Mellin Transform and Integro-Differential Equations with Logarithmic Singularity in the Kernel

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.