A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes
A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML"...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-12-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/1/40 |
| _version_ | 1797442295192092672 |
|---|---|
| author | Yusuf O. Afolabi Toheeb A. Biala Olaniyi S. Iyiola Abdul Q. M. Khaliq Bruce A. Wade |
| author_facet | Yusuf O. Afolabi Toheeb A. Biala Olaniyi S. Iyiola Abdul Q. M. Khaliq Bruce A. Wade |
| author_sort | Yusuf O. Afolabi |
| collection | DOAJ |
| description | A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo></mrow></semantics></math></inline-formula>), where time accuracy is of the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>. To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations. |
| first_indexed | 2024-03-09T12:39:46Z |
| format | Article |
| id | doaj.art-516a39137c3042a98929ae8c92353545 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T12:39:46Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-516a39137c3042a98929ae8c923535452023-11-30T22:19:22ZengMDPI AGFractal and Fractional2504-31102022-12-01714010.3390/fractalfract7010040A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded MeshesYusuf O. Afolabi0Toheeb A. Biala1Olaniyi S. Iyiola2Abdul Q. M. Khaliq3Bruce A. Wade4Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USADepartment of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37132, USADepartment of Mathematics, Clarkson University, Potsdam, NY 13699, USADepartment of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37132, USADepartment of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USAA weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo></mrow></semantics></math></inline-formula>), where time accuracy is of the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>. To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations.https://www.mdpi.com/2504-3110/7/1/40predictor-corrector schemeCaputo fractional derivativenonlinear time–space fractional equationmatrix transfergraded meshes |
| spellingShingle | Yusuf O. Afolabi Toheeb A. Biala Olaniyi S. Iyiola Abdul Q. M. Khaliq Bruce A. Wade A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes Fractal and Fractional predictor-corrector scheme Caputo fractional derivative nonlinear time–space fractional equation matrix transfer graded meshes |
| title | A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes |
| title_full | A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes |
| title_fullStr | A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes |
| title_full_unstemmed | A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes |
| title_short | A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes |
| title_sort | second order crank nicolson type scheme for nonlinear space time reaction diffusion equations on time graded meshes |
| topic | predictor-corrector scheme Caputo fractional derivative nonlinear time–space fractional equation matrix transfer graded meshes |
| url | https://www.mdpi.com/2504-3110/7/1/40 |
| work_keys_str_mv | AT yusufoafolabi asecondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes AT toheebabiala asecondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes AT olaniyisiyiola asecondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes AT abdulqmkhaliq asecondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes AT bruceawade asecondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes AT yusufoafolabi secondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes AT toheebabiala secondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes AT olaniyisiyiola secondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes AT abdulqmkhaliq secondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes AT bruceawade secondordercranknicolsontypeschemefornonlinearspacetimereactiondiffusionequationsontimegradedmeshes |