Homotopy analysis method with modified Reimann- Liouville derivative for space fractional diffusion equation

In this paper, we applied the homotopy analysis method (HAM) to construct the analytical solutions of the space fractional diffusion equations. The derivatives are defined in the Jumarie’s fractional derivative sense. The explicit solutions of the equations have been presented in the closed form by using initial conditions. Two typical examples have been discussed. The results reveal that the method is very effective and simple. On the basis of computational work and subsequent numerical results, it is worth noting that the advantage of the homotopy analysis methodology is that it displays a fast convergence of the solution. Key words: Analytical solution, fractional diffusion equation, Reimann-Liouville fractional derivative, homotopy analysis method.

[1]  Guy Jumarie,et al.  Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions , 2009, Appl. Math. Lett..

[2]  Guy Jumarie,et al.  Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative , 2009, Appl. Math. Lett..

[3]  Guy Jumarie,et al.  New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations , 2006, Math. Comput. Model..

[4]  Mark M. Meerschaert,et al.  A second-order accurate numerical approximation for the fractional diffusion equation , 2006, J. Comput. Phys..

[5]  S. Liao.  Notes on the homotopy analysis method: Some definitions and theorems , 2009 .

[6]  Z. Odibat,et al.  Analytic study on time-fractional Schrödinger equations: exact solutions by GDTM , 2008 .

[7]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[8]  M. Meerschaert,et al.  Finite difference methods for two-dimensional fractional dispersion equation , 2006 .

[9]  Shaher Momani,et al.  An explicit and numerical solutions of the fractional KdV equation , 2005, Math. Comput. Simul..

[10]  Subir Das,et al.  Solution of Fractional Vibration Equation by the Variational Iteration Method and Modified Decomposition Method , 2008 .

[11]  Shaher Momani,et al.  Analytic and approximate solutions of the space- and time-fractional telegraph equations , 2005, Appl. Math. Comput..

[12]  K. S. Chaudhuri,et al.  Application of modified decomposition method for the analytical solution of space fractional diffusion equation , 2008, Appl. Math. Comput..

[13]  Rabha W. Ibrahim,et al.  ANALYTICAL SOLUTIONS OF A FRACTIONAL OSCILLATOR BY THE DECOMPOSITION METHOD , 2007 .

[14]  G. Jumarie.  Stochastic differential equations with fractional Brownian motion input , 1993 .

[15]  Ahmed Alawneh,et al.  Variational iteration method for solving the space‐ and time‐fractional KdV equation , 2008 .

[16]  S. Liao.  On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet , 2003, Journal of Fluid Mechanics.