Analytical methods of solving hypersingular integral equations

Background. Hypersingular integral equations appear to be an actively developing area of mathematical physics. It is associated with applications of hypersingular integral equations in aerodynamics, electrodynamics, quantum physics, geophysics. Despite direct application sin physics and engineering, hypersingular integral equations occur when solving boundary problems of mathematical physics. Recently there has been published a series of works devoted to approximate methods of solving hypersingular integral equations of first and second kind on closed and open paths of integration. The interst to such methods is associated with direct applications of hypersingular integral equations in aerodynamics and electrodynamics. At the same time there are no analytical methods of solving hypersingular integral equations and polyhypersingular integral equations. The present article suggests a method of analytical solution of one class of hypersingular integral equations and polyhypersingular integral equations. This method enables to more efficiently use hypersingular integral equations in multiple applications. Materials and methods. The study used methods of the theory of singular integral equation, regular differential equations and partial derivative equations. The authors considered linear and nonlinear one-dimensional hypersingular integral equations on closed integration paths and bihyperssingular integral equations on closed smooth surfaces. The method is based on reduction of hypersingular and polyhypersingular integral equations to differential equations - regular ones and with partial derivatives. Results. The authors have developed an analytical method of solving one class of hypersingular integral equations on closed integration paths and bihypersingular integral equations on closed smooth surfaces. Conclusions. The article considers the developed method of solving hypersingular and polyhypersingulae integral equations. Whensolving applied problems this method allows to obtain solutions in the form convenient for further research. The results obtained can be used to solve problems of electrodynamics, hydrodynamics, equations of mathematical physics by the method of boundary integral equations.