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.

中文翻译：

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涉及具有附加奇异性的核的积分方程的近似解

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，该核在分子中可以具有附加奇异性。我们描述了导致此类方程的两个问题。它们是小偏差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。为了处理具有附加奇异性的核应用众所周知的弱奇异核方法，我们证明了关于具有附加奇异性核的积分方程解的逼近定理，由其核为弱奇异的积分方程的解来证明但分子是连续的。