Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph Equation

Ferreira, Milton; Rodrigues, M. Manuela; Vieira, Nelson

http://hdl.handle.net/10400.8/13320

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Authors

Ferreira, Milton

Rodrigues, M. Manuela

Vieira, Nelson

Abstract(s)

In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation where the time-fractional derivatives of orders α ∈]0,1] and β ∈]1,2] are in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS in the Fourier domain expressed in terms of a multivariate Mittag-Leffler function. 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 and of the fractional parameters α and β.

Keywords

time-fractional telegraph equation fundamental solution Caputo fractional derivative multivariate Mittag-Leffler functions H-function of two variables double Mellin-Barnes type integrals

URI

http://hdl.handle.net/10400.8/13320

Citation

Ferreira, M., Rodrigues, M.M. & Vieira, N. Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph Equation. FCAA 20, 868–894 (2017). https://doi.org/10.1515/fca-2017-0046