Numerical Solution for Cauchy and Hypersingular Integral Equations by Using Legendre Polynomials

Abstract

Singular integral equations have various applications in several basic fields of engineering mechanics newlinelike elasticity, plasticity, and aerodynamics etc. Many crack problems occurring in the field of fracture newlinemechanics such as thermoelastic stress problems around an arbitrary number of arbitrarily-located planar newlinecracks are reducible into singular integral equations or their system. In this thesis, we consider the numerical newlinesolution of two kind of singular integral equations. Firstly, the singular integral equations of first kind with newlineCauchy kernel and the system of these equations. Secondly, the hypersingular integral equations of first newlinekind and their system. newlineSingular integral equations (SIEs) with Cauchy kernel play a vital role in studying many problems of newlineaerodynamics, fracture mechanics, neutron transport, wave propagation etc. System of Cauchy singular integral newlineequations also have great importance as various problems occur in the field of aerodynamics, queuing newlinesystem analysis, electrocardiology, elasticity theory etc., can be formulated as system of Cauchy singular newlineintegral equations. The analytic solution of such equations as well as for their system, are known when newlinethese equations are dominant equations. But these analytic solutions are of limited use as it is a nontrivial newlinetask to use it practically due to the presence of singularity in the known solutions itself. Therefore, there is newlinea necessity to find their approximate solutions. newlineAnalogous to Cauchy singular integral equations, the hypersingular integral equations as well as their newlinesystem are equally important. Several problems occurring in the field of aerodynamics, aeronautics, interference newlineor interaction problems such as wing-tail surfaces problem etc., are reducible into hypersingular newlineintegral equations or their system. Similar to Cauchy singular integral equations, in case of hypersingular newlineintegral equations also, the analytical solution of these equations and their system are known only for newlinedominant equations. Further, there are many real world prob