Fundamental solution of the time‐fractional telegraph Dirac operator

Abstract

In this work, we obtain the fundamental solution (FS) of the multidimensional time-fractional telegraph Dirac operator where the 2 time-fractional derivatives of orders ∈]0, 1] and ∈]1, 2] are in the Caputo sense. Explicit integral and series representation of the FS are obtained for any dimension. We present and discuss some plots of the FS for some particular values of the dimension and of the fractional parameters and . Finally, using the FS, we study some Poisson and Cauchy problems.

First and Second Fundamental Solutions of the Time-Fractional Telegraph Equation with Laplace or Dirac Operators

“…The results presented in this paper are a generalization of the results presented in [8,9], since we incorporate the second fundamental solution coming from the second initial condition (see formulation in (9)).…”

“…The first case corresponds to a combination of the Laplace operator in space with the sum of two time-fractional derivatives of orders α ∈]0, 1] and β ∈]1, 2] in the Caputo sense, while in the second case we have the Dirac operator in space combined with a sum of the two time-fractional derivatives, using a Witt basis in the framework of Clifford algebra. The second operator factorizes the first one (see [9]). Connections between Clifford analysis and fractional calculus were recently established in the study of the stationary fractional Dirac operator (see [10,11]) and in the study of the time-fractional diffusion-wave and telegraph operators (see [7,9]).…”

“…The second operator factorizes the first one (see [9]). Connections between Clifford analysis and fractional calculus were recently established in the study of the stationary fractional Dirac operator (see [10,11]) and in the study of the time-fractional diffusion-wave and telegraph operators (see [7,9]).…”

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

“…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.…”

First and second fundamental solutions of the time-fractional telegraph equation of order 2α

“…Regarding the multidimensional case, in [5] the authors discussed and derived the solution of the time-fractional telegraph equation in R n × R + with three kinds of nonhomogeneous boundary conditions, by the method of separation of variables. Very recently, in [9,10] the authors found several representations of the fundamental solution of the time-fractional telegraph and telegraph Dirac equations in R n × R + , in terms of integrals, special functions, and series.…”