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.

摘要

本文致力于研究具有弱奇异核的第二类 Fredholm 积分方程的近似解，该核可以在分子中具有附加奇异性。我们描述了导致此类方程的两个问题。它们是最小偏差的最小化问题和熵最小化问题。当考虑涉及混合分数布朗运动的动力系统时，它们都会出现。为了对弱奇异核应用众所周知的数值方法，我们通过核为弱奇异但分子连续的积分方程的解来构建积分方程解的连续逼近，其中核包含附加奇异性. 我们证明了近似解趋于原方程的解。