General Solution for a Class of Time Fractional Partial Differential Equation

HUANG Feng-hui; GUO Bo-ling

doi:10.3879/j.issn.1000-0887.2010.07.003

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HUANG Feng-hui, GUO Bo-ling. General Solution for a Class of Time Fractional Partial Differential Equation[J]. Applied Mathematics and Mechanics, 2010, 31(7): 781-790. doi: 10.3879/j.issn.1000-0887.2010.07.003

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General Solution for a Class of Time Fractional Partial Differential Equation

doi: 10.3879/j.issn.1000-0887.2010.07.003

HUANG Feng-hui¹,
GUO Bo-ling²

1.
Department of Mathematics, School of Sciences, South China University of Technology, Guangzhou 510641, P. R. China;

Received Date: 1900-01-01
Rev Recd Date: 2010-05-25
Publish Date: 2010-07-15

Abstract

Abstract

A class of tmie fractional partial differential equation, including time fractional diffusion equation, tmie fractional reaction-diffusion equation, time fractional advection-diffusione-quation and their corresponding in teger-order partial differential equations, was considered. The fundam ental solutions for the Cauchy problem in a whole-space domain and signaling problem in a hal-fspace domain were obtained by using Fourier-Laplace trans forms and their inverse transforms. The appropriate structures for the Green functions were provided. On the other hand, the solutions in the form of a series for the in itial and boundary value p rob lem s in a bounded-space domain were derived by the Sine-Laplace or Cosine-Laplace transforms. Two examples were presented to show the application of the present technique.
- fractional differentia equation,
- Caputo fractional derivative,
- Green function,
- Laplace transform,
- Fourier transform,
- Sine(Cosine) transform

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References(8)

References

[1]	Podlubny I.Fractional Differential Equations[M].San Diego: Academic Press, 1999.
[2]	Gorenflo R, Luchko Y, Mainardi F. Wright function as scale-invariant solutions of the diffusion-wave equation[J]. J Comp Appl Math, 2000, 118(1/2): 175-191. doi: 10.1016/S0377-0427(00)00288-0
[3]	Mainardi F, Paradisi P, Gorenflo R.Probability distributions generated by fractional diffusion equations[C]Kertesz J, Kondor I. Econophysics: an Emerging Science.Dordrecht: Kluwer Academic Publishers, 1998.
[4]	Mainardi F, Luchko Y, Pagnini G.The fundamental solution of the space-time fractional diffusion equation[J].Fractional Calculus and Applied Analysis, 2001, 4(2):153-192.
[5]	Wyss W. The fractional diffusion equation[J]. J Math Phys, 1986, 27(11):2782-2785. doi: 10.1063/1.527251
[6]	Chen J, Liu F, Anh V.Analytical solution for the time-fractional telegraph equation by the method of separating variables[J]. J Math Anal Appl, 2008, 338(2):1364-1377. doi: 10.1016/j.jmaa.2007.06.023
[7]	Agrawal O P.Solution for a fractional diffusion-wave equation defined in a bounded domain[J]. Nonlinear Dynamics, 2002, 29(1/4):145-155. doi: 10.1023/A:1016539022492
[8]	Gorenflo R, Mainardi F. Fractional calculus: integral and differential equations of fractional order[C]Carpinteri A, Mainardi F. Fractals and Fractional Calculus in Continuum Mechanics. New York: Springer，1997： 223-276.