Comparison of two methods based on daubechies scale functions and legendre multiwavelets for approximate solution of Cauchy-type singular integral equation on ℝ

Abstract

Two methods based on Daubechies scale functions and Legendre multiwavelet for the approximate solution of singular integral equation of the second kind with Cauchy type on the real line ℝ are developed and compared. The integral equation considered here is of the form (Formula presented), where (Formula presented), being a regular kernel. In both of the cases, two-scale relations involving the scale functions are used for the evaluation of multiscale representation of the integral operator. Then the given integral equation is converted into a system of linear algebraic equations which can be solved easily by using library function ‘Solve[]’ available in MATHEMATICA. The convergence of the method has been proved in L2 spaces. Two examples are given and their approximate solutions obtained by the proposed methods have been compared with the available numerical results to assess the efficiency of the method developed here.