Analytical methods for solution of hypersingular and polyhypersingular integral equations

TL;DR

The paper develops a general framework to solve hypersingular and polyhypersingular integral equations by transforming them into ordinary or partial differential equations under analyticity assumptions, enabling analytical (quadrature-based) solutions. It treats linear, nonlinear, integro-differential, and bihypersingular cases and derives explicit reductions to ODEs ($p-1$ derivatives) or PDEs via the Sokhotsky-Plemel formulas, with representative examples illustrating closed-form solutions. The approach provides conditions under which integral equations and their differential counterparts are equivalent and demonstrates the feasibility of obtaining exact solutions for broad classes of singular equations. By extending these reductions to multi-variable polyhypersingular problems on product contours, the method broadens the toolbox for analytic solutions in physics and engineering contexts.

Abstract

We propose a method for transformating linear and nonlinear hypersingular integral equations into ordinary differential equations. Linear and nonlinear polyhypersingular integral equations are transformed into partial differential equations. Well known that many types of differential equations can be solved in quadratures. So, we can receive analytical solutions for many types of linear and nonlinear hypersingular and polyhypersingular integral equations.