Solution of one hypersingular integro-differential equation defined by determinants

Abstract

The paper provides an exact analytical solution to a hypersingular inregro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with vatiable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.

Author Biography

Andrei P. Shilin, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

PhD (physics and mathematics), docent; associate professor at the department of higher mathematics and mathematical physics, faculty of physics

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Published
2021-08-05
Keywords: inregro-differential equations, hypersingular integrals, generalised Sokhotsky formulas, differential equations, Riemann boundary problem
How to Cite
Shilin, A. P. (2021). Solution of one hypersingular integro-differential equation defined by determinants. Journal of the Belarusian State University. Mathematics and Informatics, 2, 17-28. https://doi.org/10.33581/2520-6508-2021-2-17-28