Generalized polynomial method for solving a Cauchy-type problem for one fractional differential equation

In this paper, we examine a Cauchy-type problem for one ordinary differential equation with Riemann–Liouville fractional derivatives. For this problem, based on the Lebesgue space of functions summable with an arbitrarily fixed degree, we propose a family of pairs of spaces of elements and right-hand sides that yield its well-posed statement. In these pairs of spaces, we develop a generalized polynomial projection method for solving the problem considered and justify it from the functional-theoretic point of view. Based on the general results obtained, we prove the convergence of the “polynomial” Galerkin method, the collocation method, and the subdomain method for solving the corresponding Cauchy-type problem.