Approximate Solution and its Convergence Analysis for Hypersingular Integral Equations

Abstract

This paper propose a residual based Galerkin method with Legendre polynomial as a basis functions to find the approximate solution of hypersingular integral equations. These equations occur quite naturally in the field of aeronautics such as problem of aerodynamics of flight vehicles and during mathematical modeling of vortex wakes behind aircraft. The analytic solution of these kind of equations is known only for a particular case ( m(x,t) =0 in Eqn (1)). Also, in these singular integral equations which occur during the formulation of many boundary value problems, the known function m(x,t) in (Eqn (1)) is not always zero. Our proposed method find the approximate solution by converting the integral equations into a linear system of algebraic equations which is easy to solve. The convergence of sequence of approximate solutions is proved and error bound is obtained theoretically. The validation of derived theoretical results and implementation of method is also shown with the aid of numerical illustrations.