Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph Equation

Abstract

In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation with time-fractional derivatives of orders α ∈]0, 1] and β ∈]1, 2] in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS expressed in terms of a multivariate MittagLeffler function in the Fourier domain. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension n and of the fractional parameters α and β.

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

“…Since is linked to K j (n), it means that this sequence is responsible for the control the lattice-behavior of the equation, ie, the discrete fractional Laplacian operator, and on the other hand, is linked to (−t ) Γ(1+ ) , which means that this term expresses the behavior of the time-fractional derivative. For example, = 2 produces (−1) t 2 (2 )! , which corresponds to the coefficient in the series of the cosine function, ie, the wave equation, and for = 1, we have (−1) t !…”

“…that corresponds to the coefficient in the series of the exponential function, ie, the heat equation. For instance, G ,2 (n, t) = ∞ ∑ =0 K (n) (−1) t 2 (2 )! represents, when reading the coefficients in such way, the wave equation combined with the discrete fractional Laplacian of order .…”

“…Fundamental solutions for many problems governed by fractional derivatives can be found in the literature. For instance, Ferreira, Rodrigues, and Vieira obtained the fundamental solution of the time‐fractional telegraph Dirac operator and the multidimensional time‐fractional telegraph equation, and Mirza and Vieru found the fundamental solutions to the advection‐diffusion equation with time‐fractional Caputo‐Fabrizio derivative. See also the references therein.…”

Fundamental solution of the time‐fractional telegraph Dirac operator

“…This operator factorizes the time‐fractional telegraph operator, which is a sum of the Laplace operator with the 2 time‐fractional derivatives already mentioned. The FS of this operator was studied recently in Ferreira et al The FS of the time‐fractional telegraph Dirac operator can be seen as a refinement of the FS of the time‐fractional telegraph operator. This opens new possibilities for the development of a fractional function theory for this operator in the context of Clifford analysis.…”

“…The structure of the paper reads as follows: In the preliminaries section, we recall some basic concepts about fractional calculus, special functions, Clifford analysis, and the Witt basis. In Section 3, we recall the integral and series representations for the FS of the time‐fractional telegraph operator in obtained in Ferreira et al In Section 4, we derive the FS for the time‐fractional telegraph Dirac operator in the form of integral and series from the representations presented in Section 3. We remark that the series representation depends on the parity of the space dimension, as it happens in Ferreira et al Connections with the time‐fractional parabolic Dirac operator are established in Section 4.…”