Exact and Approximate Solutions of Fractional Diffusion Equations with Fractional Reaction Terms

In this paper, we consider fractional reaction-diffusion equations with linear and nonlinear fractional reaction terms in a semi-infinite domain. Using q-Homotopy Analysis Method, solutions to these equations are obtained in the form of general recurrence relations. Closed form solutions in the form of the Mittag-Leffler function are perfectly obtained in the case with linear fractional reaction term due to the ability to control the auxiliary parameter h. Series solution is obtained for the case of nonlinear fractional reaction term. Numerical analysis is presented for this case to display the fast convergent rate of the series solution obtained. q-HAM is a relatively simple and powerful method and has advantages over some other methods which we discuss and demonstrate for some initial value problems.