General solutions to a class of time fractional partial differential equations

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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Correspondence to Feng-hui Huang.

Additional information

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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Key words

  • fractional differential equation
  • Caputo fractional derivative
  • Green function
  • Laplace transform
  • Fourier transform
  • sine (cosine) transform

Chinese Library Classification

  • O175.2

2000 Mathematics Subject Classification

  • 35K05