Real, Complex and Functional analysis

Solution of the hypersingular integro-differential equation on a curve located in the angular domain.

A. P. Shilin

Belarusian State University, Minsk

Abstract: We solve a new linear integro-differential equation on a closed curve on the complex plane. Some restrictions are placed on the curve location and on coefficients of the equation. The equation contains hypersingular and regular integrals. Initially, it is reduced to the Riemann – Carleman boundary value problem for analytic functions, which has a partial form and an unconventional formulation. Next, two linear differential equations with constant coefficients in domains of the complex plane with additional conditions on the solution are solved. All solvability conditions of the original equation are stated explicitly. When they are performed, the solution to the original equation is given explicitly.

Keywords: Integro-differential equation; hypersingular integral; generalised Sokhotsky formulas; Riemann – Carleman boundary value problem; linear differential equation.

UDC: 517.968.7