Advances in Difference Equations

, 2011:930297 | Cite as

Solutions to Time-Fractional Diffusion-Wave Equation in Cylindrical Coordinates

Open Access
Research Article
Part of the following topical collections:
  1. Fractional Models and their Applications

Abstract

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 Open image in new window , the Hankel transform with respect to the radial coordinate Open image in new window , the finite Fourier transform with respect to the angular coordinate Open image in new window , and the exponential Fourier transform with respect to the spatial coordinate Open image in new window . Numerical results are illustrated graphically.

Keywords

Cauchy Problem Fundamental Solution Integrodifferential Equation Percolation Cluster Angular Coordinate 

1. Introduction

The time-fractional diffusion-wave equation

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, 2, 3, 4, 5, 6, 7, 8, 9, 10] and references therein).

The fundamental solution for the fractional diffusion-wave equation in one space-dimension was obtained by Mainardi [11]. Wyss [12] obtained the solutions to the Cauchy problem in terms of Open image in new window -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 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, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In this paper, we investigate solutions to (1.1) in an infinite medium in cylindrical coordinates in the case of three spatial coordinates Open image in new window , Open image in new window , and Open image in new window .

2. Statement of the Problem

Consider the time-fractional diffusion-wave equation with a source term in cylindrical coordinates
The initial conditions are prescribed:
In (2.1), we use the Caputo fractional derivative [28, 29, 30]
where Open image in new window is the gamma function. For its Laplace transform rule, the Caputo fractional derivative requires the knowledge of the initial values of the function Open image in new window and its integer derivatives of order Open image in new window :

where Open image in new window is the transform variable.

The solution to the initial-value problem (2.1)-(2.2) can be written in the following form:

Now, we investigate the fundamental solutions Open image in new window , Open image in new window , and Open image in new window .

3. Fundamental Solution to the First Cauchy Problem

In the case of the first Cauchy problem, the initial value of a sought-for function is prescribed. Hence,

The two-dimensional Dirac delta function Open image in new window after passing to the polar coordinates takes the form Open image in new window , but for the sake of simplicity, we have omitted the multiplier Open image in new window in the solution (2.5) as well as Open image in new window in (3.2). In the initial condition (3.2), we have introduced the constant multiplier Open image in new window to obtain the nondimensional quantity Open image in new window (see (3.10)).

The solution is found using the Laplace transform with respect to time Open image in new window , the Hankel transform with respect to the radial coordinate Open image in new window , the finite Fourier transform with respect to the angular coordinate Open image in new window , and the exponential Fourier transform with respect to the spatial coordinate Open image in new window . In the transforms domain we get

where Open image in new window is the Bessel function of the first kind of order Open image in new window , the asterisk indicates the transforms, Open image in new window is the Laplace transform variable, Open image in new window is the Hankel transform variable, Open image in new window is exponential Fourier transform variable, and the integer Open image in new window is finite Fourier transform variable.

Inversion of integral transforms gives
where the prime denotes that the term corresponding to Open image in new window in the sum should be multiplied by 1/2. In (3.5), Open image in new window is the Mittag-Leffler function [28, 29, 30, 31]
The essential role of the Mittag-Leffler function in fractional calculus results from the following formula for the inverse Laplace transform [28, 29, 30]:
If the solution does not depend on the coordinate Open image in new window , then

The fundamental solution (3.8) was considered in [25] for Open image in new window .

In the case when the solution does not also depend on the angular coordinate Open image in new window , we get [17]
In calculations, we have introduced nondimensional quantities:

4. Fundamental Solution to the Second Cauchy Problem

In the case of the second Cauchy problem, the initial value of the time derivative of a sought-for function is prescribed, and for the corresponding fundamental solution we have
In this instance, the fundamental solution is expressed as
where Open image in new window is the generalized Mittag-Leffler function in two parameters Open image in new window and Open image in new window [29, 30, 31, 32]:
We have used the following formula for the inverse Laplace transform [29, 30, 31]

It is evident that (3.7) is the particular case of (4.4) corresponding to Open image in new window .

If the solution does not depend on the coordinate Open image in new window , then
In the case of axial symmetry [17],
Figures 5, 6, 7, and 8 show dependence of fundamental solution (4.2) on coordinates Open image in new window , Open image in new window , and Open image in new window , where

5. Fundamental Solution to the Source Problem

Consider the time-fractional diffusion-wave equation with a source term under zero initial conditions:
The solution is obtained using the integral transform technique and reads
If dependence of solution on the coordinate Open image in new window is not taken into account, then
In the case of axial symmetry [17],
Dependence of the solution (5.2) on the coordinates Open image in new window , Open image in new window , and Open image in new window is depicted in Figures 9, 10, and 11 with

6. Discussion

The solutions to the Cauchy and source problems for time-fractional diffusion-wave equation have been found in cylindrical coordinates. The considered equation in the case Open image in new window interpolates the Helmholtz and diffusion equation. In the case Open image in new window , the time-fractional diffusion-wave equation interpolates the standard diffusion equation and the classical wave equation.

For Open image in new window , the solutions to the fractional diffusion-wave equation feature propagating humps, underlining the proximity to the standard wave equation in contrast to the shape of curves describing the subdiffusion regime ( Open image in new window ).

For better understanding of behavior of solutions, it is worthwhile to compare the obtained results with those for delta pulse applied at the origin investigated in [32]. The Mittag-Leffler functions arising in (3.5), (4.2), and (5.2) for large values of argument are represented as

Such asymptotic results in singularities of the solution to the first and the second Cauchy problems at the point of application of the delta pulse, whereas the solution to the source problem does not have singularity. Dependence of the solution on the angular coordinate Open image in new window at some distance from the point of the delta pulse application ( Open image in new window in Figures 3 and 7) features only humps with no singularity.

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Copyright information

© Y. Z. Povstenko. 2011

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceJan Długosz UniversityCzęstochowaPoland
  2. 2.Department of Computer ScienceEuropean University of Informatics and Economics (EWSIE)WarsawPoland

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