A solution of the hypersingular integro-differential equation with determinants of the Vronsky type

Abstract

In this paper, we consider a new hypersingular integro-differential equation of arbitrary order on a closed curve located in the complex plane. The integrals in the equation are understood in the sense of the finite Hadamard part. The equation refers to linear integro-differential equations with variable coefficients of a particular form. A characteristic feature of the equation is its representation with the help of determinants close to the Vronsky ones. The method of analytical continuation, properties of determinants, and generalized Sokhotsky formulas are used for the study. The equation reduces to the Riemann boundary value problem of a jump in a certain class of functions. If the Riemann boundary problem turns out to be solvable, then one should solve linear inhomogeneous differential equations in the class of analytic functions in the domains of the complex plane. The analysis of the obtained solutions in an infinitely distant point is not evident. The study has a complete look. The conditions for the solvability of the original equation are explicitly written out. When they are fulfilled, the solution is explicitly written, to which an example is given.