# zbMATH — the first resource for mathematics

Analytical solution of a fractional diffusion equation by variational iteration method. (English) Zbl 1165.35398
Summary: In the present paper the Analytical approximate solution of a fractional diffusion equation is deduced with the help of powerful Variational Iteration method. By using an initial value, the explicit solutions of the equation for different cases have been derived, which accelerate the rapid convergence of the series solution. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented graphically.

##### MSC:
 35K57 Reaction-diffusion equations 26A33 Fractional derivatives and integrals 35A35 Theoretical approximation in context of PDEs 35C05 Solutions to PDEs in closed form 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
Full Text:
##### References:
  Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York and London · Zbl 0428.26004  Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (2003), John Willey and Sons, Inc. New York  Podlubry, J., Fractional differential equations, (1999), Academic Press San Diego, California, USA  Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003  Diethelm, K., An algorithm for the numerical solutions of differential equations of fractional order, Elec. trans. numer., 5, 1-6, (1997) · Zbl 0890.65071  He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. methods appl. mech. engrg., 167, 57-68, (1998) · Zbl 0942.76077  He, J.H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. methods appl. mech. engrg., 167, 69-73, (1998) · Zbl 0932.65143  He, J.H., Variational iteration method — a kind of nonlinear analytical technique: some examples, Internat. J. nonlinear mech., 34, 699-708, (1999) · Zbl 1342.34005  He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl. math. comput., 114, 115-123, (2000) · Zbl 1027.34009  He, J.H., Some asymptotic methods for strongly nonlinear equations, Internat. J. modern phys. B, 20, 1141-1191, (2006)  Shawagfeh, N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. math. comput., 131, 517-529, (2002) · Zbl 1029.34003  K. Diethelm, N.J. Ford, The numerical solutions of linear and nonlinear Fractional differential equations involving Fractional derivatives of several orders, Numerical Analysis Report, V.379, Manchester Centre for Computational Math., England, 2001  Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos solutions fractals, 31, 1248-1255, (2007) · Zbl 1137.65450  Saha Ray, S.; Bera, R.K., Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appl. math. comput., 174, 329-336, (2006) · Zbl 1089.65108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.