Recently, many models are formulated in terms of fractional derivatives, such as in control processing, viscoelasticity, signal processing, and anomalous diffusion. In the present paper, we further study the important properties of the Riemann‐Liouville (RL) derivative, one of mostly used fractional derivatives. Some important properties of the Caputo derivative which have not been discussed elsewhere are simultaneously mentioned. The partial fractional derivatives are also introduced. These discussions are beneficial in understanding fractional calculus and modeling fractional equations in science and engineering.

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.

In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D ^α u ( t )= f ( t , u ( t )) with a Riemann–Liouville fractional derivative via the different boundary-value problems u (0)= u ( T ), and the three-point boundary condition u (0)= β ₁ u ( η ) and u ( T )= β ₂ u ( η ), where T >0, t ∈ I =[0, T ], 0< α <1, 0< η < T , 0< β ₁ < β ₂ <1.

In this review paper, we are mainly concerned with the finite difference methods, the Galerkin finite element methods, and the spectral methods for fractional partial differential equations (FPDEs), which are divided into the time-fractional, space-fractional, and space-time-fractional partial differential equations (PDEs). Besides, fast algorithms for the FPDEs are included in order to stimulate more efficient algorithms for high-dimensional FPDEs.