Approximate solution of first kind singular integral equation with generalized kernel using Legendre multiwavelets

Abstract

In this study, numerical solution of a class of Cauchy type singular integral equations of the first kind with generalized kernels (CSIEFKGK) is obtained using Legendre multiwavelets (LMW). Instead of using complex function theory, the appropriate weight function of the solution is obtained by applying LMW-based numerical scheme. We evaluate the matrix representation of generalized kernel with and without the weight factor using recurrence relations and some elementary methods. Using the multiscale representation of the integral operator, the equation is converted into a system of linear equations. The numerical solution of CSIEFKGK is obtained by solving this systems. Finally, a number of examples including applications of crack problems are considered to illustrate the efficiency of the method developed here. Keywords Cauchy type singular integral equation of the first kind • Generalized kernel • Legendre multiwavelets • Wavelet Galerkin method • Multiscale approximation Mathematics Subject Classification 65R20 • 65D15 • 65T60 • 45E05 • 42C40 1 Introduction Singular integral equation (SIE) arises in various areas of mathematical physics and engineering. The Cauchy type singular integral equation of the first kind with the generalized kernel (CSIEFKGK) arises from the formulation of a large class of mixed BVPs in mathe-Communicated by Hui Liang.

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