Polynomial Approximations of Solutions of a Class of Fractional Order Integro-Differential Equations

A Cauchy problem for an integro-differential equation with fractional Riemann–Liouville derivatives is studied. For the given problem, based on the Lebesgue space of functions summable with an arbitrarily fixed degree, a pair of spaces of the sought elements and right-hand sides is proposed, in which the problem is well-posed in the sense of Hadamard. In this pair of spaces, a generalized polynomial projection method for solving the problem is proposed, and its theoretical and functional justification is given. In addition, an estimate of the convergence rate of approximate solutions of the equation involved to its exact solution is given.