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

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


IMPACT FACTOR 2017: 2.865
5-year IMPACT FACTOR: 3.323

CiteScore 2017: 3.06

SCImago Journal Rank (SJR) 2017: 1.967
Source Normalized Impact per Paper (SNIP) 2017: 1.954

Mathematical Citation Quotient (MCQ) 2017: 0.98

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 15, Issue 1

Issues

Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

Yuri Luchko
  • Department of Mathematics, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353, Berlin, Germany
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2011-12-29 | DOI: https://doi.org/10.2478/s13540-012-0010-7

Abstract

In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.

MSC: Primary 26A33; Secondary 35B45, 35B50, 45K05

Keywords: Caputo fractional derivative; time-fractional diffusion equation; Mittag-Leffler function; initial-boundary-value problems; maximum principle; generalized solution; spectral method

  • [1] E.G. Bazhlekova, Duhamel-type representation of the solutions of nonlocal boundary value problems for the fractional diffusion-wave equation. In: “Transform Methods and Special Functions’ Varna 1996” (Proc. 2nd Int. Workshop), Bulgarian Academy of Sciences, Sofia (1998), 32–40. Google Scholar

  • [2] K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer-Verlag, Heidelberg (2010). Google Scholar

  • [3] J.L.A. Dubbeldam, A. Milchev, V.G. Rostiashvili, and T.A. Vilgis, Polymer translocation through a nanopore: A showcase of anomalous diffusion. Physical Review E 76 (2007), 010801 (R). http://dx.doi.org/10.1103/PhysRevE.76.010801CrossrefGoogle Scholar

  • [4] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011). Web of ScienceGoogle Scholar

  • [5] A. Freed, K. Diethelm, and Yu. Luchko, Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus. NASA’s Glenn Research Center, Ohio (2002). Google Scholar

  • [6] R. Gorenflo, Yu. Luchko and S. Umarov, On the Cauchy and multi-point problems for partial pseudo-differential equations of fractional order. Fract. Calc. Appl. Anal. 3, No 3 (2000), 249–277; http://www.math.bas.bg/~fcaa Google Scholar

  • [7] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000). Google Scholar

  • [8] J. Kemppainen, Existence and uniqueness of the solution for a time-fractional diffusion equation. Fract. Calc. Appl. Anal. 14, No 3 (2011), 411–418; DOI: 10.2478/s13540-011-0025-5, hfill http://www.springerlink.com/content/1311-0454/14/3/ http://dx.doi.org/10.2478/s13540-011-0025-5CrossrefGoogle Scholar

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

  • [10] R. Klages, G. Radons, and I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim (2008). Google Scholar

  • [11] Yu. Luchko, Initial-boundary-value problems for the generalized multiterm time-fractional diffusion equation. J. Math. Anal. Appl. 374, No 2 (2011), 538–548. http://dx.doi.org/10.1016/j.jmaa.2010.08.048CrossrefGoogle Scholar

  • [12] Yu. Luchko and A. Punzi, Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations. Intern. Journal on Geomathematics 1, No 2 (2011), 257–276. http://dx.doi.org/10.1007/s13137-010-0012-8CrossrefGoogle Scholar

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

  • [14] Yu. 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] Yu. Luchko, Boundary value problems for the generalized timefractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12, No 4 (2009), 409–422; http://www.math.bas.bg/~fcaa Google Scholar

  • [16] Yu. Luchko, Operational method in fractional calculus. Fract. Calc. Appl. Anal. 2,No 4 (1999), 463–489; http://www.math.bas.bg/~fcaa Google Scholar

  • [17] Yu. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives. Acta Mathematica Vietnamica 24 (1999), 207–233. Google Scholar

  • [18] R.L. Magin, Fractional Calculus in Bioengineering: Part1, Part 2 and Part 3. Critical Reviews in Biomedical Engineering 32 (2004), 1–104, 105–193, 195–377. http://dx.doi.org/10.1615/CritRevBiomedEng.v32.10CrossrefGoogle Scholar

  • [19] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticy. World Scientific, Singapore (2010). http://dx.doi.org/10.1142/9781848163300CrossrefGoogle Scholar

  • [20] F. Mainardi, Fractional relaxation-oscillation and fractional diffusionwave phenomena. Chaos, Solitons and Fractals 7(1996), 1461–1477. http://dx.doi.org/10.1016/0960-0779(95)00125-5CrossrefGoogle Scholar

  • [21] M.M. Meerschaert, E. Nane, and P. Vellaisamy, Fractional Cauchy problems on bounded domains. Ann. Probab. 37, No 3 (2009), 979–1007. http://dx.doi.org/10.1214/08-AOP426Web of ScienceCrossrefGoogle Scholar

  • [22] R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A. Math. Gen. 37 (2004), R161–R208. http://dx.doi.org/10.1088/0305-4470/37/31/R01CrossrefGoogle Scholar

  • [23] R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations. Phys. A 278 (2000), 107–125. http://dx.doi.org/10.1016/S0378-4371(99)00503-8CrossrefGoogle Scholar

  • [24] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999). Google Scholar

  • [25] A.V. Pskhu, Partial Differential Equations of Fractional Order. Nauka, Moscow (2005) (in Russian). Google Scholar

  • [26] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). Google Scholar

  • [27] P. Pucci and J. Serrin, The Maximum Principle. Birkhäuser, Basel (2007). Google Scholar

  • [28] V.V. Uchaikin, Method of Fractional Derivatives. Artishok, Ul’janovsk (2008) (in Russian). Google Scholar

  • [29] V.S. Vladimirov, Equations of Mathematical Physics, Nauka, Moscow (1971) (in Russian). Google Scholar

About the article

Published Online: 2011-12-29

Published in Print: 2012-03-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 15, Issue 1, Pages 141–160, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-012-0010-7.

Export Citation

© 2012 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.

[3]
Hulya Kodal Sevindir and Ali Demir
Advances in Mathematical Physics, 2018, Volume 2018, Page 1
[5]
Bothayna S.H. Kashkari and Muhammed I. Syam
Results in Physics, 2018, Volume 9, Page 560
[6]
Łukasz Płociniczak and Mateusz Świtała
Journal of Mathematical Analysis and Applications, 2018, Volume 462, Number 2, Page 1425
[7]
J. L. Gracia, E. O’Riordan, and M. Stynes
Journal of Scientific Computing, 2018
[8]
[9]
Арсен Владимирович Псху and Arsen Vladimirovich Pskhu
Известия Российской академии наук. Серия математическая, 2017, Volume 81, Number 6, Page 158
[10]
Francisco J. Gaspar and Carmen Rodrigo
SIAM Journal on Scientific Computing, 2017, Volume 39, Number 4, Page A1201
[11]
Martin Stynes, Eugene O'Riordan, and José Luis Gracia
SIAM Journal on Numerical Analysis, 2017, Volume 55, Number 2, Page 1057
[13]
Kevin Burrage, Angelamaria Cardone, Raffaele D'Ambrosio, and Beatrice Paternoster
Applied Numerical Mathematics, 2017, Volume 116, Page 82
[15]
Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, and Van Thinh Nguyen
Evolution Equations and Control Theory, 2016, Volume 6, Number 1, Page 111
[16]
Yuri Luchko and Masahiro Yamamoto
Fractional Calculus and Applied Analysis, 2016, Volume 19, Number 3
[17]
Huy Tuan Nguyen, Dinh Long Le, and Van Thinh Nguyen
Applied Mathematical Modelling, 2016, Volume 40, Number 19-20, Page 8244
[18]
Zhenhai Liu, Shengda Zeng, and Yunru Bai
Fractional Calculus and Applied Analysis, 2016, Volume 19, Number 1
[19]
Rudolf Gorenflo, Yuri Luchko, and Masahiro Yamamoto
Fractional Calculus and Applied Analysis, 2015, Volume 18, Number 3
[20]
Ali Demir, Fatma Kanca, and Ebru Ozbilge
Boundary Value Problems, 2015, Volume 2015, Number 1
[21]
Živorad Tomovski and Trifce Sandev
Applied Mathematics and Computation, 2012, Volume 218, Number 20, Page 10022
[22]
Emilia Bazhlekova
Fractional Calculus and Applied Analysis, 2012, Volume 15, Number 2
[23]
Wen Chen, Jianjun Zhang, and Jinyang Zhang
Fractional Calculus and Applied Analysis, 2013, Volume 16, Number 1
[24]
Mohammed Al-Refai and Yuri Luchko
Fractional Calculus and Applied Analysis, 2014, Volume 17, Number 2
[25]
Ebru Ozbilge and Ali Demir
Journal of Inequalities and Applications, 2015, Volume 2015, Number 1
[27]
Jun-Sheng Duan, Shou-Zhong Fu, and Zhong Wang
Integral Transforms and Special Functions, 2014, Volume 25, Number 3, Page 220
[28]
Salİh Tatar, Ramazan Tınaztepe, and Süleyman Ulusoy
Applicable Analysis, 2016, Volume 95, Number 1, Page 1
[29]
Salih Tatar and Süleyman Ulusoy
Applicable Analysis, 2015, Volume 94, Number 11, Page 2233
[30]
Ebru Ozbilge and Ali Demir
Boundary Value Problems, 2014, Volume 2014, Number 1, Page 134
[31]
Shahrokh Esmaeili and Roberto Garrappa
International Journal of Computer Mathematics, 2015, Volume 92, Number 5, Page 980
[32]
Mohamed A. Hajji, Qasem M. Al-Mdallal, and Fathi M. Allan
Journal of Computational Physics, 2014, Volume 272, Page 550
[33]
H. Ye, F. Liu, V. Anh, and I. Turner
Applied Mathematics and Computation, 2014, Volume 227, Page 531

Comments (0)

Please log in or register to comment.
Log in