To the justification of the general projection method for solving linear integral equations with a fractional Riemann-Liouville integral in the principal part

Abstract

The article proposes a generalized polynomial projection method for solving linear integral equations with a fractional Riemann - Liouville integral in the main part. Fractional-integral equations have numerous applications in various applied problems, in particular, in problems of plasma diagnostics, optical and X-ray diffraction, in ultrasonic measurements in moving media, in metallurgy, biology, microscopy, seismology, in astrophysics and other fields of science.The equation we are considering is an equation of the first kind and therefore, generally speaking, refers to incorrectly set by Hadamard equations. The latter circumstance is connected with the fact that in known function spaces the fractional integral Riemann-Liouville operator is completely continuous. All this imposes its own characteristics on the construction and study of approximate methods for solving integral equations with a fractional integral operator in the main part. It should be noted that the operator of the projection method is not necessarily projective, which allows the construction of computational schemes using the methods of summation of Fourier series and interpolation polynomials. A theoretical and functional substantiation of the proposed projection method was carried out, based on the correct formulation of the equation in a pair of specially selected different Holder spaces. In particular, it proved the unique solvability of the system of linear algebraic equations of the method and the convergence of the constructed approximations to the exact solution in the norm of the space of Holder functions and, as a consequence, in the uniform metric. This also implies the substantiation of specific well-known projection methods such as the Galerkin methods, colocations, and subdomains.