时间分数阶Cable方程修正格式的误差分析

兰州理工大学学报 ›› 2023, Vol. 49 ›› Issue (1): 158-163.

时间分数阶Cable方程修正格式的误差分析

吴晓蕾^*1, 杨艳¹, 闫玉斌²

1.吕梁学院数学系, 山西吕梁 033000;
2.彻斯特大学数学系, 英国 CH24BJ

收稿日期:2021-12-31 出版日期:2023-02-28 发布日期:2023-03-21
通讯作者: 吴晓蕾(1987-),女,山西运城人,讲师. Email:568248301@qq.com
基金资助:
国家自然科学基金(11771184),山西省自然科学研究面上项目(202103021224317),山西省自然科学基金(201801D121010),山西省高校科技创新计划项目(2020L0700)

Error analysis of modified schemes for time-fractional Cable equation

WU Xiao-lei¹, YANG Yan¹, YAN Yu-bin²

1. Department of Mathematics, Lüliang University, Lüliang 033000, China;
2. School of Mathematics and Statistics, University of Chester, Chester, CH24BJ, UK

Received:2021-12-31 Online:2023-02-28 Published:2023-03-21

摘要/Abstract

摘要： 考虑时间分数阶Cable方程在修正的二阶向后差分格式下的误差分析.利用连续Laplace变换、反Laplace变换方法得到方程的准确解,类似得到空间有限元半离散解;运用Lubich的修正方法引入此分数阶微分方程的修正格式,离散的Laplace变换、反Laplace变换法得到Cable方程的时间离散解,进而讨论了时间离散下L₂范数的误差估计,得到二阶收敛阶,并用数值算例验证了定理的结论.这个结论比不修正的情形下一阶收敛阶要高.

关键词: 分数阶Cable方程, Riemann-Liouville分数阶导, Laplace变换, 非光滑数据误差估计

Abstract: Error analysis of the modified second-order backward difference scheme for the time-fractional Cable equation is carried out. By using continuous Laplace transform and inverse Laplace transform, the exact solution of the equation is obtained, and the finite element semidiscrete solution is obtained similarly. Then Lubich's correction method is used to get the modified form of the fractional differential equation. The discrete solutions of the Cable equation are obtained by means of the discrete Laplace transform and the inverse Laplace transform. Finally, the error estimates under the norm are discussed and the second order of convergence is obtained. Numerical results verification is finally performed to validate the theoretical findings discussed here, which proved that it is better than the first order convergence without modification.

Key words: fractional Cable equation, Riemann-Liouville fractional derivative, Laplace transform, Nonsmooth data error estimation

中图分类号:

O241.8

吴晓蕾, 杨艳, 闫玉斌. 时间分数阶Cable方程修正格式的误差分析[J]. 兰州理工大学学报, 2023, 49(1): 158-163.

WU Xiao-lei, YANG Yan, YAN Yu-bin. Error analysis of modified schemes for time-fractional Cable equation[J]. Journal of Lanzhou University of Technology, 2023, 49(1): 158-163.

参考文献

[1] PODLUBNY I.Fractional differential equations [M].San-Diego:Academic press,1999.
[2] DIETHELM K.The analysis of fractional differential equations:an application-oriented exposition using differential operators of Caputo type [M].Springer:Central Book Services,2010.
[3] THOMEE V.Galerkin finite element methods for parabolic problems [M].2nd ed.Berkin:Springer,2006.
[4] LUBICH C,SLOAN I H,THOMÉE V.Nonsmooth data error estimate for approximation of an evolution equationwith a positive-type memory term [J].Math Comput,1996,213:1-17.
[5] LIN Y,LI X,XU C.Finite difference/spectral approximations for the fractional cable equation [J].Math Comput,2011,80:1369-1396.
[6] HU X,ZHANG L.Implicit compact difference schemes for the fractional cable equation [J].Appl Math Model,2012,36:4027-4043.
[7] BHRAWY A H,ZAKYM A.Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation [J].Nonlinear Dyn,2015,80:101-116.
[8] LIU Y,DU Y W,LI H,et al.A two-grid finite element approximation for a nonlinear time-fractionalCable equation [J].Nonlinear Dyn,2016,85:2535-2548.
[9] LIU Y,DU Y W,LI H,et al.Some second order schemes combined with finite element method for nonlinear fractional Cabel equation [J].J Numer Algor,2019,80:533-555.
[10] QUINTANA-MURILLO J,YUSTE S B.An explicit numerical method for the fractional cable equation [J].Int J of Differ Equ,2011,2011:51-62.
[11] ZHENG Y,ZHAO Z.The discontinuous Galerkin finite element method for fractional cable equation [J].Appl Numer Math,2017,115:32-41.
[12] SAKAMOTO K,YAMAMOTO M.Initial value/boundary value problems for fractional diffusion wave equations and applications to some inverse problems [J].J Math Appl,2011,382:426-447.
[13] MARTIN S.Too much regularity may force too much uniqueness [J].Fract Calc Appl Anal,2016,19:1554-1562.
[14] STYNES M,O'RIORDAN E,GRACIA J L.Error analysis of a finite difference method on graded meshes for a time fractional diffusion equation [J].SIAM Journal on Numerical Analysis,2017,55:1057-1079.
[15] YAN Y,KHAN M,FORD N J.An analysis of the modified scheme for the time fractionalpartial differential equations with nonsmooth data [J].SIAM J Numer Anal,2018,56:210-227.
[16] YANG Y,YAN Y B,FORD N J.Some time stepping methods for fractional diffusion problems with nonsmooth data [J].Comput Methods Appl Math,2018,18:129-146.
[17] WANG Y Y,YAN Y B,YANG Y.Two high-order time discretization schemes for subdiffusion problems with nonsmooth data [J].Fract Calc Appl Anal,2020,23:1349-1380.
[18] WU X X,YAN Y Y,YAN Y Y.An analysis of the L-1 scheme for stochastic subdiffusion problem driven by integrated space-time white noise [J].Appl Numer Math,2020,157:69-87.
[19] AL-MASKARI M,KARAA S.The lumped mass FEM for a time-fractional cable equation [J].Appl Numer Math,2018,132:73-90.
[20] ZHU P,XIE L,WANG S.Nonsmooth data error estimates for FME approximation of the time fractionalcable equation [J].Appl Numer Math,2017,121:170-184.
[21] JIN B T,LI B Y,ZHOU Z.Correction of high-order BDF convolution quadrature for fractional evolution equation equations [J].SIAM J Sci Comput,2017,39:A3129-A3152.
[22] JIN B T,YAN Y,ZHOU Z.Numerical approximation of stochastic time-fractional diffusion [J].ESAIM Math Model Numer Anal,2019,53:1245-1268.