Nonlocal problem for the time-fractional generalized telegraph equation with the Prabhakar fractional derivative

TL;DR

This paper studies a nonlocal boundary value problem for a time-fractional generalized telegraph equation in a bounded domain using the Caputo-Prabhakar derivative $^{PC}D_{0t}^{\alpha,\beta,\gamma,\delta}$ and integral-type boundary conditions. A representation of the Goursat-type solution in terms of Mittag-Leffler type functions is derived and used to reduce the problem to a system of second-kind Volterra integral equations for the boundary traces. The authors prove existence and uniqueness of a regular solution and provide an explicit representation of the solution via the integral system, involving $E_{2}$-type and $\overline{F}_{E}^{(3)}$ Mittag-Leffler functions. A special parameter regime with $\alpha=1$, $0<\beta<1$, $a<0$, $b<0$, $\delta<0$, $\gamma=\beta$ yields a concrete solvability result (Theorem 2) for the nonlocal problem, illustrating the method and the role of kernel properties in well-posedness.

Abstract

In this work, nonlocal boundary value problem for the generalized telegraph equation with fractional-order derivatives is studied. Fractional differentiation is defined using the Caputo-Prabhakar operator. The equation is considered in a bounded rectangular domain of the plane with two independent variables. The nonlocal boundary condition is specified in the form of partial integral expressions of the unknown solution with respect to each variable, with given continuous kernels. Using the previously obtained representation of the solution to the Goursat problem for the studied equation in terms of Mittag-Leffler type functions, the problem is successfully reduced to a system of second-kind Volterra's integral equations with respect to the traces of the unknown solution on a part of the boundary of the region. As results, theorems of the existence and the uniqueness of the solution to the investigated problem is proved, and its representation is found in terms of the solutions to the obtained system of integral equations.