Approximate Solution and its Convergence Analysis for Hypersingular Integral Equations

Vaishali Sharma; Amit Setia

doi:10.14429/dsj.69.12479

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

DOI: https://doi.org/10.14429/dsj.69.12479

Keywords: Hypersingular kernal, Galerkin method, convergence analysis, condition number.

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.

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.

References

E.G. Ladopoulos, Singular Integral Equations, Linear and Non-linear Theory and its Applica- tions in Science and Engineering, Berlin, New York, Springer-Verlag, 2000, pp. 173-185.

H. Ashley and M. Landahl, Aerodynamics of Wings and Bodies, Dover, New York, 1985.

K.W. Mangler, Improper integrals in theoretical aerodynamics, Report no. 2424, Royal Aircraft Establishment, Farnborough, 1951.

N.V. Danilenko, A.I. Zhelannikov, M.I. Nisht and V.F. Pavlenko, Modelling of the immedi- ate trace and its effect on the aerodynamical characteristics of aircraft, Trudy WIA imeni Zhukovskogo, Vol. 1311, 1983.

V.I. Babitsky and J. Wittenburg, Foundations of Engineering Mechanics, Springer-Verlag Berlin Heidelberg, 2009.

G. Monegato, Numerical evaluation of hypersingular integrals, J. Comput. Appl. Math., 50, 1994, 9-31.

S.M. Belotserkovsky and I.K. Lifanov, Method of discrete vortices, Florida, CRC Press, 1993.

R.L. Bisplinghoff, H. Ashley and R.L. Halfman, Aeroelasticity, Addison-Wesley, Cambridge, Massachusetts, 1955.

G. Iovane, I.K. Lifanov and M.A. Sumbatyan, On direct numerical treatment of hypersingular integral equations arising in mechanics and acoustics, Acta Mech., 162, 2003, 99-110.

G.V. Ryzhakov, On the numerical method for solving a hypersingular integral Equation with the computation of the solution gradient, Diff. Equat., 49, 2013, 1168–1175.

Y.S. Chan, A.C. Fannjiang and G.H. Paulino, Integral equations with hypersingular kernels theory and applications to fracture mechanics, Int. J. Eng. Sci., 41, 2003, 683-720.

N.F. Parsons and P.A. Martin, Scattering of water waves by submerged plates using hypersin- gular integral equations, Appl. Ocean Res., 1992, 14, 313-321.

A. Chakrabarti, B.N. Mandal, U. Basu and S. Banerjea, Solution of a hypersingular integral equation of the second kind, ZAMM., 77, 1997, 319-320.

B.N. Mandal, and G.H. Bera, Approximate solution for a class of hypersingular integral equa- tions, Appl. Math. Lett., 19, 2006, 1286-1290.

Z. Chen and Y. Zhou, A new method for solving hypersingular integral equations of the first kind, Appl. Math. Lett., 24, 2011, 636-641.

A.V. Setukha, Convergence of a numerical method for solving a hypersingular integral equation on a segment with the use of piecewise linear approximations on a nonuniform grid, Diff. Equat., 53, 2017, 234–247.

M.A. Golberg and C. S. Chen, Discrete Projection Methods for Integral Equations, Computa- tional Mechanics Publications, Southampton UK and Boston USA, 1997.

P.A. Martin, End-point behaviour of solutions to hypersingular integral equations, Proceedings of the Royal Society A: Math. Phys. Eng. Sci., 432, 1991, 301-320.

A.C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Q. Appl. Math., 45, 1987, 105-122.

M.A. Golberg and J. A. Fromme, The Convergence of Several Algorithms for Solving Integral Equations with Finite Part Integrals- II, J. Math. Anal. Appl., 71, 1979, 271-286.

C.S. Kubrusly, Spectral Theory of Operators on Hilbert Spaces. Birkhuser/Springer, New York, 2012.