1. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
2. Baeumera, B., Meerschaert, M.M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010)
3. Cao, J.X., Li, C.P., Chen, Y.Q.: High-order approximation to Caputo derivatives and Caputo-type advection–diffusion equations (II). Fract. Calc. Appl. Anal. 18, 735–761 (2015)
4. Chen, M.H., Deng, W.H.: High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights. SIAM J. Sci. Comput. 37, A890–A917 (2015)
5. Chen, M.H., Deng, W.H.: High order algorithm for the time-tempered fractional Feynman–Kac equation. J. Sci. Comput. 76, 867–887 (2018)
6. Chen, C.M., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction–subdiffusion equation. Appl. Math. Comput. 198, 754–769 (2008)
7. Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85, 1603–1638 (2016)
8. Dehghan, M., Abbaszadeh, M., Deng, W.H.: Fourth-order numerical method for the space time tempered fractional diffusion–wave equation. Appl. Math. Lett. 73, 120–127 (2017)
9. Deng, W.H., Zhang, Z.J.: Numerical schemes of the time tempered fractional Feynman–Kac equation. Comput. Math. Appl. 73, 1063–1076 (2017)
10. Dimitrov, Y.: A second order approximation for the Caputo fractional derivative. J. Frac. Cal. Appl. 7, 175–195 (2016)
11. Dimitrov, Y.: Three-point approximation for the Caputo fractional derivative. Comm. Appl. Math. Comput. 31, 413–442 (2017)
12. Ding, H.F., Li, C.P.: High-order numerical approximation formulas for Riemann–Liouville (Riesz) tempered fractional derivatives: construction and application (II). Appl. Math. Lett. 86, 208–214 (2018)
13. Gajda, J., Magdziarz, M.: Fractional Fokker–Planck equation with tempered ?-stable waiting times: Langevin picture and computer simulation. Phys. Rev. E 82, 011117 (2010)
14. Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)
15. Gracia, J.L., Stynes, M.: Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems. J. Comput. Appl. Math. 273, 103–115 (2015)
16. Henry, B.I., Langlands, T.A.M., Wearne, S.L.: Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction–diffusion equations. Phys. Rev. E 74, 031116 (2006)
17. Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21, 650–678 (2017)
18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equation. Elsevier, Amsterdam (2006)
19. Li, C., Deng, W.H.: High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42, 543–572 (2016)
20. Li, C.P., Wu, R.F., Ding, H.F.: High-order approximation to Caputo derivatives and Caputo-type advection–diffusion equations (I). Commun. Appl. Ind. Math. 6, 536 (2014)
21. Li, C., Deng, W.H., Zhao, L.J.: Well posedness and numerical algorithm for the tempered fractional ordinary differential equations. Discrete Contin. Dyn. Syst. Ser. B 24, 1989–2015 (2019)
22. Li, C., Sun, X.R., Zhao, F.Q.: LDG schemes with second order implicit time discretization for a fractional sub-diffusion equation. Results Appl. Math. 4, 100079 (2019)
23. Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
24. Liu, F., Zhuang, P.H., Liu, Q.X.: The Applications and Numerical Methods of Fractional Differential Equations. Science Press, Beijing (2015)
25. Lubich, Ch.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)
26. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)
27. Meerschaert, M.M., Zhang, Y., Baeumer, B.: Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, L17403 (2008)
28. Metzler, R., Barkai, E., Klafter, J.: Deriving fractional Fokker–Planck equations from generalised master equation. Europhys. Lett. 46, 431–436 (1999)
29. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)
30. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
31. Sabzikar, F., Meerschaert, M.M., Chen, J.H.: Tempered fractional calculus. J. Comput. Phys. 293, 14–28 (2015)
32. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London (1993)
33. Schmidt, M.G.W., Sagués, F., Sokolov, I.M.: Mesoscopic description of reactions for anomalous diffusion: a case study. J. Phys. Condens. Matter. 19, 065118 (2007)
34. Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville drivative. Appl. Numer. Math. 90, 22–37 (2015)
35. Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion wave system. Appl. Numer. Math. 56, 193–209 (2006)
36. Sun, X.R., Li, C., Zhao, F.Q.: Local discontinuous Galerkin methods for the time tempered fractional diffusion equation. Appl. Math. Comput. 365, 124725 (2020)
37. Tatar, N.: The decay rate for a fractional differential equation. J. Math. Anal. Appl. 295, 303–314 (2004)
38. Wang, Z.B., Vong, S.W.: Compact difference schemes for the modified anomalous fractional subdiffusion equation and the fractional diffusion–wave equation. J. Comput. Phys. 277, 1–15 (2014)
39. Zayernouri, M., Ainsworth, M., Karniadakis, G.: Tempered fractional Sturm–Liouville eigenproblems. SIAM J. Sci. Comput. 37, A1777–A1800 (2015)
40. Zhang, Y.: Moments for tempered fractional advection–diffusion equations. J. Stat. Phys. 139, 915–939 (2010)
41. Zhang, Z.J., Deng, W.H.: Numerical approaches to the functional distribution of anomalous diffusion with both traps and flights. Adv. Comput. Math. 43, 1–34 (2017)
42. Zhang, Y.N., Sun, Z.Z.: Alternating direction implicit schemes for the two-dimensional fractional subdiffusion equation. J. Comput. Phys. 230, 8713–8728 (2011)
43. Zhang, H., Liu, F., Turner, I., Chen, S.: The numerical simulation of the tempered fractional Black– Scholes equation for European double barrier option. Appl. Math. Model. 40, 5819–5834 (2016)
