Abstract

In the one-dimensional domain it is studied the unique solvability of a class of integro-differential equations with logarithmic singularity in the kernels. The solution of the considering integro-differential equations with logarithmic singularity in the kernels on dependent of roots of corresponding characteristic equations is obtained in an explicit form. The first part of paper deals with the introduction of the Mellin transformation as a holomorphic function in the complex plane. The definition of this integral transform is given and its main properties are described in details. The second part deals with the application of the Mellin transform to the solution of one-dimensional integro-differential equations with the logarithmic singularity in the kernels. Here, in order to obtain particular solution of nonhomogeneous equations, we applied the analogies of Duhamel’s theorem. It is shown that, if the kernel of studied equation has first order logarithmic singularity of polynomial’s form, then to this equation corresponds characteristic (algebraic) equation of the third order. In this case the solution is found by means of degree of logarithmic and trigonometric functions. If the integro-differential equation has \(n\)-th order logarithmic singularity of polynomial’s form, then to this equation corresponds \(n\)-th order characteristic equation. At three main cases the solution is found in an explicit form. Also, it is established that the degree of logarithmic singularity acts to number of linearly independent solutions of given classes of integro-differential equations.

中文翻译：

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对数奇异性的Mellin变换和积分微分方程

摘要

在一维域中，研究了一类积分对数奇异性的积分微分方程的唯一可解性。以显式形式获得考虑对数奇异积分的微分方程的解，该整数取决于相应特征方程的根。本文的第一部分介绍了Mellin变换作为复平面中的全纯函数的介绍。给出了该积分变换的定义，并详细描述了其主要属性。第二部分介绍了Mellin变换在核中具有对数奇异性的一维积分微分方程解的应用。在这里，为了获得非齐次方程的特定解，我们应用了杜哈默定理的类比。结果表明，如果所研究方程的核具有多项式形式的一阶对数奇异性，则该方程对应三阶特征（代数）方程。在这种情况下，可以通过对数和三角函数的度数找到解。如果积分微分方程具有多项式形式的\（n \）次对数奇异性，则该方程对应于\（n \）次阶特征方程。在三种主要情况下，以显式形式找到解决方案。而且，已经确定对数奇异度对给定类别的积分-微分方程的线性独立解的数量起作用。