Approximate solution of linear hypersingular integral equations by the collocation method

Background. The approximate methods of solution of hypersingular integral equations are an actively developing area of calculus mathematics, which is associated, in the first place, with multiple applications of hypersingular integral equations in mechanics, aerodynamics, electrodynamics, geophysics. At the same time it is necessary to point out two circumstances: 1) an analytical solution of hypersingular integral equations is possible only in exceptional cases; 2) the range of applications of hypersingular integral equations constantly expands. These circumstances condition the topicality of building and substantiation of numerical methods of solution of hypersingular integral equations. At the present time the methods of approximate solutions of complete hypersingular integral equations in Gelder's space remain unresearched. The article is devoted to building and substantiation of approximate solutions of hypersingular integral equations by the collocation method. Materials and methods. Substatiation of solvability and convergence of the collocation method to the approximate solution of hypersingular equations is based on application of the methods of functional analysis and the approximation theory. Results. The authors suggested a modification of the collocation method for approximate solution of hypersingular integral equations and substantiated it. The researches also adduce the assessment of rapidity of convergence and extent of error. Conclusions. The authors built calculation schemes allowing to effectively solve applied problems of mechanics, aerodynamics, electrodynamics, geophysics.