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In this paper, we study the following time-fractional Feynman-Kac equation $$\begin{aligned} {_\sigma ^CD_t^{\alpha }G(x,t)}-\Delta G(x,t)=f(x,t),~~~ 0 0. \end{aligned}$$ As is well known, the optimal rate of convergence $$\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right) $$ with $$\sigma =0$$ on graded meshes has been proved in [Stynes et al., SIAM J. Numer. Anal. 55, 1057–1079 (2017)] by L1 scheme. However, there are still some significant differences when $$\sigma >0$$ . More concretely, it shall drop down to the $$\mathcal {O}\left( \tau ^{\min \{1,~r\alpha \}}\right) $$ by the implicit L1 scheme. This motivates us to design the implicit-explicit L1 scheme, which recovers a convergence rate $$\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right) $$ on graded meshes. Finally, numerical experiments are given to illustrate theoretical results.

In this paper, a class of fractional advection–dispersion equations with Caputo tempered fractional derivative are considered numerically. An efficient algorithm for the evaluation of Caputo tempered fractional derivative is proposed to sharply reduce the computational work and storage, and this is of great significance for large-scale problems. Based on the nonsmooth regularity assumptions, a semi-discrete form is obtained by finite difference method in time, and its stability and convergence are investigated. Then by finite element method, we derive the corresponding fully discrete scheme and discuss its convergence. At last, some numerical examples, based on different domains, are presented to demonstrate effectiveness of numerical schemes and confirm the theoretical analysis.

In this paper, a class of distributed-order time fractional diffusion equations (DOFDEs) on bounded domains is considered. By L1 method in temporal direction, a semi-discrete variational formulation of DOFDEs is obtained firstly. The stability and convergence of this semi-discrete scheme are discussed, and the corresponding fully discrete finite element scheme is investigated. To improve the convergence rate in time, the weighted and shifted Grunwald difference method is used. By this method, another finite element scheme for DOFDEs is obtained, and the corresponding stability and convergence are considered. And then, as a supplement, a higher order finite difference scheme of the Caputo fractional derivative is developed. By this scheme, an approach is suggested to improve the time convergence rate of our methods. Finally, some numerical examples are given for verification of our theoretical analysis.

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order $$0