Generalized quadrature for solving singular integral equations of Abel type in application to infrared tomography

TL;DR

This paper introduces generalized quadrature methods for solving Abel-type singular integral equations, reducing the problem to nonsingular linear systems and applying Tikhonov regularization to improve accuracy, demonstrated on infrared tomography.

Contribution

The paper develops new generalized quadrature techniques for Abel equations that avoid shift meshes and incorporate regularization, enhancing numerical stability and accuracy.

Findings

Regularization is effective mainly with highly noisy sources.

The methods successfully solve infrared tomography problems.

Singular integral equations exhibit self-regularization properties.

Abstract

We propose the generalized quadrature methods for numerical solution of singular integral equation of Abel type. We overcome the singularity using the analytical calculation of the singular integral expression. The problem of solution of singular integral equation is reduced to nonsingular system of linear algebraic equations without shift meshes techniques employment. We also propose generalized quadrature method for solution of Abel equation using the singular integral. Relaxed errors bounds are derived. In order to improve the accuracy we use Tikhonov regularization method. We demonstrate the efficiency of proposed techniques on infrared tomography problem. Numerical experiments show that it make sense to apply regularization in case of highly noisy sources only. That is due to the fact that singular integral equations enjoy selfregularization property.