Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by variational iteration method (VIM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present techniques. The present method performs extremely well in terms of efficiency and simplicity.
R. Metzler, E. Barkai and J. Klafter, “Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach,” Physics Review Letters, Vol. 82, No. 18, 1999, pp. 3563-3567. doi:10.1103/PhysRevLett.82.3563
R. Metzler and J. Klafter, “The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach,” Physics Reports, Vol. 339, No. 1, 2000, pp. 1-77. doi:10.1016/S0370-1573(00)00070-3
R. Metzler and J. Klafter, “The Restaurant at the End of the Random Walk: Recent Developments in the Description of Anomalous Transport by Fractional Dynamics,” Journal of Physics A, Vol. 37, No. 31, 2004, pp. 161-208.
R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto, “Fractional Calculus and Continuous-Time Finance. III. The Diffusion Limit,” Mathematical Finance, Konstanz, 2000, pp. 171-180.
F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas, “Fractional Calculus and Continuous-Time Finance II: The Waiting-Time Distribution,” Physica A, Vol. 287, no. 3-4, 2000, pp. 468-481. doi:10.1016/S0378-4371(00)00386-1
E. Scalas, R. Gorenflo and F. Mainardi, “Fractional Calculus and Continuous-Time Finance,” Physica A, Vol. 284, No. 1-4, 2000, pp. 376-384. doi:10.1016/S0378-4371(00)00255-7
M. Raberto, E. Scalas and F. Mainardi, “Waiting-Times and Returns in High Frequency Financial Data: An Empirical Study,” Physica A, Vol. 314, No. 1-4, 2002, pp. 749-755. doi:10.1016/S0378-4371(02)01048-8
D. A. Benson, S. Wheatcraft and M. M. Meerschaert, “Application of a Fractional Advection Dispersion Equation,” Water Resource Research, Vol. 36, No. 6, 2000, pp. 1403-1412. doi:10.1029/2000WR900031
B. Baeumer, M. M. Meerschaert, D. A. Benson and S. W. Wheatcraft, “Subordinated Advection-Dispersion Equation for Contaminant Transport,” Water Resource Research, Vol. 37, No. 6, 2001, pp. 1543-1550.
D. A. Benson, R. Schumer, M. M. Meerschaert and S. W. Wheatcraft, “Fractional Dispersion, Lévy Motions, and the MADE Tracer Tests,” Transport in Porous Media, Vol. 42, No. 1-2, 2001, pp. 211-240. doi:10.1023/A:1006733002131
R. Schumer, D. A. Benson, M. M. Meerschaert and S. W. Wheatcraft, “Eulerian Derivation of the Fractional Advection-Dispersion Equation,” Journal of Contaminant Hydrology, Vol. 48, No. 1-2, 2001, pp. 69-88. doi:10.1016/S0169-7722(00)00170-4
R. Schumer, D. A. Benson, M. M. Meerschaert and B. Baeumer, “Multiscaling Fractional Advection-Dispersion Equations and Their Solutions,” Water Resource Research, Vol. 39, No. 1, 2003, pp. 1022-1032.
A. Carpinteri and F. Mainardi, “Fractals and Fractional Calculus in Continuum Mechanics,” Springer-Verlag, Wien, New York, 1997, pp. 291-348.
F. Mainardi and G. Pagnini, “The Wright Functions as Solutions of the Time Fractional Diffusion Equations,” Applied Mathematics and Computation, Vol. 141, No. 1, 2003, pp. 51-62. doi:10.1016/S0096-3003(02)00320-X
O. P. Agrawal, “Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain,” Nonlinear Dynamics, Vol. 29, No. 1-4, 2002, pp. 145-155. doi:10.1023/A:1016539022492
W. R. Schneider and W. Wyss, “Fractional Diffusion and Wave Equations,” Journal of Mathematical Physics, Vol. 30, No. 1, 1989, pp. 134-144. doi:10.1063/1.528578
M. M. Meerschaert, H. Scheffler and C. Tadjeran, “Finite Difference Methods for Two-Dimensional Fractional Dispersion Equation,” Journal of Computational Physics, Vol. 211, No. 1, 2006, pp. 249-261. doi:10.1016/j.jcp.2005.05.017
C. Tadjeran, M. M. Meerschaert and H. Scheffler, “Finite Difference Methods for Two-Dimensional Fractional Dispersion Equation,” Journal of Computational Physics, Vol. 213, No. 1, 2006, pp. 205-213. doi:10.1016/j.jcp.2005.08.008
J. H. He, “Some Asymptotic Methods for Strongly Nonlinear Equations,” International Journal of Modern Physics B, Vol. 20, No. 10, 2006, pp. 1141-1199. doi:10.1142/S0217979206033796
J. H. He, “Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Media,” Computer Methods in Applied Mechanics and Engineering, Vol. 167, No. 1-2, 1998, pp. 57-68. doi:10.1016/S0045-7825(98)00108-X
J. H. He, “Variational Iteration Method for Autonomous Ordinary Differential Systems,” Applied Mathematics and Computation, Vol. 114, No. 2-3, 2000, pp. 115-123. doi:10.1016/S0096-3003(99)00104-6
J. H. He and X. H. Wu, “Construction of Solitary Solution and Compacton-Like Solution by Variational Iteration Method,” Chaos, Solitons & Fractals, Vol. 29, No. 1, 2006, pp. 108-113.
D. D. Ganji, E. M. M. Sadeghi and M. Safari, “Application of He’s Variational Iteration Method and Adomian’s Decomposition Method Method to Prochhammer Chree Equation,” International Journal of Modern Physics B, Vol. 23, No. 3, 2009, pp. 435-446. doi:10.1142/S0217979209049656
M. Safari, D. D. Ganji and M. Moslemi, “Application of He’s Variational Iteration Method and Adomian’s Decomposition Method to the Fractional Kdv-Burgers-Kuramoto Equation,” Computers and Mathematics with Applications, Vol. 58, 2009, pp. 2091-2097. doi:10.1016/j.camwa.2009.03.043
D. D. Ganji, M. Safari and R. Ghayor, “Application of He’s Variational Iteration Method and Adomian’s Decomposition Method to Sawada-Kotera-Ito Seventh-Order Equation,” Numerical Methods for Partial Differential Equations, Vol. 27, No. 4, 2011, pp. 887-897. doi:10.1002/num.20559
M. Safari, D. D. Ganji and E. M. M. Sadeghi, “Application of He’s Homotopy Perturbation and He’s Variational Iteration Methods for Solution of Benney-Lin Equation,” International Journal of Computer Mathe- matics, Vol. 87, No. 8, 2010, pp. 1872-1884. doi:10.1080/00207160802524770
I. Podlubny, “Fractional Differential Equations,” Aca-demic Press, San Diego, 1999.
K. B. Oldham and J. Spanier, “The Fractional Calculus,” Academic Press, New York and London, 1974.