Search Advanced

AIMS Mathematics

2021, Volume 6, Issue 12: 13187-13209. doi: 10.3934/math.2021762
Previous Article Next Article
Research article

A high order numerical method for solving Caputo nonlinear fractional ordinary differential equations

  • School of Data Science and Information Engineering, Guizhou Minzu University, 550025 Guiyang, China
  • Received: 01 August 2021 Accepted: 09 September 2021 Published: 15 September 2021

  • MSC : 26A33, 65L12, 65M06, 65M12

  • In this paper, we construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations. Firstly, we use the piecewise Quadratic Lagrange interpolation method to construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations, and then analyze the local truncation error of the high order numerical scheme. Secondly, based on the local truncation error, the convergence order of $ 3-\theta $ order is obtained. And the convergence are strictly analyzed. Finally, the numerical simulation of the high order numerical scheme is carried out. Through the calculation of typical problems, the effectiveness of the numerical algorithm and the correctness of theoretical analysis are verified.

    Citation: Xumei Zhang, Junying Cao. A high order numerical method for solving Caputo nonlinear fractional ordinary differential equations[J]. AIMS Mathematics, 2021, 6(12): 13187-13209. doi: 10.3934/math.2021762

    Related Papers:

  • In this paper, we construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations. Firstly, we use the piecewise Quadratic Lagrange interpolation method to construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations, and then analyze the local truncation error of the high order numerical scheme. Secondly, based on the local truncation error, the convergence order of $ 3-\theta $ order is obtained. And the convergence are strictly analyzed. Finally, the numerical simulation of the high order numerical scheme is carried out. Through the calculation of typical problems, the effectiveness of the numerical algorithm and the correctness of theoretical analysis are verified.


    加载中


    [1] C. W. Lv, C. J. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput., 38 (2016), A2699–A2724. doi: 10.1137/15M102664X
    [2] J. Y. Cao, C. J. Xu, Z. Q. Wang, A high order finite difference/spectral approximations to the time fractional diffusion equations, Adv. Mater. Res., 875 (2014), 781–785.
    [3] J. Y. Cao, C. J. Xu, A high order schema for the nmerical solution of the fractional ordinary differential equations, J. Comput. Phys., 238 (2013), 154–168. doi: 10.1016/j.jcp.2012.12.013
    [4] H. L. Liao, W. McLean, J. W. Zhang, A discrete gronwall inequality with applications to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal., 57 (2019), 218–237. doi: 10.1137/16M1175742
    [5] W. P. Bu, L. Ji, Y. F. Tang, J. Zhou, Space-time finite element method for the distributed-order time fractional reaction diffusion equations, Appl. Numer. Math., 152 (2020), 446–465. doi: 10.1016/j.apnum.2019.11.010
    [6] G. H. Gao, Z. Z. Sun, H. W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259 (2014), 33–50. doi: 10.1016/j.jcp.2013.11.017
    [7] C. P. Xie, S. M. Fang, Finite difference scheme for time fractional diffusion equation with fractional boundary conditions, Math. Method. Appl. Sci., 43 (2020), 3473–3487. doi: 10.1002/mma.6132
    [8] C. P. Li, M. Cai, High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations: revisited, Numer. Funct. Anal. Optim., 38 (2017), 861–890. doi: 10.1080/01630563.2017.1291521
    [9] J. Alessandra, S. M. Paola, On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations, AIMS Mathematics, 6 (2021), 9109–9125. doi: 10.3934/math.2021529
    [10] Y. Zhang, X. B. Bao, L. B. Liu, Analysis of a finite difference scheme for a nonlinear Caputo fractional differential equation on an adaptive grid, AIMS Mathematics, 6 (2021), 8611–8624. doi: 10.3934/math.2021500
    [11] Z. J. Meng, M. X. Yi, J. Huang, L. Song, Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials, Appl. Math. Comput., 336 (2018), 454–464.
    [12] S. Abbas, D. Mehdi, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59 (2010), 1326–1336. doi: 10.1016/j.camwa.2009.07.006
    [13] M. R. Eslahchi, M. Dehghan, M. Parvizi, Application of the collocation method for solving nonlinear fractional integro-differential equations, J. Comput. Appl. Math., 257 (2014), 105–128. doi: 10.1016/j.cam.2013.07.044
    [14] Y. N. Zhang, Z. Z. Sun, X. Zhao, Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal., 50 (2012), 1535–1555. doi: 10.1137/110840959
    [15] W. H. Luo, C. P. Li, T. Z. Huang, X. M. Gu, G. C. Wu, A high-order accurate numerical scheme for the Caputo derivative with applications to fractional diffusion problems, Numer. Funct. Anal. Optim., 39 (2018), 600–622. doi: 10.1080/01630563.2017.1402346
    [16] S. Lee, H. Kim, B. Jang, A fast and high-order numerical method for nonlinear fractional-order differential equations with non-singular kernel, Appl. Numer. Math., 163 (2021), 57–76. doi: 10.1016/j.apnum.2021.01.013
    [17] N. Khadijeh, D. Raziyeh, Galerkin finite element method for nonlinear fractional differential equations, Numer. Algorithms, 88 (2021), 113–141. doi: 10.1007/s11075-020-01032-2
    [18] Y. L. Guo, Z. Q. Wang, An hp-version Chebyshev collocation method for nonlinear fractional differential equations, Appl. Numer. Math., 158 (2020), 194–211. doi: 10.1016/j.apnum.2020.08.003
    [19] Z. D. Gu, Spectral collocation method for nonlinear Riemann-Liouville fractional differential equations, Appl. Numer. Math., 157 (2020), 654–669. doi: 10.1016/j.apnum.2020.07.003
    [20] I. Podlubny, Fractional differential equations, New York: Academic Press, 1999.
    [21] H. Xi, Y. Gu, Generalized finite difference method for electroelastic analysis of three-dimensional piezoelectric structures, Appl. Math. Lett., 117 (2021), 107084. doi: 10.1016/j.aml.2021.107084
    [22] Y. Gu, C. M. Fan, Z. J. Fu, Localized method of fundamental solutions for three-dimensional elasticity problems: theory, Adv. Appl. Math. Mech., 13 (2021), 1520–1534. doi: 10.4208/aamm.OA-2020-0134
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
2.2 3

Metrics

Article views(1557) PDF downloads(112) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(6)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog