Integro-differential equations with singular and hypersingular integrals

Journal of the Belarusian State University. Mathematics and Informatics

2019

The linear equation on the curve located on the complex plane is studied. The equation contains the desired function, its derivatives of the first and second orders, as well as hypersingular integrals with the desired function. The coefficients of the equation have a special structure. The equation is reduced to the Riemann boundary value problem for analytic functions and two second order linear differential equations. The boundary value problem is solved by Gakhov formulas, and the differential equations are solved by the method of variation of arbitrary constants. The solution of the original equation is constructed in quadratures. The result is formulated as a theorem. An example is given.

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“…Тогда соотношения (4), (5) станут линейными дифференциальными уравнениями 2-го порядка. Начнем с решения уравнения (4). Ему можно придать вид…”

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Explicit solution of one hypersingular integro-differential equation of the second order

Journal of the Belarusian State University. Mathematics and Informatics

2019

A hypersingular integro-differential equation of the Euler type

Vescì Akademìì navuk Belarusì. Seryâ fizika-matematyčnyh navuk

2020

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.

On the solution of one integro-differential equation with singular and hypersingular integrals

Vescì Akademìì navuk Belarusì. Seryâ fizika-matematyčnyh navuk

2020

A linear integro-differential equation of the first order given on a closed curve located on the complex plane is studied. The coefficients of the equation have a special structure. The equation contains a singular integral, which can be understood as the main value by Cauchy, and a hypersingular integral which can be understood as the end part by Hadamard. The analytical continuation method is applied. The equation is reduced to a sequential solution of the Riemann boundary value problem and two linear differential equations. The Riemann problem is solved in the class of analytic functions with special points. Differential equations are solved in the class of analytical functions on the complex plane. The conditions for the solvability of the original equation are explicitly given. The solution of the equation when these conditions are fulfilled is also given explicitly. Examples are considered. A non-obvious special case is analyzed.