Solution of one hypersingular integro-differential equation defined by determinants

The paper provides an exact analytical solution to a hypersingular inregro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with vatiable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.

LINEAR CONJUGATION PROBLEM WITH MULTIPOINT NONLOCAL CONDITION FOR A PARABOLIC-HYPERBOLIC EQUATION IN CYLINDRICAL DOMAIN

The linear conjugation problem with multipoint nonlocal condition in the time variable for a mixed parabolic-hyperbolic equation of the second order in a cylindrical domain, which is Cartesian product of the time segment and the spatial multidimensional torus, is investigated. The conditions of the existence and uniqueness of а solution to the problem in the scale of Sobolev spaces are obtained. It has been proved that these conditions fulfill for almost all (with respect to the Lebesgue measure) values of the left node of the multipoint condition.

A hypersingular integro-differential equation of the Euler type

In this paper, we study an integro-differential equation on a closed curve located on the complex plane. The integrals included in the equation are understood as a finite part by Hadamard. The coefficients of the equation have a particular structure. The analytical continuation method is applied. The equation is reduced to a boundary value linear conjugation problem for analytic functions and linear Euler differential equations in the domains of the complex plane. Solutions of the Euler equations, which are unambiguous analytical functions, are sought. The conditions of solvability of the initial equation are given explicitly. The solution of the initial equation obtained under these conditions is also given explicitly. Examples are considered.