Estimation of errors of quadrature formula for singular integrals of Cauchy type with special forms

TL;DR

This paper analyzes the errors of quadrature formulas for singular Cauchy-type integrals, demonstrating that spline-based approximation achieves optimal convergence with minimal smoothness requirements for the density function.

Contribution

It introduces error estimates for quadrature formulas using linear spline interpolation for singular integrals of Cauchy type, highlighting the advantages of spline approximation over other methods.

Findings

Established convergence rates for quadrature errors with spline approximation.

Showed spline methods require less smoothness of the density function.

Compared spline approximation favorably to other approximation techniques.

Abstract

In this work, we consider the singular integrals of Cauchy type of the forms $$\ds J(f,x)= \frac{\sqrt{1-x^2}}{\pi}\int_{-1}^1\frac{f(t)}{\sqrt{1-t^2}(t-x)}\,dt, -1<x<1 and $$\ds \Phi(f,z)= -\frac{\sqrt{z^2-1}}{\pi}\int_{-1}^1\frac{f(t)}{\sqrt{1-t^2}(t-z)}\,dt, \ \ \qquad z\notin [-1,1].$$ which are understood as Cauchy principal value integrals. Quadrature formulas (QFs) for singular integrals (SIs) \re{eq1} and \re{eq2} are of the forms $$\ds J(f,x)= \sum_{k=0}^{N}A_k(x)f(t_k)+ R_N(f,x), \ \ \qquad-1<x<1. and \ds \Phi(f,z)= \sum_{k=0}^{N}B_k(z)f(t_k)+ R_N^*(f,z), z\notin [-1,1]$$ where $z$ is complex variable with $|Re(z)|>1$. With the help of linear spline interpolation, we have proved the rate of convergence of the errors of QFs \re{eq3} and \re{eq4} for different classes (i.e. $H^\a([-1,1],K), C^{m,\a}[-1,1], W^r[-1,1]$) of density function $f(t)$. It is shown that approximation by spline possesses more advantages than other kinds of approximation: it requires the minimum smoothness of density function $f(x)$ to get good order of decreasing errors.