Time-fractional telegraph equation of distributed order in higher dimensions with Hilfer fractional derivatives

Abstract

In this paper, we consider the time-fractional telegraph equation of distributed order in higher spatial dimensions, where the time derivative is in the sense of Hilfer, thus interpolating between the Riemann-Liouville and the Caputo fractional derivatives. By employing the techniques of the Fourier, Laplace, and Mellin transforms, we obtain a representation of the solution of the Cauchy problem associated with the equation in terms of convolutions involving functions that are Laplace integrals of Fox H-functions. Fractional moments of the first fundamental solution are computed and for the special case of double-order distributed it is analyzed in detail the asymptotic behavior of the second-order moment, by application of the Tauberian Theorem. Finally, we exhibit plots of the variance showing its behavior for short and long times, and for different choices of the parameters along small dimensions.

Keywords

• time-fractional telegraph equation
• distributed order
• hilfer fractional derivative
• integral transforms
• fox h-function
• fractional moments
• tauberian theorem