A Mixed Element Algorithm Based on the Modified <i>L</i>1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model
In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupl...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-12-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/5/4/274 |
| _version_ | 1797504519909670912 |
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| author | Jinfeng Wang Baoli Yin Yang Liu Hong Li Zhichao Fang |
| author_facet | Jinfeng Wang Baoli Yin Yang Liu Hong Li Zhichao Fang |
| author_sort | Jinfeng Wang |
| collection | DOAJ |
| description | In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>1</mn></mrow></semantics></math></inline-formula>-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> error estimates are performed and the feasibility is validated by the calculated data. |
| first_indexed | 2024-03-10T04:05:45Z |
| format | Article |
| id | doaj.art-db6e23a30b514ce6ade8983aaf8263f3 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T04:05:45Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-db6e23a30b514ce6ade8983aaf8263f32023-11-23T08:24:39ZengMDPI AGFractal and Fractional2504-31102021-12-015427410.3390/fractalfract5040274A Mixed Element Algorithm Based on the Modified <i>L</i>1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave ModelJinfeng Wang0Baoli Yin1Yang Liu2Hong Li3Zhichao Fang4School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, 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, ChinaSchool of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, ChinaIn this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>1</mn></mrow></semantics></math></inline-formula>-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> error estimates are performed and the feasibility is validated by the calculated data.https://www.mdpi.com/2504-3110/5/4/274fourth-order fractional diffusion-wave equationmodified <i>L</i>1-formulamixed element methoda priori error estimates |
| spellingShingle | Jinfeng Wang Baoli Yin Yang Liu Hong Li Zhichao Fang A Mixed Element Algorithm Based on the Modified <i>L</i>1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model Fractal and Fractional fourth-order fractional diffusion-wave equation modified <i>L</i>1-formula mixed element method a priori error estimates |
| title | A Mixed Element Algorithm Based on the Modified <i>L</i>1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model |
| title_full | A Mixed Element Algorithm Based on the Modified <i>L</i>1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model |
| title_fullStr | A Mixed Element Algorithm Based on the Modified <i>L</i>1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model |
| title_full_unstemmed | A Mixed Element Algorithm Based on the Modified <i>L</i>1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model |
| title_short | A Mixed Element Algorithm Based on the Modified <i>L</i>1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model |
| title_sort | mixed element algorithm based on the modified i l i 1 crank nicolson scheme for a nonlinear fourth order fractional diffusion wave model |
| topic | fourth-order fractional diffusion-wave equation modified <i>L</i>1-formula mixed element method a priori error estimates |
| url | https://www.mdpi.com/2504-3110/5/4/274 |
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