Nonlocal problem for the time-fractional hyperbolic-type equation with the Prabhakar fractional derivative

In this paper, we investigate a nonlocal boundary value problem for a time-fractional hyperbolic-type partial differential equation involving fractional derivatives of regularized Prabhakar. The fractional differentiation is defined through the regularized Prabhakar operator, which provides a flexible framework for modeling memory effects with non-singular kernels. The equation is considered on a bounded rectangular domain in the plane with respect to two independent variables. The boundary conditions are nonlocal and are prescribed in the form of partial integral expressions of the unknown solution along each spatial variable, where the corresponding kernels are assumed to be continuous. Building upon previously obtained representation formulas for the solution of the associated Goursat problem in terms of Mittag-Leffler type functions, the original boundary value problem is transformed into a coupled system of Volterra integral equations of the second kind for the traces of the solution on a portion of the boundary. This reduction allows us to apply classical methods of integral equations to analyze the problem. By employing appropriate estimates for the regularized Prabhakar kernels and the properties of the resulting integral operators, we rigorously establish the existence and uniqueness of the solution to the nonlocal boundary value problem. Furthermore, an explicit representation of the solution is derived in terms of the solutions of the obtained system of integral equations. The results demonstrate that the regularized Prabhakar framework provides a robust and analytically tractable approach for treating time-fractional hyperbolic problems with nonlocal boundary interactions.