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Mathematical analysis and efficient finite element approximation for variable-order time-fractional reaction-diffusion equation with nonsingular kernel. (English) Zbl 1530.35348

MSC:

35R11 Fractional partial differential equations
35K57 Reaction-diffusion equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI

References:

[1] BensonD, SchumerR, MeerschaertMM, WheatcraftSW. Fractional dispersion, Lévy motions, and the MADE tracer tests. Transp Porous Media. 2001;42:211‐240.
[2] BouchaudJ, GeorgesA. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys Rep. 1990;195:127‐293.
[3] MeerschaertMM, SikorskiiA. Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics; 2011.
[4] MetzlerR, KlafterJ. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000;339:1‐77. · Zbl 0984.82032
[5] DengW. Finite element method for the space and time fractional Fokker-Planck equation. SIAM J Numer Anal. 2009;47(1):204‐226. · Zbl 1416.65344
[6] ErvinV, HeuerN, RoopJ. Regularity of the solution to 1‐D fractional order diffusion equations. Math Comp. 2018;87:2273‐2294. · Zbl 1394.65145
[7] JiC, DaiW, SunZ. Numerical schemes for solving the time‐fractional dual‐phase‐lagging heat conduction model in a double‐layered nanoscale thin film. J Sci Comput. 2019;81:1767‐1800. · Zbl 1434.65120
[8] JinB, LazarovR, ZhouZ. Numerical methods for time‐fractional evolution equations with nonsmooth data: a concise overview. Comput Methods Appl Mech Eng. 2019;346:332‐358. · Zbl 1440.65138
[9] LiaoH, McLeanW, ZhangJ. A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J Numer Anal. 2019;57(1):218‐237. · Zbl 1414.65008
[10] LinY, XuC. Finite difference/spectral approximations for the time‐fractional diffusion equation. J Comput Phys. 2007;225(2):1533‐1552. · Zbl 1126.65121
[11] LiuY, DuY, LiH, LiuF, WangY. Some second‐order θ schemes combined with finite element method for nonlinear fractional cable equation. Numer Algorithms. 2019;80(2):533‐555. · Zbl 1433.65218
[12] LiuY, DuY, LiH, WangJ. A two‐grid finite element approximation for a nonlinear time‐fractional Cable equation. Nonlinear Dynam. 2016;85(4):2535‐2548. · Zbl 1349.65429
[13] ZhouZ, GongW. Finite element approximation of optimal control problems governed by time fractional diffusion equation. Comput Math Appl. 2016;71:301‐318. · Zbl 1443.65235
[14] CaputoM, FabrizioM. A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl. 2015;1:1‐13.
[15] BaleanuD, JajarmiA, MohammadiH, RezapourS. A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Solitons Fract. 2020;134:109705. · Zbl 1483.92041
[16] BaleanuD, MohammadiH, RezapourS. A fractional differential equation model for the COVID‐19 transmission by using the Caputo-Fabrizio derivative. Adv Differ Equ. 2020:299. https://doi.org/10.1186/s13662‐020‐02762‐2 · Zbl 1485.37075 · doi:10.1186/s13662‐020‐02762‐2
[17] BaleanuD, FernandezA. On fractional operators and their classifications. Math. 2019;7:830.
[18] Cruz‐DuarteJM, Rosales‐GarciaJ, Correa‐CelyCR, Garcia‐PerezA, Avina‐CervantesJG. A closed form expression for the Gaussian‐based Caputo-Fabrizio fractional derivative for signal processing applications. Commun Nonlinear Sci Numer Simul. 2018;61:138‐148. · Zbl 1470.94036
[19] AtanganaA, BaleanuD. Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer. J Engrg Mech. 2017;143:D4016005.
[20] AtanganaA, BaleanuD. New fractional derivatives with nonlocal and non‐singular kernel: theory and application to heat transfer model. Therm Sci. 2016;20(2):763‐769.
[21] BaleanuD, AydognSM, MohammadiH, RezapourS. On modelling of epidemic childhood diseases with the Caputo-Fabrizio derivative by using the Laplace Adomian decomposition method. Alex Eng J. 2020;59:3029‐3039.
[22] JenaRM, ChakravertyS, BaleanuD. SIR epidemic model of childhood diseases through fractional operators with Mittag-Leffler and exponential kernels. Math Comput Simulat. 2021;182:514‐534. · Zbl 1524.92096
[23] ZhangM, LiuY, LiH. High‐order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo‐Fabrizio fractional derivative. Numer Meth PDEs. 2019;35:1588‐612. · Zbl 1416.82040
[24] LorenzoCF, HartleyTT. Variable order and distributed order fractional operators. Nonlinear Dynam. 2002;29:57‐98. · Zbl 1018.93007
[25] AtanganaA. On the stability and convergence of the time‐fractional variable order telegraph equation. J Comput Phys. 2015;293:104‐114. · Zbl 1349.65263
[26] ChenC, LiuH, ZhengX, WangH. A two‐grid MMOC finite element method for nonlinear variable‐order time‐fractional mobile/immobile advection‐diffusion equations. Comput Math Appl. 2020;79(9):2771‐2783. · Zbl 1437.65180
[27] LiZ, WangH, XiaoR, YangS. A variable‐order fractional differential equation model of shape memory polymers. Chaos Solitons Frac. 2017;102:473‐485.
[28] SunH, ZhangY, ChenW, ReevesD. Use of a variable‐index fractional‐derivative model to capture transient dispersion in heterogeneous media. J Contaminant Hydrology. 2014;157:47‐58.
[29] SunH, ChangA, ZhangY, ChenW. A review on variable‐order fractional differential equations: mathematical foundations, physical models, numerical methods and applications. Fract Calc Appl Anal. 2019;22:27‐59. · Zbl 1428.34001
[30] SunH, ChenW, ChenY. Variable‐order fractional differential operators in anomalous diffusion modeling. Physica A: Stat Mech Appl. 2009;388:4586‐4592.
[31] WangH, ZhengX. Wellposedness and regularity of the variable‐order time‐fractional diffusion equations. J Math Anal Appl. 2019a;475:1778‐1802. · Zbl 1516.35477
[32] WangH, ZhengX. Analysis and numerical solution of a nonlinear variable‐order fractional differential equation. Adv Comput Math. 2019b;45:2647‐2675.
[33] ZhengX, WangH. An error estimate of a numerical approximation to a hidden‐memory variable‐order space‐time fractional diffusion equation. SIAM J Numer Anal. 2020;58:2492‐2514. · Zbl 1450.65135
[34] ZhengX, WangH. Optimal‐order error estimates of finite element approximations to variable‐order time‐fractional diffusion equations without regularity assumptions of the true solutions. IMA J Numer Anal. 2020. https://doi.org/10.1093/imanum/draa013 · Zbl 07528285 · doi:10.1093/imanum/draa013
[35] TarasovV. Caputo-Fabrizio operator in terms of integer derivatives: memory or distributed lag?Comput Appl Math. 2019;38:113. · Zbl 1438.26016
[36] GhanbariB, Gómez‐AguilarJF. Modeling the dynamics of nutrient‐phytoplankton‐zooplankton system with variable‐order fractional derivatives. Chaos Solitons Fract. 2018;116:114‐120. · Zbl 1442.92128
[37] Solís‐PérezJE, Gómez‐AguilarJF, AtanganaA. Novel numerical method for solving variable‐order fractional differential equations with power, exponential and Mittag-Leffler laws. Chaos Solitons Fract. 2018;114:175‐185. · Zbl 1415.65148
[38] ZhengX, WangH, FuH. Well‐posedness of fractional differential equations with variable‐order Caputo-Fabrizio derivative. Chaos Solitons Fract. 2020;138:109966. · Zbl 1490.35533
[39] JiangS, ZhangJ, ZhangQ, ZhangZ. Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun Comput Phys. 2017;21:650‐678. · Zbl 1488.65247
[40] KeR, NgMK, SunH. A fast direct method for block triangular Toeplitz‐like with tri‐diagonal block systems from time‐fractional partial differential equations. J Comput Phys. 2015;303:203‐211. · Zbl 1349.65404
[41] WangH, BasuTS. A fast finite difference method for two‐dimensional space‐fractional diffusion equations. SIAM J Sci Comput. 2012;34:A2444‐A2458. · Zbl 1256.35194
[42] WangH, WangK, SircarT. A direct \(O(N \log^2 N)\) finite difference method for fractional diffusion equations. J Comput Phys. 2010;229:8095‐8104. · Zbl 1198.65176
[43] FuH, WangH. A preconditioned fast parareal finite difference method for space‐time fractional partial differential equation. J Sci Comput. 2019;78:1724‐1743. · Zbl 1415.65190
[44] WuS, ZhouT. Parareal algorithms with local time‐integrators for time fractional differential equations. J Comput Phys. 2018;358:135‐149. · Zbl 1422.65473
[45] XuQ, HesthavenJS, ChenF. A parareal method for time‐fractional differential equations. J Comput Phys. 2015;293:173‐183. · Zbl 1349.65220
[46] FangZ, SunH, WangH. A fast method for variable‐order Caputo fractional derivative with applications to time‐fractional diffusion equations. Comput Math Appl. 2020;80(5):1443‐1458. · Zbl 1447.65022
[47] JiaJ, ZhengX, FuH, DaiP, WangH. A fast method for variable‐order space‐fractional diffusion equations. Numer Algor. 2020;85:1519‐1540. https://doi.org/10.1007/s11075-020#x0002D;00875#x0002D;z · Zbl 1456.65132 · doi:10.1007/s11075-020#x0002D;00875#x0002D;z
[48] AdamsRA, FournierJJF. Sobolev Spaces. San Diego: Elsevier; 2003.
[49] EvansLC. Partial Differential Equations Graduate Studies in Mathematics, Vol. 19. Rhode Island: American Mathematical Society; 1998. · Zbl 0902.35002
[50] SakamotoK, YamamotoM. Initial value/boundary value problems for fractional diffusion‐wave equations and applications to some inverse problems. J Math Anal Appl. 2011;382:426‐447. · Zbl 1219.35367
[51] ThoméeV. Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics 1054, Springer‐Verlag: New York; 1984. · Zbl 0528.65052
[52] LuchkoY. Initial‐boundary‐value problems for the one‐dimensional time‐fractional diffusion equation. Fract Calc Appl Anal. 2012;15:141‐160. · Zbl 1276.26018
[53] StynesM, O’RiordanE, GraciaJL. Error analysis of a finite difference method on graded mesh for a time‐fractional diffusion equation. SIAM J Numer Anal. 2017;55:1057‐1079. · Zbl 1362.65089
[54] HackbuschW. Integral Equations: Theory and Numerical Treatment International Series of Numerical Mathematics, Vol. 120. Birkhäuser Verlag: Basel; 1995. · Zbl 0823.65139
[55] StynesM. Fractional‐order derivatives defined by continuous kernels are too restrictive. Appl Math Lett. 2018;85:22‐26. · Zbl 1401.26017
[56] KachiaK, Solís‐PérezJE, Gómez‐AguilarJF. Chaos in a three‐cell population cancer model with variable‐order fractional derivative with power, exponential and Mittag-Leffler memories. Chaos Solitons Fract. 2020;140:110177. · Zbl 1495.92030
[57] ZhouH, YangS, ZhangS. Modeling non‐Darcian flow and solute transport in porous media with the Caputo-Fabrizio derivative. Appl Math Model. 2019;68:603‐615. · Zbl 1481.76234
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