Nonlinear Dynamics

, Volume 29, Issue 1–4, pp 145–155 | Cite as

Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain

  • Om P. Agrawal

Abstract

A general solution is given for a fractional diffusion-wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. The finite sine transform technique is used to convert a fractional differential equation from a space domain to a wavenumber domain. Laplace transform is used to reduce the resulting equation to an ordinary algebraic equation. Inverse Laplace and inverse finite sine transforms are used to obtain the desired solutions. The response expressions are written in terms of the Mittag–Leffler functions. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Two examples are presented to show the application of the present technique. Results show that for fractional time derivatives of order 1/2 and 3/2, the system exhibits, respectively, slow diffusion and mixed diffusion-wave behaviors.

fractional derivative fractional order diffusion-wave equation Laplace transform bounded domain solution for fractional diffusion-wave equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mainardi, F., 'Fractional calculus: Some basic problems in continuum and statistical mechanics', in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds.), Springer-Verlag, New York, 1997, pp. 291–348.Google Scholar
  2. 2.
    Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.Google Scholar
  3. 3.
    Nigmatullin, R. R., 'To the theoretical explanation of the universal response', Physica B 123, 1984, 739–745.Google Scholar
  4. 4.
    Nigmatullin, R. R., 'Realization of the generalized transfer equation in a medium with fractal geometry', Physica B 133, 1986, 425–430.Google Scholar
  5. 5.
    Wyss, W., 'The fractional diffusion equation', Journal of Mathematical Physics 27, 1986, 2782–2785.Google Scholar
  6. 6.
    Schneider, W. R. and Wyss, W., 'Fractional diffusion and wave equations', Journal of Mathematical Physics 30, 1989, 134–144.Google Scholar
  7. 7.
    Sanz-serna, J. M., 'A numerical method for a partial integro-differential equation', SIAM Numerical Analysis 25, 1988, 319–327.Google Scholar
  8. 8.
    Fujita, Y., 'Integrodifferential equation which interpolates the heat equation and the wave equation', Osaka Journal of Mathematics 27, 1990, 309–321.Google Scholar
  9. 9.
    Fujita,Y., 'Integrodifferential equation which interpolates the heat equation and the wave equation. II', Osaka Journal of Mathematics 27, 1990, 797–804.Google Scholar
  10. 10.
    Ginoa, M., Cerbelli, S., and Roman, H. E., 'Fractional diffusion equation and relaxation in complex viscoelastic materials', Physica A 191, 1992, 449–453.Google Scholar
  11. 11.
    Roman, H. E. and Alemany, P. A., 'Continuous-time random walks and the fractional diffusion equation', Journal of Physics A 27, 1994, 3407–3410.Google Scholar
  12. 12.
    Mbodje, B. and Montseny, G., 'Boundary fractional derivative control of the wave equation', IEEE Transactions on Automatic Control 40, 1995,-378–382.Google Scholar
  13. 13.
    Mainardi, F. and Paradisi, P., 'Model of diffusive waves in viscoelasticity based on fractional calculus', in Proceedings of the IEEE Conference on Decision and Control, Vol. 5, O. R. Gonzales, IEEE, New York, 1997, pp. 4961–4966.Google Scholar
  14. 14.
    Gorenflo, R., Luchko, Y., and Mainardi, F., 'Wright functions as scale-invariant solutions of the diffusion-wave equation', Journal of Computational and Applied Mathematics 118, 2000, 175–191.Google Scholar
  15. 15.
    Agrawal, O. P., 'A general solution for the fourth-order fractional diffusion-wave equation', Fractional Calculation and Applied Analysis 3, 2000, 1–12.Google Scholar
  16. 16.
    Agrawal, O. P., 'A general solution for a fourth-order fractional Diffusion-wave equation defined in a bounded domain', Computers & Structures 79, 2001, 1497–1501.Google Scholar
  17. 17.
    Metzler, R. and Klafter, J., 'Boundary value problems for fractional Diffusion equations', Physica A 278, 2000, 107–125.Google Scholar
  18. 18.
    Hilfer, R., 'Fractional diffusion based on Riemann-Liouville fractional Derivatives', Journal of Physical Chemistry B 104, 2000, 3914–3917.Google Scholar
  19. 19.
    Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach, Longhorne, PA, 1993.Google Scholar
  20. 20.
    Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.Google Scholar
  21. 21.
    Gorenflo, R. and Mainardi, F., 'Fractional calculus: Integral and differential equations of fractional order', in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds.), Springer-Verlag, New York, 1997, pp. 223–276.Google Scholar
  22. 22.
    Rossikhin, Y. A. and Shitikova, M. V., 'Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids', Applied Mechanics Reviews 50, 1997, 15–67.Google Scholar
  23. 23.
    Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.Google Scholar
  24. 24.
    Butzer, P. L. and Westphal, U., 'An introduction to fractional calculus', in Applications of Fractional Calculus in Physics, R. Hilfer (ed.), World Scientific, Singapore, 2000, pp. 1–85.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Om P. Agrawal
    • 1
  1. 1.Mechanical Engineering and Energy ProcessesSouthern Illinois UniversityCarbondaleU.S.A

Personalised recommendations