References
- A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys. 280 (2015), pp. 424–438.
- A. Alsaedi, B. Ahmad, and M. Kirane, Maximum principle for certain generalized time and space-fractional diffusion equations, Quart. Appl. Math. 73 (2015), pp. 163–175.
- W. Cao, F. Zeng, Z. Zhang, and G. Karniadakis, Implicit–explicit difference schemes for fractional differential equations with non-smooth solutions, SIAM J. Sci. Comput. 38 (2016), pp. A3070–A3093.
- K. Cheng, C. Wang, and S. Wise, An energy stable BDF2 Fourier pesudo-spectral numerical scheme for the square phase field crystal equation, Comm. Comput. Phys. 26 (2019), pp. 1335–1364.
- G. Dee and W. Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett. 60 (1998), pp. 2641–2644.
- N. Ford, M. Morgado, and M. Rebelo, Nonpolynomial collocation approximation of solutions to fractional differential equations, Fract. Calc. Appl. Anal. 16 (2013), pp. 874–891.
- G. Gao, Z. Sun, and H. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its application, J. Comput. Phys. 259 (2014), pp. 33–50.
- S. Godoy and L. Garcia-Colin, From the quantum random walk to classical mesoscopic diffusion in crystalline solids, Phys. Rev. E 53 (1996), pp. 5779–5785.
- R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlinear Dyn. 29 (2002), pp. 129–143.
- D. Hou and C. Xu, Highly efficient and energy dissipative schemes for the time fractional Allen–Cahn equation, SIAM J. Sci. Comput. 43 (2021), pp. A3305–A3327.
- D. Hou and C. Xu, A second order energy dissipative scheme for time fractional L2 gradient flows using SAV approach, J. Sci. Comput. 90 (2022), pp. 1–22. doi:10.1007/s10915-021-01667-w
- B. Ji, H. Liao, Y. Gong, and L. Zhang, Adaptive linear second-order energy stable schemes for time-fractional Allen–Cahn equation with volume constraint, Commun. Nonlinear Sci. Numer. Simul.90 (2020), pp. 105366.
- B. Ji, X. Zhu, and H. Liao, Energy stability of variable-step L1-type schemes for time-fractional Cahn–Hilliard model, arXiv:2201.00920v1, 2022.
- S. Jiang, J. Zhang, Q. Zhang, and Z. Zhang, Fast evaluation of the Caputo fractional derivative and its application to fractional diffusion equations, Commun. Comput. Phys. 21 (2017), pp. 650–678.
- N. Laskin, Fractional market dynamics, Physica A 287 (2000), pp. 482–492.
- H. Liao, T. Tang, and T. Zhao, An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen–Cahn equation, SIAM J. Sci. Comput. 43 (2021), pp. A3503–A3526.
- H. Liao, T. Tang, and T. Zhou, A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen–Cahn equations, J. Comput. Phys. 414 (2020), pp. 109473.
- H. Liao, T. Tang, and T. Zhou, Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators, arXiv:2011.13383v1, 2020.
- H. Liao, X. Zhu, and J. Wang, An adaptive L1 time-stepping scheme preserving a compatiable energy law for time-fractional Allen–Cahn equation. Numer. Math. Theory Methods Appl., arXiv:2102.07577v1, 2021.
- Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), pp. 1533–1522.
- Y. Liu, Y. Du, H. Li, J. Li, and S. He, A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative, Comput. Math. Appl 70 (2015), pp. 2474–2492.
- Y. Liu, Z. Fang, H. Li, and S. He, A mixed finite element method for a time-fractional fourth-order partial differential equation, Appl. Math. Comput. 243 (2014), pp. 703–717.
- C. Quan, T. Tang, and J. Jiang, How to define dissipation-preserving energy for time-fractional phase-field equations, CSIAM-AM 1 (2020), pp. 478–490.
- C. Quan, T. Tang, and J. Jiang, Numerical energy dissipation for time-fractional phase-field equations, arXiv:2019.06178v1, 2020.
- C. Quan and B. Wang, Energy stable L2 schemes for time fractional phase-field equations, J. Comput. Phys. 458 (2022), p. 111085.
- J. Shen, Z. Sun, and W. Cao, A finite difference scheme on graded meshes for time-fractional nonlinear Korteweg–de Vries equation, Appl. Math. Comput. 361 (2019), pp. 752–765.
- J. Shen, Z. Sun, and R. Du, Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time, East Asian J. Appl. Math. 8 (2018), pp. 834–858.
- J. Shen, J. Xu, and J. Yang, The Salar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys. 353 (2018), pp. 407–416.
- J. Shen and X. Yang, Numerical approximations of Allen–Cahn and Cahn–Hilliard equations, Discrete Contin. Dyn. Syst. 28 (2010), pp. 1669–1691.
- V. Srivastava and K. Rai, A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues, Math. Comput. Model. 51 (2010), pp. 616–624.
- M. Stynes, E. O'riordan, and J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079.
- H. Sun, Z. Sun, and R. Du, A linearized second-order difference scheme for the nonlinear time-fractional fourth-order reaction–diffusion equation, Numer. Math. Theor. Meth. Appl. 12 (2019), pp. 1168–1190.
- Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math.56 (2006), pp. 193–209.
- T. Tang, H. Yu, and T. Zhou, On energy dissipation theory and numerical stability for time-fractional Hase field equations, SIAM J. Sci. Comput. 41 (2019), pp. A3757–A3778.
- C. Wang, X. Wang, and S. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst. 28 (2010), pp. 405–423.
- L. Wei and Y. He, Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems, Appl. Math. Modell. 38 (2014), pp. 1511–1522.
- S. Yuste and J. Quintana-Murillo, A finite difference method with non-uniform timesteps for fractional diffusion equation, Comput. Phys. Commun. 183 (2012), pp. 2594–2600.
- F. Zeng, C. Li, F. Liu, and I. Turner, Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM J. Sci. Comput. 37 (2015), pp. A55–A78.
- Z. Zhang and Z. Qiao, An adaptive time-stepping strategy for the Cahn–Hilliard equation, Comm. Comput. Phys. 11 (2012), pp. 1261–1278.
- Z. Zhang, F. Zeng, and G.E. Karniadakis, Optimal error estimates of spectral Petrov–Galerkin and collocation methods for initial value problems of fractional differential equations, SIAM J. Numer. Anal. 53 (2015), pp. 2074–2096.
- L. Zhao and W. Deng, High order finite difference methods on non-uniform meshes for space fractional operators, Adv. Comput. Math. 42 (2016), pp. 425–468.