Hypersingular Integro-Differential Equation Containing Polynomials and Their Derivatives in Coefficients

Abstract

A new linear integro-differential equation is considered on a closed curve located on the complex plane. The integrals in the equation are understood in the sense of a finite part, according to Hadamard. The order of the equation can be higher than one. The coefficients of the equation have a special structure. A characteristic of the coefficients is that they contain two arbitrary polynomials and their derivatives. Solvability conditions are explicitly stated. Whenever they are satisfied, an exact analytical solution is given. Generalized Sokhotsky formulas, the theory of Riemann boundary value problems, methods for solving linear differential equations, and the properties of analytic functions of a complex variable are used. An example is given.

Keywords

• integro-differential equation
• hypersingular integral
• Riemann boundary problem
• linear differential equation
• analytic function