An analytic solution for the space–time fractional advection–dispersion equation using the optimal homotopy asymptotic method

We present an analytic algorithm to solve the space–time fractional advection–dispersion equation (FADE) based on the optimal homotopy asymptotic method (OHAM), which has the advantage of controlling the region and rate of convergence of the solution series via several auxiliary parameters over the traditional homotopy analysis method (HAM) having only one auxiliary parameter. Furthermore, our proposed algorithm gives better results compared to the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) in the sense that fewer iterations are required to get a sufficiently accurate solution and the solution has a greater radius of convergence. We find that the iterations obtained by the proposed method converge to the numerical/exact solution of the ADE as the fractional orders α,β,γα,β,γ tend to their integral values. Numerical examples are given to illustrate the proposed algorithm. The figures and tables show the superiority of the OHAM over the HAM.