Abstract
A class of time fractional partial differential equations is considered, which includes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier-Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine-Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.
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References
Podlubny, I. Fractional Differential Equations, Academic Press, San Diego, California (1999)
Gorenflo, R., Luchko, Y., and Mainardi, F. Wright function as scale-invariant solutions of the diffusion-wave equation. J. Comput. Appl. Math. 118, 175–191 (2000)
Mainardi, F., Paradisi, P., and Gorenflo, R. Probability distributions generated by fractional diffusion equations.Workshop on Econophysics: an Energing Science (eds. Kertesz, J. and Kondor, I.), Kluwer Academic Publishers, Dordrecht (1998)
Mainardi, F., Luchko, Y., and Pagnini, V. The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153–192 (2001)
Wyss, W. The fractional diffusion equation. J. Math. Phys. 27(11), 2782–2785 (1986)
Chen, J., Liu, F., and Anh, V. Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338(2), 1364–1377 (2008)
Agrawal, O. P. Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynamics 29, 145–155 (2002)
Gorenflo, R. and Mainardi, F. Fractional calculus: integral and differential equations of fractional order. Fractals and Fractional Calculus in Continuum Mechanics (eds. Carpinteri, A. and Mainardi, F.), Springer, New York, 223–276 (1997)
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Contributed by Bo-ling GUO
Project supported by the Fundamental Research Funds for the Central Universities, South China University of Technology (No. 2009ZM0050), the Research Foundation for the Doctoral Program of Higher Education of China (No. 20070561040), and the Natural Science Foundation of Guangdong Province of China (No. 07300823)
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Huang, Fh., Guo, Bl. General solutions to a class of time fractional partial differential equations. Appl. Math. Mech.-Engl. Ed. 31, 815–826 (2010). https://doi.org/10.1007/s10483-010-1316-9
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DOI: https://doi.org/10.1007/s10483-010-1316-9
Key words
- fractional differential equation
- Caputo fractional derivative
- Green function
- Laplace transform
- Fourier transform
- sine (cosine) transform