An Explicit Numerical Method for the Fractional Cable Equation
An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Gr...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2011/231920 |
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| _version_ | 1832558180799873024 |
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| author | J. Quintana-Murillo S. B. Yuste |
| author_facet | J. Quintana-Murillo S. B. Yuste |
| author_sort | J. Quintana-Murillo |
| collection | DOAJ |
| description | An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicted very accurately the conditions under which the present explicit method is stable. This was thoroughly checked by means of extensive numerical integrations. |
| format | Article |
| id | doaj-art-67844ce06f524e538f4553f0ee4cef78 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-67844ce06f524e538f4553f0ee4cef782025-02-03T01:32:59ZengWileyInternational Journal of Differential Equations1687-96431687-96512011-01-01201110.1155/2011/231920231920An Explicit Numerical Method for the Fractional Cable EquationJ. Quintana-Murillo0S. B. Yuste1Departamento de Física, Universidad de Extremadura, 06071 Badajoz, SpainDepartamento de Física, Universidad de Extremadura, 06071 Badajoz, SpainAn explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicted very accurately the conditions under which the present explicit method is stable. This was thoroughly checked by means of extensive numerical integrations.http://dx.doi.org/10.1155/2011/231920 |
| spellingShingle | J. Quintana-Murillo S. B. Yuste An Explicit Numerical Method for the Fractional Cable Equation International Journal of Differential Equations |
| title | An Explicit Numerical Method for the Fractional Cable Equation |
| title_full | An Explicit Numerical Method for the Fractional Cable Equation |
| title_fullStr | An Explicit Numerical Method for the Fractional Cable Equation |
| title_full_unstemmed | An Explicit Numerical Method for the Fractional Cable Equation |
| title_short | An Explicit Numerical Method for the Fractional Cable Equation |
| title_sort | explicit numerical method for the fractional cable equation |
| url | http://dx.doi.org/10.1155/2011/231920 |
| work_keys_str_mv | AT jquintanamurillo anexplicitnumericalmethodforthefractionalcableequation AT sbyuste anexplicitnumericalmethodforthefractionalcableequation AT jquintanamurillo explicitnumericalmethodforthefractionalcableequation AT sbyuste explicitnumericalmethodforthefractionalcableequation |