Approximate solution of the integral equations involving kernel with additional singularity

Abstract

The article 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 deviations and the entropy minimization problem. Both of them appear when considering a dynamical system involving a mixed fractional Brownian motion. In order to apply well-known numerical methods for weakly singular kernels, we build the continuous approximation of the solution of an 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 prove that the approximated solutions tend to the solution of the original equation. We demonstrate numerically how our methods work being applied to our specific integral equations.