Analytical Methods of Solving Hypersingular Integral Equations

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.

Regular, Singular and Hypersingular Integrals over Fractal Contours

“…Analytical methods for solving singular and hypersingular integral equations are known only for very special cases (see [1] for singular integral equations and [2], Ref. [3] for hypersingular integral equations).…”

“…The numerical approach to the evaluation of SI and HI over fractals is different. Even though there are few papers devoted to solutions of SIE and HIE on the prefractals of the Cantor perfect set and the Sierpinski "carpet" [14,15], and the research on existence and uniqueness of SIE solutions on fractals is being actively developed [16], the works devoted to generic numerical methods for calculation of SI and HI and solution of SIE and HIE over fractals are unknown to the authors.…”

“…The most natural way is to calculate the Stieltjes integral (1) using known quadrature formulas, which is applied for the integral (2). Using (2), however, requires an explicit form of the function z = g(x).…”

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

“…These equations are solved mainly numerically on the basis of approximate approaches. A good overview of the relevant results is provided in this long article [5], which specifically states that "there are currently a few disparate results only which are dedicated to analytical methods for solving hypersingular integral equations" [5] (p. 248). It is mostly related to hypersingular integro-differential equations.…”

“…Theorem. For Equation (5) to be solvable, it is necessary and sufficient that conditions (11) are satisfied (when γ < −l), (21), ( 22), ( 26)-(29) and the system (30) are solvable. If Equation (5) turns out to be solvable, then its solution has the form t…”