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Publication
Eigenfunctions of the time‐fractional diffusion‐wave operator
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Abstract(s)
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 $\beta \in ]1,2[$ defined in the Caputo sense. The integral representations are obtained in form of the inverse Fourier-Bessel transform and as double contour integrals of the Mellin-Barnes type. Concerning series expansions, the eigenfunctions are expressed as the double generalized hypergeometric series for any $\beta \in ]1,2[$ and as Kamp\'{e} de F\'{e}riet and Lauricella series in two variables for the rational values of $\beta$. The limit cases $\beta=1$ (diffusion operator) and $\beta=2$ (wave operator) as well as an intermediate case $\beta=\frac{3}{2}$ 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 $\beta$ and the spatial dimension $n$.
Description
Keywords
Time-fractional diffusion-wave operator Eigenfunctions; Caputo fractional derivatives; Generalized hypergeometric series. Eigenfunctions Caputo fractional derivatives Generalized hypergeometric series
Citation
Ferreira, M., Luchko, Yu., Rodrigues, M.M., Vieira, N., Eigenfunctions of the time‐fractional diffusion‐wave operator. Math Meth Appl Sci. 2021; 44(2): 1713–1743. https://doi.org/10.1002/mma.6874
Publisher
Wiley Online Library