Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative

doi:10.1088/1674-1056/ab3af3

中国物理B ›› 2019, Vol. 28 ›› Issue (10): 100201-100201.doi: 10.1088/1674-1056/ab3af3

• SPECIAL TOPIC—Recent advances in thermoelectric materials and devices • 下一篇

Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative

Jiahui Hu(胡嘉卉), Jungang Wang(王俊刚), Yufeng Nie(聂玉峰), Yanwei Luo(罗艳伟)

1 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, China;
2 College of Science, Henan University of Technology, Zhengzhou 450001, China

收稿日期:2019-06-15 修回日期:2019-07-23 出版日期:2019-10-05 发布日期:2019-10-05
通讯作者: Yufeng Nie E-mail:yfnie@nwpu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant No. 11471262) and Henan University of Technology High-level Talents Fund, China (Grant No. 2018BS039).

Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative

Jiahui Hu(胡嘉卉)^1,2, Jungang Wang(王俊刚)¹, Yufeng Nie(聂玉峰)¹, Yanwei Luo(罗艳伟)²

1 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, China;
2 College of Science, Henan University of Technology, Zhengzhou 450001, China

Received:2019-06-15 Revised:2019-07-23 Online:2019-10-05 Published:2019-10-05
Contact: Yufeng Nie E-mail:yfnie@nwpu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant No. 11471262) and Henan University of Technology High-level Talents Fund, China (Grant No. 2018BS039).

摘要/Abstract

摘要： The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the non-local time-space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman-Kac equation. The proposed difference schemes have the q-th (q=1,2,3,4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.

关键词: backward fractional Feynman-Kac equation, fractional substantial derivative, compact finite difference scheme, numerical inversion of Laplace transforms

Abstract: The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the non-local time-space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman-Kac equation. The proposed difference schemes have the q-th (q=1,2,3,4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.

Key words: backward fractional Feynman-Kac equation, fractional substantial derivative, compact finite difference scheme, numerical inversion of Laplace transforms

中图分类号: (Numerical approximation and analysis)

02.60.-x

02.60.Cb (Numerical simulation; solution of equations) 02.70.Bf (Finite-difference methods)

胡嘉卉, 王俊刚, 聂玉峰, 罗艳伟. Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative[J]. 中国物理B, 2019, 28(10): 100201-100201.

Jiahui Hu(胡嘉卉), Jungang Wang(王俊刚), Yufeng Nie(聂玉峰), Yanwei Luo(罗艳伟). Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative[J]. Chin. Phys. B, 2019, 28(10): 100201-100201.

参考文献 40

[30]	Li X and Xu C 2009 SIAM J. Numer. Anal. 47 2108 doi: 10.1137/080718942
[1]	Comtet A, Desbois J and Texier C 2005 J. Phys. A: Math. Gen. 38 R341
[31]	Deng W, Chen M and Barkai E 2015 J. Sci. Comput. 62 718 doi: 10.1007/s10915-014-9873-6
[2]	Foltin G, Oerding K, Rácz Z, Workman R and Zia R 1994 Phys. Rev. E 50 R639
[32]	Chen M and Deng W 2013 ESAIM Math. Model. Numer. Anal. 49 373
[3]	Hummer G and Szabo A 2001 Proc. Natl. Acad. Sci. USA 98 3658 doi: 10.1073/pnas.071034098
[4]	Baule A and Friedrich R 2006 Phys. Lett. A 350 167 doi: 10.1016/j.physleta.2005.10.017
[33]	Sun Z 2009 The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations (Beijing: Science Press) p. 126
[5]	Majumdar S and Bray A 2002 Phys. Rev. E 65 051112 doi: 10.1103/PhysRevE.65.051112
[34]	Liao H and Sun Z 2010 Numer. Methods Partial Differential Equations 26 37 doi: 10.1002/num.20414
[35]	Chen M and Deng W 2014 Commun. Comput. Phys. 16 516 doi: 10.4208/cicp.120713.280214a
[6]	Yor M 2001 Exponential Functionals of Brownian Motion and Related Processes (Springer Science & Business Media) pp. 182-203
[7]	Kac M 1949 Trans. Am. Math. Soc. 65 1 doi: 10.1090/S0002-9947-1949-0027960-X
[36]	Abate J 1995 ORSA J. Comput. 7 36 doi: 10.1287/ijoc.7.1.36
[8]	Rueangkham N and Modchang C 2016 Chin. Phys. B 25 048201 doi: 10.1088/1674-1056/25/4/048201
[37]	Barkai E 2001 Phys. Rev. E 63 046118 doi: 10.1103/PhysRevE.63.046118
[9]	Hu W and Shao Y 2014 Acta Phys. Sin. 63 238202 (in Chinese)
[38]	Barkai E, Metzler R and Klafter J 2000 Phys. Rev. E 61 132
[10]	Abdelwahed H, El-Shewy E and Mahmoud A 2017 Chin. Phys. Lett. 34 035202 doi: 10.1088/0256-307X/34/3/035202
[39]	Metzler R and Klafter J 2000 Phys. Rep. 339 1 doi: 10.1016/S0370-1573(00)00070-3
[11]	Agmon N 1984 J. Chem. Phys. 81 3644 doi: 10.1063/1.448113
[40]	Metzler R, Barkai E and Klafter J 1999 Phys. Rev. Lett. 82 3653 doi: 10.1103/PhysRevLett.82.3653
[12]	Carmi S, Turgeman L and Barkai E 2010 J. Stat. Phys. 141 1071 doi: 10.1007/s10955-010-0086-6
[13]	Carmi S and Barkai E 2011 Phys. Rev. E 84 061104 doi: 10.1103/PhysRevE.84.061104
[14]	Turgeman L, Carmi S and Barkai E 2009 Phys. Rev. Lett. 103 190201 doi: 10.1103/PhysRevLett.103.190201
[15]	Friedrich R, Jenko F, Baule A and Eule S 2006 Phys. Rev. Lett. 96 230601 doi: 10.1103/PhysRevLett.96.230601
[16]	Chen M, Deng W and Wu Y 2013 Appl. Numer. Math. 70 22 doi: 10.1016/j.apnum.2013.03.006
[17]	Chen S, Liu F, Zhuang P and Anh V 2009 Appl. Math. Model. 33 256 doi: 10.1016/j.apm.2007.11.005
[18]	Gao G and Sun Z 2011 J. Comput. Phys. 230 586 doi: 10.1016/j.jcp.2010.10.007
[19]	Meerschaert M and Tadjeran C 2006 Appl. Numer. Math. 56 80 doi: 10.1016/j.apnum.2005.02.008
[20]	Sun Z and Wu X 2006 Appl. Numer. Math. 56 193 doi: 10.1016/j.apnum.2005.03.003
[21]	Deng W 2008 SIAM J. Numer. Anal. 47 204
[22]	Ervin V and Roop J 2006 Numer. Methods Partial Differential Equations 22 558 doi: 10.1002/num.20112
[23]	Jiang Y and Ma J 2011 J. Comput. Appl. Math. 235 3285 doi: 10.1016/j.cam.2011.01.011
[24]	Liu Y, Li H, Gao W, He S and Fang Z 2014 Sci. World J. 2014 141467
[25]	Liu Y, Fang Z, Li H and He S 2014 Appl. Math. Comput. 243 703
[26]	Liu F, Zhuang P, Turner I, Burrage K and Anh V 2014 Appl. Math. Model. 38 3871 doi: 10.1016/j.apm.2013.10.007
[27]	Feng L, Zhuang P, Liu F and Turner I 2015 Appl. Math. Comput. 257 52
[28]	Zhao J, Li H, Fang Z and Liu Y 2019 Mathematics 7 600 doi: 10.3390/math7070600
[29]	Li C, Zeng F and Liu F 2012 Fract. Calc. Appl. Anal. 15 383
[30]	Li X and Xu C 2009 SIAM J. Numer. Anal. 47 2108 doi: 10.1137/080718942
[31]	Deng W, Chen M and Barkai E 2015 J. Sci. Comput. 62 718 doi: 10.1007/s10915-014-9873-6
[32]	Chen M and Deng W 2013 ESAIM Math. Model. Numer. Anal. 49 373
[33]	Sun Z 2009 The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations (Beijing: Science Press) p. 126
[34]	Liao H and Sun Z 2010 Numer. Methods Partial Differential Equations 26 37 doi: 10.1002/num.20414
[35]	Chen M and Deng W 2014 Commun. Comput. Phys. 16 516 doi: 10.4208/cicp.120713.280214a
[36]	Abate J 1995 ORSA J. Comput. 7 36 doi: 10.1287/ijoc.7.1.36
[37]	Barkai E 2001 Phys. Rev. E 63 046118 doi: 10.1103/PhysRevE.63.046118
[38]	Barkai E, Metzler R and Klafter J 2000 Phys. Rev. E 61 132
[39]	Metzler R and Klafter J 2000 Phys. Rep. 339 1 doi: 10.1016/S0370-1573(00)00070-3
[40]	Metzler R, Barkai E and Klafter J 1999 Phys. Rev. Lett. 82 3653 doi: 10.1103/PhysRevLett.82.3653

Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative

Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative

RichHTML

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献 40

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	Junqi Lai(赖君奇), Bowen Chen(陈博文), Zhiwei Xing(邢志伟), Xuefei Li(李雪飞), Shulong Lu(陆书龙), Qi Chen(陈琪), and Liwei Chen(陈立桅). Quantitative measurement of the charge carrier concentration using dielectric force microscopy[J]. 中国物理B, 2023, 32(3): 37202-037202.
[2]	Ming-Jing Du(杜明婧), Bao-Jun Sun(孙宝军), and Ge Kai(凯歌). Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics[J]. 中国物理B, 2023, 32(3): 30202-030202.
[3]	Yan Zhou(周岩), Peiyu Ji(季佩宇), Maoyang Li(李茂洋), Lanjian Zhuge(诸葛兰剑), and Xuemei Wu(吴雪梅). Influence of magnetic field on power deposition in high magnetic field helicon experiment[J]. 中国物理B, 2023, 32(2): 25205-025205.
[4]	Jian Li(李健) and Hai-Qing Lin(林海青). Gauss quadrature based finite temperature Lanczos method[J]. 中国物理B, 2022, 31(5): 50203-050203.
[5]	Jun Li(李军) and Dao-Xin Yao(姚道新). Superconductivity in octagraphene[J]. 中国物理B, 2022, 31(1): 17403-017403.
[6]	Kun Wang(王焜), Xuehua Hu(胡学华), Sayipjamal Dulat, and Bai-Song Xie(谢柏松). Effect of symmetrical frequency chirp on pair production[J]. 中国物理B, 2021, 30(6): 60204-060204.
[7]	. [J]. 中国物理B, 2021, 30(2): 20202-0.
[8]	何林泽, 张满超, 吴春旺, 谢艺, 吴伟, 陈平形. Effects of imperfect pulses on dynamical decoupling using quantum trajectory method[J]. 中国物理B, 2018, 27(12): 120303-120303.
[9]	徐文静, 徐伟, 蔡力. Heteroclinic cycles in a new class of four-dimensional discontinuous piecewise affine systems[J]. 中国物理B, 2018, 27(11): 110201-110201.
[10]	马吉超, 魏高峰, 刘丹丹. Improved reproducing kernel particle method for piezoelectric materials[J]. 中国物理B, 2018, 27(1): 10201-010201.
[11]	庞明宝, 任泊宁. Effects of rainy weather on traffic accidents of a freeway using cellular automata model[J]. 中国物理B, 2017, 26(10): 108901-108901.
[12]	吴意, 马永其, 冯伟, 程玉民. Topology optimization using the improved element-free Galerkin method for elasticity[J]. 中国物理B, 2017, 26(8): 80203-080203.
[13]	傅浩, 周炜恩, 钱旭, 宋松和, 张利英. Conformal structure-preserving method for damped nonlinear Schrödinger equation[J]. 中国物理B, 2016, 25(11): 110201-110201.
[14]	杨志强, 刘世伟, 孙毅. Statistical second-order two-scale analysis and computation for heat conduction problem with radiation boundary condition in porous materials[J]. 中国物理B, 2016, 25(9): 90202-090202.
[15]	卢佳, 周怀春. Finite-difference time-domain modeling of curved material interfaces by using boundary condition equations method[J]. 中国物理B, 2016, 25(9): 90203-090203.