J4 ›› 2011, Vol. 46 ›› Issue (8): 23-30.

• Articles • Previous Articles     Next Articles

Fundamental solutions of fractional-in-space diffusion equation with  Riesz fractional derivative in two and three dimensions

WANG Xue-bin   

  1. Department of Mathematics and Computer, Wuyi University, Wuyishan 354300, Fujian, China
  • Received:2011-03-12 Online:2011-08-20 Published:2011-09-08
  • About author:WANG Xuebin(1976- ), Male, Lecturer, his research mainly focuses on fractional calculus.Email:wxbnp@163.com
  • Supported by:

    Supported by the Natural Science Foundation of Fujian Province (2008J0204); Fujian Provincial Department of Education Category the Projects(JA09242); Wuyi University Special Research Fund for Young Teachers(xq201022)


The fundamental solutions of fractional-in-space diffusion equation is considered with Riesz fractional derivative (RFDE) in two and three dimensions. The existing definitions of the fractional Laplacian (two dimensions and three dimensions) are investigated and discussed by using eigenfunction expansion, and the relations between fractional Laplacian and Riesz fractional derivative are given. Finally,  the fundamental solutions of homogeneous and non-homogeneous RFDE with an initial and boundary condition is derived on a finite domain  using a spectral representation.

Key words: Riesz fractional derivative; fractional-in-space diffusion equation; Rimann-Liouville fractional derivative

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