ABSTRACT
Fredholm integral equation of second kind with Cauchy singular kernels in both bounded and unbounded domain have been studied in this chapter. Singular integral equations (SIEs) with Cauchy type kernel arise in boundary integral formulation of a large class of mixed boundary value problems of mathematical physics, especially when two-dimensional problems are considered, and also arise in some contact and fracture problems in solid mechanics, mainly crack problems in elasticity. A variety of wavelet basis has been used to approximate the solution of singular integral equation with Cauchy type kernel having variable and fixed singularities within its domain. Uniqueness and existence of solution of the integral equation are discussed in brief. Evaluation of integrals for getting multiscale regularization (MSR) of the Cauchy singular integral operator is presented. MSR of Cauchy singular integral operator is given. These results have been used in the subsequent part of this chapter to convert the singular integral equation with Cauchy-type kernel to a system of linear algebraic equations which can be solved efficiently by using any solver e.g., ”NSolve” available in MATHEMATICA. The following topics are discussed here.
Basis comprising truncated scale functions in Daubechies family, Evaluation of matrix elements, Multiwavelet family, Cauchy singular integral equation with constant and variable coefficients, Equation of first kind, Approximation of Hilbert transform and solution of integral equation of second kind in R in the basis generated by scale functions in Coiflet family and autocorrelation function.
