Computational solutions of unified fractional reaction–diffusion equations with composite fractional time derivative

• We do apply fractional calculus to reaction–diffusion equations.
• Give solutions of these fractional derivative type equations suitable for numerical analysis and simulation.
• Application of Fourier and Laplace transform techniques for handling fractional differential equations.

This paper deals with the investigation of the computational solutions of an unified fractional reaction–diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz–Feller fractional derivative and adding the function ϕ(x,t)ϕ(x,t) which is a nonlinear function governing reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space–time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz–Feller space fractional derivatives are also discussed.