Approximate solution of the integral equations involving kernel with additional singularity

TL;DR

This paper develops methods for approximately solving Fredholm integral equations with kernels that have both weak and additional singularities, relevant for dynamical systems involving fractional Brownian motion.

Contribution

It introduces a theorem for approximating solutions of integral equations with complex singular kernels using solutions with simpler weakly singular kernels.

Findings

Theorem proving approximation of solutions with additional singularity by weakly singular kernel solutions

Numerical demonstrations validating the proposed approximation methods

Application to integral equations arising in dynamical systems with fractional Brownian motion

Abstract

The paper is devoted to the approximate solutions of the Fredholm integral equations of the second kind with the weak singular kernel that can have additional singularity in the numerator. We describe two problems that lead to such equations. They are the problem of minimization of small deviation and the entropy minimization problem. Both of them appear when considering dynamical system involving mixed fractional Brownian motion. In order to deal with the kernel with additional singularity applying well-known methods for weakly singular kernels, we prove the theorem on the approximation of solution of integral equation with the kernel containing additional singularity by the solutions of the integral equations whose kernels are weakly singular but the numerator is continuous. We demonstrate numerically how our methods work being applied to our specific integral equations.