Approximate Solution and its Convergence Analysis for Hypersingular Integral Equations

  • Vaishali Sharma Department of Mathematics, BITS Pilani - K K Birla Goa Campus, Goa - 403 726
  • Amit Setia Department of Mathematics, BITS Pilani - K K Birla Goa Campus, Goa - 403 726
Keywords: Hypersingular kernal, Galerkin method, convergence analysis, condition number.


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.                            

Author Biographies

Vaishali Sharma, Department of Mathematics, BITS Pilani - K K Birla Goa Campus, Goa - 403 726

Dr Vaishali Sharma has completed her PhD (Applied Mathematics) from BITS Pilani KK Birla Goa Campus, in 2018. Presently, she is working as visiting faculty in the department of mathematics in BITS-Pilani, Goa-Campus. Her areas of interest include study of numerical solution of singular integral equations and its applications.

Amit Setia, Department of Mathematics, BITS Pilani - K K Birla Goa Campus, Goa - 403 726

Dr Amit Setia has completed his PhD (Applied Mathematics) from IIT Roorkee, in 2009. Currently serving as an Assistant Professor in the Department of Mathematics, BITS Pilani KK Birla Goa Campus. His areas of interest include modelling of some problems of fracture mechanics of special kind of materials known as piezoelectric materials, study of numerical solutions of singular integral equations, fractional order differential and integral equations.




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How to Cite
Sharma, V., & Setia, A. (2019). Approximate Solution and its Convergence Analysis for Hypersingular Integral Equations. Defence Science Journal, 69(2), 173-178.