Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


IMPACT FACTOR 2018: 3.514
5-year IMPACT FACTOR: 3.524

CiteScore 2018: 3.44

SCImago Journal Rank (SJR) 2018: 1.891
Source Normalized Impact per Paper (SNIP) 2018: 1.808

Mathematical Citation Quotient (MCQ) 2018: 1.08

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 18, Issue 3

Issues

High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)

Jianxiong Cao / Changpin Li / YangQuan Chen
Published Online: 2015-05-23 | DOI: https://doi.org/10.1515/fca-2015-0045

Abstract

In this paper, we first establish a high-order numerical algorithm for α-th (0 < α < 1) order Caputo derivative of a given function f(t), where the convergence rate is (4 − α)-th order. Then by using this new formula, an improved difference scheme with high order accuracy in time to solve Caputo-type fractional advection-diffusion equation with Dirichlet boundary conditions is constructed. Finally, numerical examples are carried out to confirm the efficiency of the constructed algorithm.

Keywords : Caputo derivative; advection-diffusion equation; difference scheme; stability; convergence

References

  • [1] A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280 (2015), 424-438.Google Scholar

  • [2] J. X. Cao, C. P. Li, Finite difference scheme for the time-space fractional diffusion equations. Centr. Eur. J. Phys. 11, No 10 (2013), 1440-1456.Web of ScienceGoogle Scholar

  • [3] C. Chen, F. Liu, I. Turner, V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, No 2 (2007), 886-897.Google Scholar

  • [4] J. X. Cao, C. P. Li, Y. Q. Chen, Compact differencemethod for solving the fractional reaction-subdiffusion equation with neumann boundary value condition. Int. J. Comput. Math. 92, No 1 (2015), 167-180.CrossrefWeb of ScienceGoogle Scholar

  • [5] G. Gao, Z. Sun, H. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259 (2014), 33-50.Web of ScienceGoogle Scholar

  • [6] B. Jin, R. Lazarov, Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal.; First Publ. online: Jan. 24, 2015; doi: 10.1093/imanum/dru063, 2015.CrossrefGoogle Scholar

  • [7] I. Karatay, N. Kale, S. R. Bayramoglu, A new difference scheme for time fractional heat equations based on the Crank-Nicholson method. Fract. Calc. Appl. Anal. 16, No 4 (2013), 892-910; DOI: 10.2478/s13540-013-0055-2; http://www.degruyter.com/view/j/fca.2013.16.issue-4/s13540-013-0055-2/s13540-013-0055-2.xml; http://link.springer.com/article/10.2478/s13540-013-0055-2.CrossrefGoogle Scholar

  • [8] C. P. Li, R. F. Wu, H. F. Ding, High-order approximation to Caputo derivative and Caputo-type advection-diffusion equation (I). Commun. Appl. Ind. Math. 7, No 1 (2015), In press.Google Scholar

  • [9] C. Lubich, Discretized fractional calculus. SIAM J. Math. Anal. 17, No 3 (1986), 704-719.CrossrefGoogle Scholar

  • [10] C. P. Li, A. Chen, J. J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230, No 9 (2011), 3352-3368.Web of ScienceGoogle Scholar

  • [11] K. B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York, 2006.Google Scholar

  • [12] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.Google Scholar

  • [13] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, B. M. Vinagre Jara, Matrix approach to discrete fractional calculus II: Partial fractional differential equations. J. Comput. Phys. 228, No 8 (2009), 3137-3153.Google Scholar

  • [14] E. Sousa, Numerical approximations for fractional diffusion equations via splines. Comput. Math. Appl. 62, No 3 (2011), 938-944.CrossrefGoogle Scholar

  • [15] E. Sousa, How to approximate the fractional derivative of order 1 < α ≤ 2. Int. J. Bifurcation Chaos. 22, No 4 (2012).CrossrefWeb of ScienceGoogle Scholar

  • [16] F. H. Zeng, C. P. Li, F. W. Liu, I. Turner, The use of finite difference/ element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35, No 6 (2013), A2976-A3000.Web of ScienceCrossrefGoogle Scholar

  • [17] F. H. Zeng, C. P. Li, F. Liu, I. Turner, Numerical algorithms for timefractional subdiffusion equation with second-order accuracy. SIAM. J. Sci. Comput. 37, No 1 (2015), A55-A78. CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2014-12-01

Published Online: 2015-05-23

Published in Print: 2015-06-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 18, Issue 3, Pages 735–761, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2015-0045.

Export Citation

© Diogenes Co., Sofia.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Hengfei Ding and Changpin Li
Numerical Methods for Partial Differential Equations, 2019
[2]
Yongliang Zhao, Peiyong Zhu, Xianming Gu, Xile Zhao, and Huanyan Jian
Journal of Physics: Conference Series, 2019, Volume 1324, Page 012030
[3]
R. Mokhtari and F. Mostajeran
Communications on Applied Mathematics and Computation, 2019
[4]
Yanting Zhao, Yiheng Wei, Jiachang Wang, and Yong Wang
SSRN Electronic Journal , 2018
[5]
Z. Soori and A. Aminataei
Applied Numerical Mathematics, 2019, Volume 144, Page 21
[6]
Haixiang Zhang, Xuehua Yang, and Da Xu
Numerical Algorithms, 2019, Volume 80, Number 3, Page 849
[7]
Xuhao Li and Patricia J. Y. Wong
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2019, Volume 99, Number 5, Page e201800094
[11]
Riccardo Fazio, Alessandra Jannelli, and Santa Agreste
Applied Sciences, 2018, Volume 8, Number 6, Page 960
[12]
Omar Abu Arqub
International Journal of Numerical Methods for Heat & Fluid Flow, 2018, Volume 28, Number 4, Page 828
[13]
Sadia Arshad, Jianfei Huang, Abdul Q.M. Khaliq, and Yifa Tang
Journal of Computational Physics, 2017
[14]
Changpin Li and An Chen
International Journal of Computer Mathematics, 2017, Page 1
[15]
Hengfei Ding and Changpin Li
Numerical Methods for Partial Differential Equations, 2017, Volume 33, Number 5, Page 1754
[17]
Changpin Li and Min Cai
Numerical Functional Analysis and Optimization, 2017, Volume 38, Number 7, Page 861
[18]
Yuping Ying, Yanping Lian, Shaoqiang Tang, and Wing Kam Liu
Computer Methods in Applied Mechanics and Engineering, 2017, Volume 317, Page 42
[19]
Kamlesh Kumar, Rajesh K. Pandey, and Shiva Sharma
Journal of Computational and Applied Mathematics, 2017, Volume 315, Page 287
[20]
Changpin Li, Qian Yi, and An Chen
Journal of Computational Physics, 2016, Volume 316, Page 614
[21]
Hefeng Li, Jianxiong Cao, and Changpin Li
Journal of Computational and Applied Mathematics, 2016, Volume 299, Page 159

Comments (0)

Please log in or register to comment.
Log in