Generalized space–time fractional diffusion equation with composite fractional time derivative

We investigate the solution of space–time fractional diffusion equations with a generalized Riemann–Liouville time fractional derivative and Riesz–Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and the Fox H -function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grünwald–Letnikov approximation are also used to solve the space–time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space–time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space–time fractional diffusion equations with a singular term of the form δ ( x ) ⋅ t − β Γ ( 1 − β ) ( β > 0 ) .