Hypersingular integro-differential equation with recurrent relations in coefficients

Abstract

A new hypersingular integro-differential equation is considered on a closed curve located on the complex plane. The equation refers to linear equations with variable coefficients of a special kind. A characteristic feature is the presence of constant multipliers in the coefficients, given by some recurrent relations. The equation is first reduced to solving the Riemann boundary value problem on the original curve. A class of functions is established for solving the Riemann problem, after which this problem is solved. Next, it is necessary to solve two linear differential equations of arbitrary order for analytical functions in two different regions of the complex plane. The corresponding fundamental systems of solutions are found, after which the method of variation of arbitrary constants is used for the solution. Restrictions are imposed on the obtained solutions of differential equations in order to achieve their analyticity. As a result, all the resulting solvability conditions of the original equation are written explicitly. The solution of the original equation after solving the differential equations can be written explicitly. Solved the example.