Projection methods for solving hypersingular integral equations in fractals

Background. Approximate methods for solving hypersingular integral equations are an actively developing area of calculus amthematics. It relates to multiple applications of hypersingular equations in aerodynamics, electrodynamics, physics, and also to one circumstance - analytical solutions of hypersingular integral equations are possible only in exceptional cases. Besides direct applications in physics and engineering, hypersingular integral equations of first kind occur in approximate solution of boundary problems of mathematical physics. Recently, the interest to studying analytical and numerical methods for solving hypersingular integral equations has significantly increased due to active application of methods of fractal geometry in radio engineering and radiolocation. It has turned out that one of main methods of fractal antennas modeling is hypersingular integral equations. The present work suggests and substantiates spline-collocation methods of zero and first orders for solving hypersingular integral equations on fractals. Materials and methods. The study used methods of functional analysis and approximation theory. The authors considered linear one- and two-dimensional hypersingular integral equations on fractals. For determinancy in the case of a one-dimensional integral the research took the Cantor perfect set as an area of integration, and in the case of two-dimensional one - the Sierpinski carpet. The authors built projection computing schemes, substantiated on the basis of the analysis of logarithmic norms of the corresponding matrixes. Results. The authors built three computing schemes for solving hypersingular integral equations on fractals of various types and dimensionality and obtained estimates of rapidity of convergence and error of the said schemes. The generated schemes appear to be models for building and substantiation of computing schemes on fractals of various nature. Conclusions. The authors built and substantiated computing schemes of approx.imate solution of hypersingular integral equations of first and second kinds on model fractals. Model fractals are represented as the Cantor perfect set and the Sierpinski carpet. The obtained results may be used in fractal antennas modeling.