Mathematics

A boundary value problem on the semi-axis for an ordinary differential equation with a fractional Caputo derivative

I. E. Egorov^a, E. D. Fedotov<sup>b

a</sup> North-Eastern Federal University named after M. K. Ammosov, Yakutsk
^b Yakutsk Branch of the Regional Scientific and Educational Mathematical Center "Far Eastern Center of Mathematical Research"

Abstract: The paper considers the unique solvability of a boundary value problem on the semiaxis for a higher-order ordinary di erential equation with a fractional Caputo derivative and constant coe cients in the class of bounded functions, where the order of the fractional Caputo derivative lies in the interval (0, 1). Higher orders of the fractional derivative are obtained by composing fractional Caputo derivatives. A special case of the fractional Caputo derivative for integer orders of the derivative coincides with the classical concept of the derivative and the problem under consideration becomes a classical boundary value problem on the half-axis for a higher-order ordinary di erential equation. For the equation under consideration, a fundamental system of solutions in the class of bounded functions is constructed. Conditions of the Lopatinsky type for boundary operators are obtained under which the boundary value problem is uniquely solvable in the class of bounded functions.

Keywords: Caputo derivative, boundary value problem, solution, estimate.

UDC: 517.95

Received: 15.02.2023
Accepted: 29.05.2023

DOI: 10.25587/SVFU.2023.49.50.003