Volume 12, Issue 4
A Linearized Second-Order Difference Scheme for the Nonlinear Time-Fractional Fourth-Order Reaction-Diffusion Equation

Hong Sun, Zhi-zhong Sun & Rui Du

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 1168-1190.

Published online: 2019-06

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  • Abstract

This paper presents a second-order linearized finite difference scheme for the nonlinear time-fractional fourth-order reaction-diffusion equation. The temporal Caputo derivative is approximated by $L2$-$1_\sigma$ formula with the approximation order of $\mathcal{O}(\tau^{3-\alpha}).$ The unconditional stability and convergence of the proposed scheme are proved by the discrete energy method. The scheme can achieve the global second-order numerical accuracy both in space and time. Three numerical examples are given to verify the numerical accuracy and efficiency of the difference scheme.


  • Keywords

Fractional differential equation, Caputo derivative, high order equation, nonlinear, linearized, difference scheme, convergence, stability.

  • AMS Subject Headings

65M06, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-12-1168, author = {Sun , Hong and Sun , Zhi-zhong and Du , Rui }, title = {A Linearized Second-Order Difference Scheme for the Nonlinear Time-Fractional Fourth-Order Reaction-Diffusion Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {4}, pages = {1168--1190}, abstract = {

This paper presents a second-order linearized finite difference scheme for the nonlinear time-fractional fourth-order reaction-diffusion equation. The temporal Caputo derivative is approximated by $L2$-$1_\sigma$ formula with the approximation order of $\mathcal{O}(\tau^{3-\alpha}).$ The unconditional stability and convergence of the proposed scheme are proved by the discrete energy method. The scheme can achieve the global second-order numerical accuracy both in space and time. Three numerical examples are given to verify the numerical accuracy and efficiency of the difference scheme.


}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2017-0144}, url = {http://global-sci.org/intro/article_detail/nmtma/13219.html} }
TY - JOUR T1 - A Linearized Second-Order Difference Scheme for the Nonlinear Time-Fractional Fourth-Order Reaction-Diffusion Equation AU - Sun , Hong AU - Sun , Zhi-zhong AU - Du , Rui JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1168 EP - 1190 PY - 2019 DA - 2019/06 SN - 12 DO - http://dor.org/10.4208/nmtma.OA-2017-0144 UR - https://global-sci.org/intro/nmtma/13219.html KW - Fractional differential equation, Caputo derivative, high order equation, nonlinear, linearized, difference scheme, convergence, stability. AB -

This paper presents a second-order linearized finite difference scheme for the nonlinear time-fractional fourth-order reaction-diffusion equation. The temporal Caputo derivative is approximated by $L2$-$1_\sigma$ formula with the approximation order of $\mathcal{O}(\tau^{3-\alpha}).$ The unconditional stability and convergence of the proposed scheme are proved by the discrete energy method. The scheme can achieve the global second-order numerical accuracy both in space and time. Three numerical examples are given to verify the numerical accuracy and efficiency of the difference scheme.


Hong Sun, Zhi-zhong Sun & Rui Du. (2019). A Linearized Second-Order Difference Scheme for the Nonlinear Time-Fractional Fourth-Order Reaction-Diffusion Equation. Numerical Mathematics: Theory, Methods and Applications. 12 (4). 1168-1190. doi:10.4208/nmtma.OA-2017-0144
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