Polynomial approximations of solutions to a class of fractional-order integro-differential equationsr

Yu. R. Agachev, A. V. Guskova

Kazan Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia

Abstract: In this paper, 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 correctly posed according to 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, and also an estimate is given of the rate of convergence of the approximate solution of the equation under consideration to its exact solution.

Keywords: Cauchy problem, integro-differential equation, fractional derivative, Lebesgue space, correct setting, projection method, polynomial approximation, convergence of the method.

UDC: 517.51: 517.968: 519.642

Received: 13.07.2024
Revised: 18.11.2024
Accepted: 18.12.2024

DOI: 10.26907/0021-3446-2025-10-18-29