Eigenfunctions of the time‐fractional diffusion‐wave operator - Ferreira, Milton (2022)
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 β ∈]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 β ∈]1, 2[ and as Kampé de Fériet and Lauricella series in two variables for the rational values of β. The limit cases =1 (diffusion operator) and =2 (wave operator) as well as an intermediate case =32 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 β and the spatial dimension n.