References
- C. Celik and M. Duman . Crank–Nicolson method for the fractional diffusion equation with the riesz fractional derivative . J. Comput. Phys. , 231 : 1743 – 1750 , 2012 . http://dx.doi.org/10.1016/j.jcp.2011.11.008 .
- W. Chen and S. Holm . Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency . J. Acoustical Soc. America , 115 : 1424 – 1430 , 2004 . http://dx.doi.org/10.1121/1.1646399 .
- Y. Fujita . Cauchy problems of fractional order and stable processes . Japan J. Appl. Math. , 7 3 : 459 – 476 , 1990 . http://dx.doi.org/10.1007/BF03167854 .
- M.M. Khader . On the numerical solutions for the fractional diffusion equation . Commun. Nonlinear Sci. Numer. Simul. , 16 : 2535 – 2542 , 2011 . http://dx.doi.org/10.1016/j.cnsns.2010.09.007 .
- Khader , M.M. , Sweilam , N.H. and Mahdy , A.M.S. 2011 . An efficient numerical method for solving the fractional diffusion equation . J. Appl. Math. Bioinform. , 1 : 1 – 12 .
- Kilbas , A.A. , Srivastava , H.M. and Trujillo , J.J. 2006 . Theory and Applications of Fractional Differential Equations , Amsterdam : Elsevier .
- T.A.M. Langlands and B.I. Henry . The accuracy and stability of an implicit solution method for the fractional diffusion equation . J. Comput. Phys. , 205 : 719 – 736 , 2005 . http://dx.doi.org/10.1016/j.jcp.2004.11.025 .
- Y. Lin and C. Xu . Finite difference/spectral approximations for the time fractional diffusion equation . J. Comput. Phys. , 225 : 1533 – 1552 , 2007 . http://dx.doi.org/10.1016/j.jcp.2007.02.001 .
- Mainardi , F. 1995 . “ Fractional diffusive waves in viscoeslactic solids ” . In Non-linear Waves in Solids , Edited by: Wegner , J.L. and Norwood , F.R. 93 – 97 . Fairfield , NJ : ASME/AMR .
- S. Momami and Z. Odibat . Analytical approach to linear fractional partial differential equations arising in fluid mechanics . Phys. Lett. A , 355 : 271 – 279 , 2006 . http://dx.doi.org/10.1016/j.physleta.2006.02.048 .
- K. Mustapha and W. Mclean . Piece-linear, discontinuous Galerkin method for a fractional diffusion equation . Numer. Algorithms , 6 : 159 – 184 , 2011 . http://dx.doi.org/10.1007/s11075-010-9379-8 .
- Oldham , K.B. and Spanier , J. 1974 . The Fractional Calculus , New York : Academic .
- Podlubny , I. 1999 . Fractional Differential Equations , San Diego : Academic Press .
- Prenter , P.M. 1975 . Splines and Variational Methods , New York : John Wiley .
- J. Quintana-Murillo and S.B. Yuste . An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form . J. Comput. Nonlinear Dynam. , 6 : 021014 , 2011 . http://dx.doi.org/10.1115/1.4002687 .
- S.S. Ray . Exact solutions for time-fractional diffusion-wave equations by decomposition method . Phys. Scr. , 75 : 53 – 61 , 2007 . http://dx.doi.org/10.1088/0031-8949/75/1/008 .
- K. Rektorys . Variational Methods in Mathematics, Science and Engineering . D. Reidel Publishing Company, second ed., P.O. Box 17, 3300 Dordrecht-Holland , pp. 161 – 162 , 1977 .
- H.G. Sun , W. Chen and K.Y. Sze . A semi-analytical finite element method for a class of time-fractional diffusion equations . arXiv:1109.0641v1 [math-ph] , 2011 .
- Sweilam , N.H. , Khader , M.M. and Mahdy , A.M.S. 2012 . Crank–Nicolson finite difference method for solving time-fractional diffusion equation . J. Fract. Calc. Appl. (JFCA) , 2 : 1 – 9 .
- S.B. Yuste . Weighted average finite difference methods for fractional diffusion equations . J. Comput. Phys. , 216 : 264 – 274 , 2006 . http://dx.doi.org/10.1016/j.jcp.2005.12.006 .