Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations
In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-04-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/4/201 |
| _version_ | 1827672469968781312 |
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| author | Deng Wang Yang Liu Hong Li Zhichao Fang |
| author_facet | Deng Wang Yang Liu Hong Li Zhichao Fang |
| author_sort | Deng Wang |
| collection | DOAJ |
| description | In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results. |
| first_indexed | 2024-03-10T04:08:47Z |
| format | Article |
| id | doaj.art-028f237474f74e22816828d7ecb1bf50 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T04:08:47Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-028f237474f74e22816828d7ecb1bf502023-11-23T08:15:28ZengMDPI AGFractal and Fractional2504-31102022-04-016420110.3390/fractalfract6040201Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential EquationsDeng Wang0Yang Liu1Hong Li2Zhichao Fang3School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, ChinaSchool of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, ChinaSchool of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, ChinaSchool of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, ChinaIn this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results.https://www.mdpi.com/2504-3110/6/4/201nonlinear fourth-order fractional integro-differential equationWSGI approximationBDF2mixed finite element method |
| spellingShingle | Deng Wang Yang Liu Hong Li Zhichao Fang Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations Fractal and Fractional nonlinear fourth-order fractional integro-differential equation WSGI approximation BDF2 mixed finite element method |
| title | Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations |
| title_full | Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations |
| title_fullStr | Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations |
| title_full_unstemmed | Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations |
| title_short | Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations |
| title_sort | second order time stepping scheme combined with a mixed element method for a 2d nonlinear fourth order fractional integro differential equations |
| topic | nonlinear fourth-order fractional integro-differential equation WSGI approximation BDF2 mixed finite element method |
| url | https://www.mdpi.com/2504-3110/6/4/201 |
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