Eigenfunctions of the time‐fractional diffusion‐wave operator

Abstract

In this paper, we present some new integral and series representations for the eigenfunctions of the multidimensional time-fractional diffusion-wave operator with the time-fractional derivative of order ∈]1, 2[ defined in the Caputo sense. The integral representations are obtained in form of the inverse Fourier-Bessel transform and as a double contour integrals of the Mellin-Barnes type. Concerning series expansions, the eigenfunctions are expressed as the double generalized hypergeometric series for any ∈]1, 2[ and as Kampé de Fériet and Lauricella series in two variables for the rational values of. The limit cases = 1 (diffusion operator) and = 2 (wave operator) as well as an intermediate case = 3 2 are studied in detail. Finally, we provide several plots of the eigenfunctions to some selected eigenvalues for different particular values of the fractional derivative order and the spatial dimension n.

Time-fractional diffusion equation with $$\psi $$-Hilfer derivative

“…where f is the Fourier transform of the function f . Inverting the ψ-Laplace transform, taking into account (11) and using (10), we have…”

“…Furthermore, fractional derivatives give us a more accurate interpretation of anomalous diffusion phenomena when compared to the corresponding integer derivatives. Our motivation is to continue our previous work related to this subject (see [8][9][10][11]31]) and to present a unified approach to the time-fractional diffusion equation using the ψ-Hilfer derivative. As a by-product, we obtain the solution of the Cauchy problem associated with our equation in terms of convolution integrals involving Fox H-functions.…”