Comparison of Two Methods Based on Daubechies Scale Functions and Legendre Multiwavelets for Approximate Solution of Cauchy-Type Singular Integral Equation on ${\mathbb {R}}$

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 $\mathbb {R}$ are developed and compared. The integral equation considered here is of the form $u(x)+\lambda \int _{-\infty }^{\infty } K(x,t)u(t)dt=f(x),\ x\in \mathbb {R}$, where $K(x,t)=\frac{1}{t-x}+h(x,t)$, h(x, t) 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 $L^2$ 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.