Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

6 Issues per year


The journal celebrates now its 20 years!

IMPACT FACTOR 2016: 2.034
5-year IMPACT FACTOR: 2.359

CiteScore 2016: 2.18

SCImago Journal Rank (SJR) 2016: 1.372
Source Normalized Impact per Paper (SNIP) 2016: 1.492

Mathematical Citation Quotient (MCQ) 2016: 0.61

Online
ISSN
1314-2224
See all formats and pricing
More options …

Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder

Yuriy Povstenko
  • Institute of Mathematics and Computer Science, Jan Długosz University in Częstochowa, al.Armii Krajowej 13/15, 42-200, Częstochowa, Poland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2011-06-29 | DOI: https://doi.org/10.2478/s13540-011-0026-4

Abstract

The time-fractional diffusion-wave equation is considered in an infinite cylinder in the case of three spatial coordinates r, ϕ and z. The Caputo fractional derivative of the order 0 < α ≤ 2 is used. Several examples of problems with Dirichlet and Neumann boundary conditions at a surface of the cylinder are solved using the integral transforms technique. Numerical results are illustrated graphically.

MSC: Primary 35R11; Secondary 26A33, 35K05, 45K05

Keywords: fractional calculus; Mittag-Leffler functions; fractional partial differential equations; diffusion-wave equation

  • [1] H. Berens, U. Westphal, A Cauchy problem for a generalized wave equation. Acta Sci. Math. (Szeged) 29, No 1–2 (1968), 93–106. Google Scholar

  • [2] Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27, No 2 (1990), 309–321. Google Scholar

  • [3] A.S. Galitsyn, A.N. Zhukovsky, Integral Transforms and Special Functions in Heat Conduction Problems. Kiev, Naukova Dumka (1976) (In Russian). Google Scholar

  • [4] R. Gorenflo, F. Mainardi, Fractional oscillations and Mittag-Leffler functions. Preprint PR-A-96-14, Fachbereich Mathematik und Informatik, Freie Universität Berlin (1996), 1–22. Google Scholar

  • [5] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order. In: A. Carpinteri, F. Mainardi (Eds.): Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien (1997), 223–276. Google Scholar

  • [6] R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, No 1–4 (2002), 129–143. http://dx.doi.org/10.1023/A:1016547232119CrossrefGoogle Scholar

  • [7] A. Hanyga, Multidimensional solutions of space-time-fractional diffusion equations. Proc. R. Soc. Lond. A 458, No 2018 (2002), 429–450. http://dx.doi.org/10.1098/rspa.2001.0893CrossrefGoogle Scholar

  • [8] A. Hanyga, Multidimensional solutions of time-fractional diffusionwave equations. Proc. R. Soc. Lond. A 458, No 2020 (2002), 933–957. http://dx.doi.org/10.1098/rspa.2001.0904CrossrefGoogle Scholar

  • [9] X.Y. Jiang, M.Y. Xu, The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems. Physica A 389, No 17 (2010), 3368–3374. http://dx.doi.org/10.1016/j.physa.2010.04.023CrossrefWeb of ScienceGoogle Scholar

  • [10] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006). Google Scholar

  • [11] A.N. Kochubei, Fractional order diffusion. Diff. Equations 26, (1990), 485–492. Google Scholar

  • [12] E.K. Lenzi, L.R. da Silva, A.T. Silva, L.R. Evangelista, M.K. Lenzi, Some results for a fractional diffusion equation with radial symmetry in a confined region. Physica A 388, No 6 (2009), 806–810. http://dx.doi.org/10.1016/j.physa.2008.11.030CrossrefGoogle Scholar

  • [13] E.K. Lenzi, R. Rossato, M.K. Lenzi, L.R. da Silva, G. Gonçalves, Fractional diffusion equation and external forces: solutions in a confined region. Z. Naturforsch. 65a, No 5 (2010), 423–430. Google Scholar

  • [14] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351, No 1 (2009) 218–223. http://dx.doi.org/10.1016/j.jmaa.2008.10.018CrossrefGoogle Scholar

  • [15] Y. Luchko, Some uniqueness and existence results for the initialboundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59, No 5 (2010) 1766–1772. http://dx.doi.org/10.1016/j.camwa.2009.08.015CrossrefWeb of ScienceGoogle Scholar

  • [16] F. Mainardi, The fundamental solutions for the fractional diffusionwave equation. Appl. Math. Lett. 9, No 6 (1996), 23–28. http://dx.doi.org/10.1016/0893-9659(96)00089-4CrossrefGoogle Scholar

  • [17] F. Mainardi, Fractional relaxation-oscillation and fractional diffusionwave phenomena. Chaos, Solitons & Fractals 7, No 9 (1996), 1461–1477. http://dx.doi.org/10.1016/0960-0779(95)00125-5CrossrefGoogle Scholar

  • [18] B.N. Narahari Achar, J.W. Hanneken, Fractional radial diffusion in a cylinder. J. Mol. Liq. 114, No 1–3 (2004), 147–151. Google Scholar

  • [19] N. Özdemir, D. Karadeniz, Fractional diffusion-wave problem in cylindrical coordinates. Phys. Lett. A 372, No 38 (2008), 5968–5972. http://dx.doi.org/10.1016/j.physleta.2008.07.054CrossrefGoogle Scholar

  • [20] N. Özdemir, O.P. Agrawal, D. Karadeniz, B.B. Iskender, Axissymmetric fractional diffusion-wave problem: Part I — Analysis, In: 6th Euromech Nonlinear Dynamics Conference (ENOC-2008), Saint Petersburg, Russia, 30 June–4 July 2008. Google Scholar

  • [21] N. Özdemir, D. Karadeniz, B.B. Iskender, Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A 373, No 2 (2009), 221–226. http://dx.doi.org/10.1016/j.physleta.2008.11.019Web of ScienceCrossrefGoogle Scholar

  • [22] N. Özdemir, O.P. Agrawal, D. Karadeniz, B.B. Iskender, Fractional optimal control problem of an axis-symmetric diffusion-wave propagation. Phys. Scr. T 136 (2009), 014024. http://dx.doi.org/10.1088/0031-8949/2009/T136/014024Web of ScienceCrossrefGoogle Scholar

  • [23] I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999). Google Scholar

  • [24] A.D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman & Hall/CRC, Boca Raton (2002). Google Scholar

  • [25] Y.Z. Povstenko, Fractional heat conduction equation and associated thermal stresses. J. Thermal Stresses 28, No 1 (2005), 83–102. http://dx.doi.org/10.1080/014957390523741CrossrefGoogle Scholar

  • [26] Y.Z. Povstenko, Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation, Int. J. Engng Sci. 43, No 11–12 (2005), 977–991. http://dx.doi.org/10.1016/j.ijengsci.2005.03.004CrossrefGoogle Scholar

  • [27] Y.Z. Povstenko, Two-dimensional axisymmentric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation. Int. J. Solids Struct. 44, No 7-8 (2007), 2324–2348. http://dx.doi.org/10.1016/j.ijsolstr.2006.07.008CrossrefWeb of ScienceGoogle Scholar

  • [28] Y.Z. Povstenko, Fractional radial diffusion in a cylinder. J. Mol. Liq. 137, No 1–3 (2008), 46–50. http://dx.doi.org/10.1016/j.molliq.2007.03.006CrossrefGoogle Scholar

  • [29] Y.Z. Povstenko, Thermoelasticity which uses fractional heat conduction equation. J. Math. Sci. 162, No 2 (2009), 296–305. http://dx.doi.org/10.1007/s10958-009-9636-3CrossrefGoogle Scholar

  • [30] Y.Z. Povstenko, Theory of thermoelasticity based on the space-timefractional heat conduction equation. Phys. Scr. T 136 (2009) 014017. http://dx.doi.org/10.1088/0031-8949/2009/T136/014017CrossrefGoogle Scholar

  • [31] Y.Z. Povstenko, Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dyn. 59, No 4 (2010), 593–605. http://dx.doi.org/10.1007/s11071-009-9566-0CrossrefGoogle Scholar

  • [32] H. Qi, J. Liu, Time-fractional radial diffusion in hollow geometries. Meccanica 45, No 4 (2010), 577–583. http://dx.doi.org/10.1007/s11012-009-9275-2Web of ScienceCrossrefGoogle Scholar

  • [33] W.R. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30, No 1 (1989), 134–144. http://dx.doi.org/10.1063/1.528578CrossrefGoogle Scholar

  • [34] I.N. Sneddon, The Use of Integral Transforms. McGraw-Hill, New York (1972). Google Scholar

  • [35] A.A. Voroshilov, A.A. Kilbas, The Cauchy problem for the diffusionwave equation with the Caputo partial derivative. Diff. Equations 42, No 5 (2006), 638–649. http://dx.doi.org/10.1134/S0012266106050041CrossrefGoogle Scholar

  • [36] W. Wyss, The fractional diffusion equation. J. Math. Phys. 27, No 11 (1986), 2782–2785. http://dx.doi.org/10.1063/1.527251CrossrefGoogle Scholar

About the article

Published Online: 2011-06-29

Published in Print: 2011-09-01


Citation Information: Fractional Calculus and Applied Analysis, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-011-0026-4.

Export Citation

© 2011 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[2]
Yuriy Povstenko, Tamara Kyrylych, and Grażyna Rygał
Entropy, 2017, Volume 19, Number 5, Page 203
[3]
Lin Liu, Liancun Zheng, and Fawang Liu
Journal of Statistical Mechanics: Theory and Experiment, 2017, Volume 2017, Number 4, Page 043208
[4]
Lin Liu, Liancun Zheng, Fawang Liu, and Xinxin Zhang
Nonlinear Dynamics, 2017, Volume 89, Number 1, Page 213
[5]
Lin Liu, Liancun Zheng, and Fawang Liu
Journal of Molecular Liquids, 2017, Volume 233, Page 326
[7]
S. Chen and X.Y. Jiang
Computers & Mathematics with Applications, 2017, Volume 73, Number 6, Page 1172
[9]
Yuriy Povstenko
Fractional Calculus and Applied Analysis, 2016, Volume 19, Number 4
[10]
Yuriy Povstenko and Joanna Klekot
Boundary Value Problems, 2016, Volume 2016, Number 1
[12]
Yuriy Povstenko
Entropy, 2013, Volume 15, Number 10, Page 4122
[13]
[14]
Lin Liu, Liancun Zheng, Fawang Liu, and Xinxin Zhang
Communications in Nonlinear Science and Numerical Simulation, 2016, Volume 38, Page 45
[15]
[17]
Davood Rostamy and Kobra Karimi
Fractional Calculus and Applied Analysis, 2012, Volume 15, Number 4
[19]
Yuriy Povstenko
Fractional Calculus and Applied Analysis, 2014, Volume 17, Number 1
[22]
Y. Povstenko
The European Physical Journal Special Topics, 2013, Volume 222, Number 8, Page 1767
[24]
Yuriy Povstenko
Applied Mathematics and Computation, 2015, Volume 257, Page 327
[26]
Ming-Sheng Hu, Ravi P. Agarwal, and Xiao-Jun Yang
Abstract and Applied Analysis, 2012, Volume 2012, Page 1
[27]
Y. Z. Povstenko
International Journal of Differential Equations, 2012, Volume 2012, Page 1

Comments (0)

Please log in or register to comment.
Log in