
Numerical examples
Example 1 Consider the integral equation:with exact solution y(t)=cos(2πt), we choose r=1, N=1, thereforeso by solving the linear system of (I−KD)Y=X we obtainthereforeand this solution is exact solution. Example 2 Consider the integral equation:with exact solution y(t)=et, results are shown in Table 1 (with r=2, N=5). Example 3 Consider the integral equation:
References (5)
Introduction functional analysis with applications
(1978)- K. Maleknejad, M.Tavassoli Kajani, Solving second kind integral equations by Galerkin methods with hybrid Legandre and...
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