Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation
The article proposes a nonlocal explicit finite-difference scheme for the numerical solution of a nonlinear, ordinary differential equation with a derivative of a fractional variable order of the Gerasimov–Caputo type. The questions of approximation, convergence, and stability of this scheme are stu...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-07-01
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| Series: | Mathematical and Computational Applications |
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| Online Access: | https://www.mdpi.com/2297-8747/26/3/55 |
| _version_ | 1797518327394861056 |
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| author | Roman Parovik Dmitriy Tverdyi |
| author_facet | Roman Parovik Dmitriy Tverdyi |
| author_sort | Roman Parovik |
| collection | DOAJ |
| description | The article proposes a nonlocal explicit finite-difference scheme for the numerical solution of a nonlinear, ordinary differential equation with a derivative of a fractional variable order of the Gerasimov–Caputo type. The questions of approximation, convergence, and stability of this scheme are studied. It is shown that the nonlocal finite-difference scheme is conditionally stable and converges to the first order. Using the fractional Riccati equation as an example, the computational accuracy of the numerical method is analyzed. It is shown that with an increase in the nodes of the computational grid, the order of computational accuracy tends to unity, i.e., to the theoretical value of the order of accuracy. |
| first_indexed | 2024-03-10T07:28:18Z |
| format | Article |
| id | doaj.art-bb42b76cdaa448f48f26c2cbe2443790 |
| institution | Directory Open Access Journal |
| issn | 1300-686X 2297-8747 |
| language | English |
| last_indexed | 2024-03-10T07:28:18Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematical and Computational Applications |
| spelling | doaj.art-bb42b76cdaa448f48f26c2cbe24437902023-11-22T14:07:09ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472021-07-012635510.3390/mca26030055Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order EquationRoman Parovik0Dmitriy Tverdyi1Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences, 684034 Kamchatskiy Kray, RussiaFaculty of Physics and Mathematics, Vitus Bering Kamchatka State University, 683032 Petropavlovsk-Kamchatskiy, RussiaThe article proposes a nonlocal explicit finite-difference scheme for the numerical solution of a nonlinear, ordinary differential equation with a derivative of a fractional variable order of the Gerasimov–Caputo type. The questions of approximation, convergence, and stability of this scheme are studied. It is shown that the nonlocal finite-difference scheme is conditionally stable and converges to the first order. Using the fractional Riccati equation as an example, the computational accuracy of the numerical method is analyzed. It is shown that with an increase in the nodes of the computational grid, the order of computational accuracy tends to unity, i.e., to the theoretical value of the order of accuracy.https://www.mdpi.com/2297-8747/26/3/55fractional variable order derivativeexplicit finite-difference schemestabilityconvergencecomputational accuracy |
| spellingShingle | Roman Parovik Dmitriy Tverdyi Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation Mathematical and Computational Applications fractional variable order derivative explicit finite-difference scheme stability convergence computational accuracy |
| title | Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation |
| title_full | Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation |
| title_fullStr | Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation |
| title_full_unstemmed | Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation |
| title_short | Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation |
| title_sort | some aspects of numerical analysis for a model nonlinear fractional variable order equation |
| topic | fractional variable order derivative explicit finite-difference scheme stability convergence computational accuracy |
| url | https://www.mdpi.com/2297-8747/26/3/55 |
| work_keys_str_mv | AT romanparovik someaspectsofnumericalanalysisforamodelnonlinearfractionalvariableorderequation AT dmitriytverdyi someaspectsofnumericalanalysisforamodelnonlinearfractionalvariableorderequation |