Solutions to Time-Fractional Diffusion-Wave Equation in Cylindrical Coordinates
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Research Article Solutions to Time-Fractional Diffusion-Wave Equation in Cylindrical Coordinates Y. Z. Povstenko1, 2 1 2
Institute of Mathematics and Computer Science, Jan Długosz University, 42200 Cze¸stochowa, Poland Department of Computer Science, European University of Informatics and Economics (EWSIE), 03741 Warsaw, Poland
Correspondence should be addressed to Y. Z. Povstenko, [email protected] Received 8 December 2010; Accepted 6 February 2011 Academic Editor: J. J. Trujillo Copyright q 2011 Y. Z. Povstenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Nonaxisymmetric solutions to time-fractional diffusion-wave equation with a source term in cylindrical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time t, the Hankel transform with respect to the radial coordinate r, the finite Fourier transform with respect to the angular coordinate ϕ, and the exponential Fourier transform with respect to the spatial coordinate z. Numerical results are illustrated graphically.
1. Introduction The time-fractional diffusion-wave equation ∂α u aΔu ∂tα
1.1
is a mathematical model of important physical phenomena ranging from amorphous, colloid, glassy, and porous materials through fractals, percolation clusters, random, and disordered media to comb structures, dielectrics and semiconductors, polymers, and biological systems see 1–10 and references therein. The fundamental solution for the fractional diffusion-wave equation in one spacedimension was obtained by Mainardi 11. Wyss 12 obtained the solutions to the Cauchy problem in terms of H-functions using the Mellin transform. Schneider and Wyss 13 converted the diffusion-wave equation with appropriate initial conditions into the integrodifferential equation and found the corresponding Green functions in terms of Fox
2
Advances in Difference Equations 1.25 α 0.5
1 0.75
α1
0.5
Gf
0.25 0 −0.25 α 1.5
−0.5 −0.75
α 1.7
−1 −1.25
0
0.5
1
1.5
2
2.5
r/ρ
Figure 1: Dependence of nondimensional fundamental solution Gf r, ϕ, z, ρ, φ, ζ, t on the radial coordinate r for φ 0, ζ 0, z 0, ϕ 0, and κ 0.5.
functions. Fujita 14 treated integrodifferential equation which interpolates the diffusion equation and the wave equation. Hanyga 15 studied Green functions and propagator functions in one, two, and three dimensions. Previously, in studies concerning time-fractional diffusion-wave equation in cylindrical coordinates, only one or two spatial coordinates have been considered 16–27. In this paper, we investigate solutions to 1.1 in an infinite medium in cylindrical coordinates in the case of three spatial coordinates r, ϕ, and z.
2. Statement of the Problem Consider the time-fractional diffusion-wave equation with a source term in cylindrical coordinates ∂α u ∂2 u 1 ∂u 1 ∂2 u ∂2 u Q r, ϕ, z, t ,
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