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

摘要

This paper deals with the investigation of the computational solutions of anunified fractional reaction-diffusion equation, which is obtained from thestandard diffusion equation by replacing the time derivative of first order bythe generalized fractional time-derivative defined by Hilfer (2000), the spacederivative of second order by the Riesz-Feller fractional derivative and addingthe function phi(x,t) which is a nonlinear function overning reaction. Thesolution is derived by the application of the Laplace and Fourier transforms ina compact and closed form in terms of the H-function. The main result obtainedin this paper provides an elegant extension of the fundamental solution for thespace-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 obtainedexplicitly. Fractional order moments of the distribution are deduced. At theend, mild extensions of the derived results associated with a finite number ofRiesz-Feller space fractional derivatives are also discussed.