Boundary value problem for a differential-difference equation with a fractional derivative

L.M. Vidzizheva, D.A. Kanametova

Abstract. The work is devoted to the study of a differential-difference equation with a fractional derivative of order not exceeding one. For the equation under consideration, a boundary value problem is posed and solved on a manifold that is a countable union of intervals. To solve the problem, we used an analogue of the Green function method, adapted for differential-difference equations. A general representation of the solution to the equation under study has been found, a fundamental solution has been constructed in terms of the Prabhakar function, its properties have been studied, and a theorem on the existence and uniqueness of a solution to the problem under study has been proven.

Keywords: fractional derivative, McKendrick – Von Foerster equation, fractional integration operator, fractional differentiation operator, differential-difference equation, Riemann – Liouville integral, difference operators, Prabhakar function, Mittag-Leffler function

For citation. Vidzizheva L.M., Kanametova D.A. Boundary value problem for a differential-difference equation with a fractional derivative. News of the Kabardino-Balkarian Scientific Center of RAS. 2024. Vol. 26. No. 4. Pp. 130–144. DOI: 10.35330/1991-6639-2024-26-4-130-144