A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method
The present manuscript include, finite difference method and quartic B-spline based differential quadrature method (FDM-DQM) to obtain the numerical solutions for the nonlinear Schr¨odinger (NLS) equation. For this purpose, firstly Schrödinger equation has been converted into coupled real value diff...
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| Format: | Article |
| Language: | English |
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Balikesir University
2019-07-01
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| Series: | An International Journal of Optimization and Control: Theories & Applications |
| Subjects: | |
| Online Access: | http://www.ijocta.org/index.php/files/article/view/709 |
| _version_ | 1797913266817597440 |
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| author | Ali Başhan |
| author_facet | Ali Başhan |
| author_sort | Ali Başhan |
| collection | DOAJ |
| description | The present manuscript include, finite difference method and quartic B-spline based differential quadrature method (FDM-DQM) to obtain the numerical solutions for the nonlinear Schr¨odinger (NLS) equation. For this purpose, firstly Schrödinger equation has been converted into coupled real value differential equations and then they have been discretized using special type of classical finite difference method namely, Crank-Nicolson scheme. After that, Rubin and Graves linearization techniques have been utilized and differential quadrature method has been applied. So, partial differential equation turn into algebraic equation system. Next, in order to be able to test the accuracy of the newly hybrid method, the error norms L2 and L? as well as the two lowest invariants I1 and I2 have been calculated. Besides those, the relative changes in those invariants have been given. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison has clearly indicated that the currently utilized method, namely FDM-DQM, is an effective and efficient numerical scheme
and allowed us to propose to solve a wide range of nonlinear equations. |
| first_indexed | 2024-04-10T12:10:16Z |
| format | Article |
| id | doaj.art-8c5a88b7c1384eafbc0ddc068aa3b8e6 |
| institution | Directory Open Access Journal |
| issn | 2146-0957 2146-5703 |
| language | English |
| last_indexed | 2024-04-10T12:10:16Z |
| publishDate | 2019-07-01 |
| publisher | Balikesir University |
| record_format | Article |
| series | An International Journal of Optimization and Control: Theories & Applications |
| spelling | doaj.art-8c5a88b7c1384eafbc0ddc068aa3b8e62023-02-15T16:16:03ZengBalikesir UniversityAn International Journal of Optimization and Control: Theories & Applications2146-09572146-57032019-07-019210.11121/ijocta.01.2019.00709A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature methodAli Başhan0Zonguldak Bulent Ecevit UniversityThe present manuscript include, finite difference method and quartic B-spline based differential quadrature method (FDM-DQM) to obtain the numerical solutions for the nonlinear Schr¨odinger (NLS) equation. For this purpose, firstly Schrödinger equation has been converted into coupled real value differential equations and then they have been discretized using special type of classical finite difference method namely, Crank-Nicolson scheme. After that, Rubin and Graves linearization techniques have been utilized and differential quadrature method has been applied. So, partial differential equation turn into algebraic equation system. Next, in order to be able to test the accuracy of the newly hybrid method, the error norms L2 and L? as well as the two lowest invariants I1 and I2 have been calculated. Besides those, the relative changes in those invariants have been given. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison has clearly indicated that the currently utilized method, namely FDM-DQM, is an effective and efficient numerical scheme and allowed us to propose to solve a wide range of nonlinear equations.http://www.ijocta.org/index.php/files/article/view/709Partial differential equationsDifferential quadrature methodQuartic B-SplinesSchr¨odinger equation. |
| spellingShingle | Ali Başhan A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method An International Journal of Optimization and Control: Theories & Applications Partial differential equations Differential quadrature method Quartic B-Splines Schr¨odinger equation. |
| title | A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method |
| title_full | A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method |
| title_fullStr | A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method |
| title_full_unstemmed | A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method |
| title_short | A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method |
| title_sort | mixed methods approach to schrodinger equation finite difference method and quartic b spline based differential quadrature method |
| topic | Partial differential equations Differential quadrature method Quartic B-Splines Schr¨odinger equation. |
| url | http://www.ijocta.org/index.php/files/article/view/709 |
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