Line element method of solving singular integral equations

The work in this paper is concerned with application of a very simple numerical technique called line element method to solve singular integral equations with weakly singular kernel and hypersingular kernel. Of the integral equations considered here are first kind Abel integral equation and integral equation with log kernel and hypersingular integral equations of first and second kind. In this method, the range of integration as well as the interval of definition of the integral equation are discretised into finite number of small subintervals and the unknown function satisfying the integral equation is assumed to be constant in each small subinterval. This reduces the integral equations to a system of algebraic equations which is then solved to obtain the unknown function in each subinterval. The method is illustrated with examples. It is observed that a very accurate result is obtained by applying this method. The error analysis for this method is also given.

[1]  Paul A. Martin,et al.  Scattering of water waves by submerged plates using hypersingular integral equations , 1992 .

[2]  Carlos Alberto Brebbia,et al.  The Boundary Element Method for Engineers , 1978 .

[3]  B. N. Mandal,et al.  Applied Singular Integral Equations , 2011 .

[4]  Paul A. Martin,et al.  Exact Solution of a Simple Hypersingular Integral Equation , 1992 .

[5]  B. N. Mandal,et al.  Water Wave Scattering by Barriers , 2000 .

[6]  Jeng-Tzong Chen,et al.  A Practical Guide to Boundary Element Methods with the Software Library BEMLIB , 2002 .

[7]  A DIRECT BOUNDARY ELEMENT APPROACH WITH LINEAR BOUNDARY ELEMENTS FOR THE COMPRESSIBLE FLUID FLOW AROUND , 2007 .

[8]  Gernot Beer,et al.  Introduction to Finite and Boundary Element Methods for Engineers , 1993 .

[9]  Glaucio H. Paulino,et al.  Integral equations with hypersingular kernels––theory and applications to fracture mechanics , 1999, math/9906058.

[10]  Sean F. Wu,et al.  The Boundary Element Method in Acoustics , 2000 .

[11]  Carlos Alberto Brebbia,et al.  Boundary Elements: An Introductory Course , 1989 .

[12]  Sudeshna Banerjea,et al.  Solution of a Hypersingular Integral Equation of the Second Kind , 1997 .

[13]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[14]  Sudeshna Banerjea,et al.  On a weakly singular integral equation and its application , 2008, Appl. Math. Lett..

[15]  A. Kaya,et al.  On the solution of integral equations with strongly singular kernels , 1985 .

[16]  John T. Katsikadelis,et al.  Boundary Elements: Theory and Applications , 2002 .