On the Convergence Problem of One-Dimensional Hypersingular Integral Equations

Suzan J Obaiys

doi:10.1155/2013/974751

Abstract

We develop the expansion method of singular integral equation (SIE) for hypersingular integral equation (HSIE). Relating the hypersingular integrals to Cauchy principal-value integrals, we interpolate the kernel and the density functions to the truncated Chebyshev series of the second kind. The corresponding convergence results for the functions <svg style="vertical-align:-3.56265pt;width:103.45px;" id="M1" height="20.549999" version="1.1" viewBox="0 0 103.45 20.549999" width="103.45" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,16.05)"><path id="x1D453" d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
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Highlights

The integral equation is defined as an equation with an unknown function that appears under the integral sign
We develop the expansion method of singular integral equation (SIE) for hypersingular integral equation (HSIE)
A wealth of the literature on applications related to the numerical evaluation of hypersingular integral equations HSIEs could be found in [5,6,7,8,9,10]

Summary

Introduction

The integral equation is defined as an equation with an unknown function that appears under the integral sign. These equations could be classified by the order of singularity [1]. A wealth of the literature on applications related to the numerical evaluation of hypersingular integral equations HSIEs could be found in [5,6,7,8,9,10]. This paper focuses on onedimensional singular integral equations (SIEs) found in various mixed boundary value problems of mathematical physics and engineering such as isotropic elastic bodies involving cracks, aerodynamics, hydrodynamics, elasticity, and other related areas.

Description of the General Method

Convergence Rates

Numerical Example

Conclusion