International Journal of Differential Equations

On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis

D. Goos, G. Reyero, S. Roscani, and E. Santillan Marcus

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the time-fractional derivative in the Caputo sense of order α(0, 1). Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when α1 of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation.

Article information

Source
Int. J. Differ. Equ., Volume 2015 (2015), Article ID 439419, 14 pages.

Dates
Received: 10 July 2015
Accepted: 31 August 2015
First available in Project Euclid: 20 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1484881444

Digital Object Identifier
doi:10.1155/2015/439419

Mathematical Reviews number (MathSciNet)
MR3413056

Zentralblatt MATH identifier
1336.35358

Citation

Goos, D.; Reyero, G.; Roscani, S.; Santillan Marcus, E. On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis. Int. J. Differ. Equ. 2015 (2015), Article ID 439419, 14 pages. doi:10.1155/2015/439419. https://projecteuclid.org/euclid.ijde/1484881444


Export citation

References

  • J. R. Cannon, The One-Dimensional Heat Equation, Cambridge University Press, Cambridge, UK, 1984.
  • A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964.MR0181836
  • D. V. Widder, The Heat Equation, Academic Press, New York, NY, USA, 1975.MR0466967
  • K. Diethelm, The Analysis of Fractional Differential Equations: An Application Oriented Exposition Using Differential Operators of Caputo Typ, Springer, 2010.MR2680847
  • Y. Fujita, “Integrodifferential equations which interpolates the heat equation and a wave equation,” Osaka Journal of Mathematics, vol. 27, pp. 309–321, 1990.
  • A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.MR2218073
  • Y. Luchko, “Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1766–1772, 2010.MR2595950
  • F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001.MR1829592
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.MR1658022
  • W. Wyss, “The fractional diffusion equation,” Journal of Mathematical Physics, vol. 27, no. 11, pp. 2782–2785, 1986.MR861345
  • F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010.MR2676137
  • S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211–255, 2004.MR2047909
  • V. R. Voller, F. Falcini, and R. Garra, “Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects,” Physical Review E, vol. 87, no. 4, Article ID 042401, 2013.
  • Y. Povstenko, “Fractional heat conduction in a semi-infinite composite body,” Communications in Applied and Industrial Mathematics, vol. 6, no. 1, p. e-482, 2014.
  • F. Mainardi and M. Tomirotti, “On a special function arising in the time fractional diffusion-wave equation,” in Transform Methods and Special Functions, Sofia 1994, P. Rusev, I. Dimovski, and V. Kiryakova, Eds., pp. 171–183, Science Culture Technology, Singapore, 1995.
  • E. M. Wright, “The generalized Bessel function of order greater than one,” Quarterly Journal of Mathematics. Oxford. Second Series, vol. 11, pp. 36–48, 1940.MR0003875
  • S. Roscani and E. Santillan Marcus, “Two equivalent Stefan's problems for the time-fractional diffusion equation,” Fractional Calculus and Applied Analysis, vol. 16, no. 4, pp. 802–815, 2013.MR3124337
  • J. Liu and M. Xu, “Some exact solutions to Stefan problems with fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 351, no. 2, pp. 536–542, 2009.MR2473959
  • S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser, 2004.MR2093219
  • R. Gorenflo, Y. Luchko, and F. Mainardi, “Analytical properties and applications of the Wright function,” Fractional Calculus & Applied Analysis, vol. 2, no. 4, pp. 383–414, 1999.MR1752379
  • H. Brézis, Análisis Funcional Teoría y aplicaciones. Versión española de Juan Ramón Esteban, Alianza Editorial, Madrid, Spain, 1984.
  • F. Mainardi, “On the initial value problem for the fractional diffusion-wave equation,” in Proceeding of VIIth International Conference on Waves and Stability in Continuous Media WASCOM, Bologna, Italy, 4–7 October 1993, S. Rionero and T. Ruggeri, Eds., pp. 246–251, World Scientific, Singapore, 1994.
  • Y. Lin and C. Xu, “Finite difference/spectral approximations for the time-fractional diffusion equation,” Journal of Computational Physics, vol. 225, no. 2, pp. 1533–1552, 2007.MR2349193
  • W. Chen, L. Ye, and H. Sun, “Fractional diffusion equations by the Kansa method,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1614–1620, 2010.MR2595933
  • M. Javidi and B. Ahmad, “Numerical solution of fractional partial differential equations by numerical Laplace inversion technique,” Advances in Difference Equations, vol. 2013, article 375, 2013.MR3337229 \endinput