References
- Hilfer R. Applications of fractional calculus in physics. Singapore: World Scientific; 2000.
- Magin RL. Fractional calculus in bioengineering. Redding: Begell House; 2006.
- Ortigueira MD. Fractional calculus for scientists and engineers. New York: Springer; 2011.
- Li C, Zeng F. Numerical methods for fractional calculus. Boca Raton: Chapman & Hall/CRC; 2015.
- Li C, Cai M. Theory and numerical approximations of fractional integrals and derivatives. Philadelphia: SIAM; 2019.
- Liu F, Zhuang P, Liu Q. Numerical methods of fractional partial differential equations and applications. Beijing: Chinese Science Press; 2015.
- Sun Z, Gao G. Finite difference methods for fractional differential equations. 2nd ed. Beijing: Chinese Science Press; 2021.
- Lin Y, Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys. 2007;225:1533–1552.
- Sun Z, Wu X. A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math. 2006;56:193–209.
- Zhao Y, Shen C, Qu M, et al. Finite element methods for fractional diffusion equations. Int J Model Simul Sci Comput. 2020;11:2030001.
- Li J, Huang Y, Lin Y. Developing finite element methods for Maxwell's equations in a cole-cole dispersive medium. SIAM J Sci Comput. 2011;33(6):3153–3174.
- Li C, Zhao Z, Chen Y. Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput Math Appl. 2011;62:855–875.
- Feng L, Liu F, Turner I, et al. Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains. Appl Math Model. 2018;59:441–463.
- Feng L, Liu F, Turner I. Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains. Commun Nonlinear Sci Numer Simul. 2019;70:354–371.
- Jiang Y, Ma J. High-order finite element methods for time-fractional partial differential equations. J Comput Appl Math. 2011;235(11):3285–3290.
- Liu Y, Du Y, Li H, et al. A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative. Comput Math Appl. 2015;70:2474–2492.
- Liu Y, Du Y, Li H, et al. Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem. Comput Math Appl. 2015;70:573–591.
- Jin B, Lazarov R, Zhou Z. An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J Numer Anal. 2016;36(1):197–221.
- Li M, Huang C, Jiang F. Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes. Appl Anal. 2016;96(8):1269–1284.
- Zhao Y, Chen P, Bu W, et al. Two mixed finite element methods for time-fractional diffusion equations. J Sci Comput. 2017;70(1):407–428.
- Li D, Liao H, Sun W, et al. Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems. Commun Comput Phys. 2018;24(1):86–103.
- Jiang Y, Xu X. A monotone finite volume method for time fractional Fokker-Planck equations. Sci China Math. 2019;62(4):783–794.
- Ewing R, Lazarov R, Lin Y. Finite volume element approximations of nonlocal reactive flows in porous media. Numer Methods Partial Differ Equ. 2000;16(3):285–311.
- Chatzipantelidis P, Lazarov RD, Thomée V. Error estimates for a finite volume element method for parabolic equations in convex polygonal domains. Numer Methods Partial Differ Equ. 2004;20(5):650–674.
- Zhang Z. Error estimates of finite volume element method for the pollution in groundwater flow. Numer Methods Partial Differ Equ. 2009;25(2):259–274.
- Carstensen C, Dond AK, Nataraj N, et al. Three first-order finite volume element methods for Stokes equations under minimal regularity assumptions. SIAM J Numer Anal. 2018;56(4):2648–2671.
- Zhang T, Li Z. An analysis of finite volume element method for solving the Signorini problem. Appl Math Comput. 2015;270:830–841.
- Luo Z, Xie Z, Shang Y, et al. A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems. J Comput Appl Math. 2011;235(8):2098–2111.
- Bank RE, Rose DJ. Some error estimates for the box methods. SIAM J Numer Anal. 1987;24(4):777–787.
- Li Y, Li R. Generalized difference methods on arbitrary quadrilateral networks. J Comput Math. 1999;17(6):653–672.
- Li R, Chen Z, Wu W. Generalized difference methods for differential equations: numerical analysis of finite volume methods. New York: Marcel Dekker; 2000.
- Sayevand K, Arjang F. Finite volume element method and its stability analysis for analyzing the behavior of sub-diffusion problems. Appl Math Comput. 2016;290:224–239.
- Karaa S, Mustapha K, Pani AK. Finite volume element method for two-dimensional fractional subdiffusion problems. IMA J Numer Anal. 2017;37(2):945–964.
- Karaa S, Pani AK. Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data. ESAIM: M2AN. 2018;52:773–801.
- Zhao J, Fang Z, Li H, et al. A Crank-Nicolson finite volume element method for time fractional Sobolev equations on triangular grids. Mathematics. 2020;8:1591.
- Zhao J, Fang Z, Li H, et al. Finite volume element method with the WSGD formula for nonlinear fractional mobile/immobile transport equations. Adv Differ Equ. 2020;2020:688.
- Adams R. Sobolev spaces. New York: Academic Press; 1975.
- Sakamoto K, Yamamoto M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J Math Anal Appl. 2011;382:426–447.
- Stynes M. Too much regularity may force too much uniqueness. Fract Calc Appl Anal. 2016;19:1554–1562.
- Ervin V, Heuer N, Roop J. Regularity of the solution to 1-D fractional order diffusion equations. Math Comput. 2018;87:2273–2294.
- Jin B, Li B, Zhou Z. Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J Sci Comput. 2017;39:A3129–A3152.
- Stynes M, O'Riordan E, Gracia J L. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J Numer Anal. 2017;55:1057–1079.
- Zeng F, Zhang Z, Karniadakis G E. Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Comput Methods Appl Mech Eng. 2017;327:478–502.
- Liao H, Li D, Zhang J. Sharp error estimate of the nonuniform L1 formula for linear reaction–subdiffusion equations. SIAM J Numer Anal. 2018;56:1112–1133.
- Zheng X, Wang H. An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J Numer Anal. 2020;58:330–352.
- Yin B, Liu Y, Li H, et al. Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions. J Sci Comput. 2020;84:424.
- Yin B, Liu Y, Li H, et al. Approximation methods for the distributed order calculus using the convolution quadrature. Discrete Contin Dyn Syst Ser B. 2021;26:1447–1468.