Nonlocal problem for multi-parametric integral-differential equation

Abstract

This paper investigates a nonlocal boundary value problem for a multi-parametric integral-differential equation involving a Caputo–Prabhakar type operator in a bounded rectangular domain. This operator generalizes both the classical Caputo fractional derivative and the Prabhakar integral, incorporating memory effects through kernels expressed in terms of generalized Mittag-Leffler functions. The nonlocal conditions are prescribed by partial integral expressions involving the unknown function and given continuous kernels, supplemented by a boundary condition along a characteristic line. Using a known representation of the solution to the corresponding Goursat problem in terms of bivariate and trivariate Mittag-Leffler-type functions, we reduce the problem to a system of Volterra integral equations of the second kind for the unknown boundary traces. Two cases, depending on whether an auxiliary constant vanishes, are analysed separately. In the degenerate case, a first-kind Volterra equation arises and is transformed into a second-kind one by differentiation. Based on the boundedness properties of the special functions involved, sufficient conditions ensuring the existence and uniqueness of the solution are established, and an explicit rep-resentation of the solution is obtained through the derived integral system.

The authors' abstract, as published at the source. BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS, 2026 · DOI ↗