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

Abstract

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.

Keywords

• integro-differential equation
• hypersingular integral
• riemann boundary value problem
• linear differential equation