## 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.

This is a preview of subscription content, access via your institution.

## References

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.Oldham, K. B. and Spanier, J.,

*The Fractional Calculus*, Academic Press, New York, 1974.Nigmatullin, R. R., 'To the theoretical explanation of the universal response',

*Physica B***123**, 1984, 739–745.Nigmatullin, R. R., 'Realization of the generalized transfer equation in a medium with fractal geometry',

*Physica B***133**, 1986, 425–430.Wyss, W., 'The fractional diffusion equation',

*Journal of Mathematical Physics***27**, 1986, 2782–2785.Schneider, W. R. and Wyss, W., 'Fractional diffusion and wave equations',

*Journal of Mathematical Physics***30**, 1989, 134–144.Sanz-serna, J. M., 'A numerical method for a partial integro-differential equation',

*SIAM Numerical Analysis***25**, 1988, 319–327.Fujita, Y., 'Integrodifferential equation which interpolates the heat equation and the wave equation',

*Osaka Journal of Mathematics***27**, 1990, 309–321.Fujita,Y., 'Integrodifferential equation which interpolates the heat equation and the wave equation. II',

*Osaka Journal of Mathematics***27**, 1990, 797–804.Ginoa, M., Cerbelli, S., and Roman, H. E., 'Fractional diffusion equation and relaxation in complex viscoelastic materials',

*Physica A***191**, 1992, 449–453.Roman, H. E. and Alemany, P. A., 'Continuous-time random walks and the fractional diffusion equation',

*Journal of Physics A***27**, 1994, 3407–3410.Mbodje, B. and Montseny, G., 'Boundary fractional derivative control of the wave equation',

*IEEE Transactions on Automatic Control***40**, 1995,-378–382.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.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.Agrawal, O. P., 'A general solution for the fourth-order fractional diffusion-wave equation',

*Fractional Calculation and Applied Analysis***3**, 2000, 1–12.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.Metzler, R. and Klafter, J., 'Boundary value problems for fractional Diffusion equations',

*Physica A***278**, 2000, 107–125.Hilfer, R., 'Fractional diffusion based on Riemann-Liouville fractional Derivatives',

*Journal of Physical Chemistry B***104**, 2000, 3914–3917.Samko, S. G., Kilbas, A. A., and Marichev, O. I.,

*Fractional Integrals and Derivatives-Theory and Applications*, Gordon and Breach, Longhorne, PA, 1993.Miller, K. S. and Ross, B.,

*An Introduction to the Fractional Calculus and Fractional Differential Equations*, Wiley, New York, 1993.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.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.Podlubny, I.,

*Fractional Differential Equations*, Academic Press, New York, 1999.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.

## Author information

### Authors and Affiliations

## Rights and permissions

## About this article

### Cite this article

Agrawal, O.P. Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain.
*Nonlinear Dynamics* **29**, 145–155 (2002). https://doi.org/10.1023/A:1016539022492

Issue Date:

DOI: https://doi.org/10.1023/A:1016539022492

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