- Research Article
- Open Access

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

- Y.Z. Povstenko
^{1, 2}Email author

**2011**:930297

https://doi.org/10.1155/2011/930297

© Y. Z. Povstenko. 2011

**Received:**8 December 2010**Accepted:**6 February 2011**Published:**28 February 2011

## 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 , the Hankel transform with respect to the radial coordinate , the finite Fourier transform with respect to the angular coordinate , and the exponential Fourier transform with respect to the spatial coordinate . Numerical results are illustrated graphically.

## Keywords

- Cauchy Problem
- Fundamental Solution
- Integrodifferential Equation
- Percolation Cluster
- Angular Coordinate

## 1. Introduction

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 space-dimension was obtained by Mainardi [11]. Wyss [12] obtained the solutions to the Cauchy problem in terms of -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 [16–27]. In this paper, we investigate solutions to (1.1) in an infinite medium in cylindrical coordinates in the case of three spatial coordinates , , and .

## 2. Statement of the Problem

where is the transform variable.

Now, we investigate the fundamental solutions , , and .

## 3. Fundamental Solution to the First Cauchy Problem

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

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

The fundamental solution (3.8) was considered in [25] for .

## 4. Fundamental Solution to the Second Cauchy Problem

It is evident that (3.7) is the particular case of (4.4) corresponding to .

## 5. Fundamental Solution to the Source Problem

## 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 interpolates the Helmholtz and diffusion equation. In the case , the time-fractional diffusion-wave equation interpolates the standard diffusion equation and the classical wave equation.

For , 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 ( ).

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 at some distance from the point of the delta pulse application ( in Figures 3 and 7) features only humps with no singularity.

## Authors’ Affiliations

## References

- Mainardi F:
**Fractional calculus: some basic problems in continuum and statistical mechanics.**In*Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), CISM Courses and Lectures*.*Volume 378*. Edited by: Carpinteri A, Mainardi F. Springer, Vienna, Austria; 1997:291-348.View ArticleGoogle Scholar - Pękalski A, Sznajd-Weron K (Eds):
*Anomalous Diffusion: From Basics to Applications*. Springer, Berlin, Germany; 1999.MATHGoogle Scholar - Metzler R, Klafter J:
**The random walk's guide to anomalous diffusion: a fractional dynamics approach.***Physics Reports*2000,**339**(1):1-77. 10.1016/S0370-1573(00)00070-3MathSciNetView ArticleMATHGoogle Scholar - Zaslavsky GM:
**Chaos, fractional kinetics, and anomalous transport.***Physics Reports*2002,**371**(6):461-580. 10.1016/S0370-1573(02)00331-9MathSciNetView ArticleMATHGoogle Scholar - West BJ, Bologna M, Grigolini P:
*Physics of Fractal Operators, Institute for Nonlinear Science*. Springer, New York, NY, USA; 2003:x+354.View ArticleGoogle Scholar - Metzler R, Klafter J:
**The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics.***Journal of Physics A*2004,**37**(31):R161-R208. 10.1088/0305-4470/37/31/R01MathSciNetView ArticleMATHGoogle Scholar - Povstenko YZ:
**Fractional heat conduction equation and associated thermal stress.***Journal of Thermal Stresses*2005,**28**(1):83-102.MathSciNetView ArticleGoogle Scholar - Magin RL:
*Fractional Calculus in Bioengineering*. Begell House Publishers, Connecticut, Mass, USA; 2006.Google Scholar - Gafiychuk VV, Datsko BYo:
**Pattern formation in a fractional reaction-diffusion system.***Physica A*2006,**365**(2):300-306. 10.1016/j.physa.2005.09.046View ArticleGoogle Scholar - Uchaikin VV:
*Method of Fractional Derivatives*. Artishock, Ulyanovsk, Russia; 2008.Google Scholar - Mainardi F:
**The fundamental solutions for the fractional diffusion-wave equation.***Applied Mathematics Letters*1996,**9**(6):23-28. 10.1016/0893-9659(96)00089-4MathSciNetView ArticleMATHGoogle Scholar - Wyss W:
**The fractional diffusion equation.***Journal of Mathematical Physics*1986,**27**(11):2782-2785. 10.1063/1.527251MathSciNetView ArticleMATHGoogle Scholar - Schneider WR, Wyss W:
**Fractional diffusion and wave equations.***Journal of Mathematical Physics*1989,**30**(1):134-144. 10.1063/1.528578MathSciNetView ArticleMATHGoogle Scholar - Fujita Y:
**Integrodifferential equation which interpolates the heat equation and the wave equation.***Osaka Journal of Mathematics*1990,**27**(2):309-321.MathSciNetMATHGoogle Scholar - Hanyga A:
**Multidimensional solutions of time-fractional diffusion-wave equations.***The Royal Society of London. Proceedings. Series A*2002,**458**(2020):933-957. 10.1098/rspa.2001.0904MathSciNetView ArticleMATHGoogle Scholar - Povstenko YZ:
**Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation.***International Journal of Engineering Science*2005,**43**(11-12):977-991. 10.1016/j.ijengsci.2005.03.004MathSciNetView ArticleMATHGoogle Scholar - Povstenko YZ:
**Two-dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation.***International Journal of Solids and Structures*2007,**44**(7-8):2324-2348. 10.1016/j.ijsolstr.2006.07.008MathSciNetView ArticleMATHGoogle Scholar - Achar BNN, Hanneken JW:
**Fractional radial diffusion in a cylinder.***Journal of Molecular Liquids*2004,**114**(1-3):147-151. 10.1016/j.molliq.2004.02.012View ArticleGoogle Scholar - Povstenko YZ:
**Fractional radial diffusion in a cylinder.***Journal of Molecular Liquids*2008,**137**(1-3):46-50. 10.1016/j.molliq.2007.03.006View ArticleMATHGoogle Scholar - Özdemir N, Karadeniz D:
**Fractional diffusion-wave problem in cylindrical coordinates.***Physics Letters A*2008,**372**(38):5968-5972. 10.1016/j.physleta.2008.07.054MathSciNetView ArticleMATHGoogle Scholar - Özdemir N, Agrawal OP, Karadeniz D, Iskender BB:
**Axis-symmetric fractional diffusion-wave problem: part I—analysis.***Proceedings of the 6th Euromech Nonlinear Dynamics Conference (ENOC '08), June-July 2008, Saint Petrsburg, Russia*Google Scholar - Lenzi EK, da Silva LR, Silva AT, Evangelista LR, Lenzi MK:
**Some results for a fractional diffusion equation with radial symmetry in a confined region.***Physica A*2009,**388**(6):806-810. 10.1016/j.physa.2008.11.030View ArticleGoogle Scholar - Özdemir N, Karadeniz D, İskender BB:
**Fractional optimal control problem of a distributed system in cylindrical coordinates.***Physics Letters A*2009,**373**(2):221-226. 10.1016/j.physleta.2008.11.019MathSciNetView ArticleMATHGoogle Scholar - Özdemir N, Agrawal OP, Karadeniz D, Iskender BB:
**Fractional optimal control problem of an axis-symmetric diffusion-wave propagation.***Physica Scripta T*2009,**136:**-5.Google Scholar - Jiang X, Xu M:
**The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems.***Physica A*2010,**389**(17):3368-3374. 10.1016/j.physa.2010.04.023MathSciNetView ArticleGoogle Scholar - Lenzi EK, Rossato R, Lenzi MK, da Silva LR, Gonçalves G:
**Fractional diffusion equation and external forces: solutions in a confined region.***Zeitschrift für Naturforschung Section A*2010,**65**(5):423-430.Google Scholar - Qi H, Liu J:
**Time-fractional radial diffusion in hollow geometries.***Meccanica*2010,**45**(4):577-583. 10.1007/s11012-009-9275-2MathSciNetView ArticleMATHGoogle Scholar - Gorenflo R, Mainardi F:
**Fractional calculus: integral and differential equations of fractional order.**In*Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), CISM Courses and Lectures*.*Volume 378*. Edited by: Carpinteri A, Mainardi F. Springer, Vienna, Austria; 1997:223-276.View ArticleGoogle Scholar - Kilbas AA, Srivastava HM, Trujillo JJ:
*Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies*.*Volume 204*. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar - Podlubny I:
*Fractional Differential Equations, Mathematics in Science and Engineering*.*Volume 198*. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar - Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG:
*Higher Transcendental Functions. Vol. III*. McGraw-Hill, New York, NY, USA; 1955:xvii+292.MATHGoogle Scholar - Povstenko YZ:
**Fundamental solutions to three-dimensional diffusion-wave equation and associated diffusive stresses.***Chaos, Solitons & Fractals*2008,**36**(4):961-972. 10.1016/j.chaos.2006.07.031MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.