Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method
This paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo se...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/3/127 |
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| author | Adel R. Hadhoud Abdulqawi A. M. Rageh Taha Radwan |
| author_facet | Adel R. Hadhoud Abdulqawi A. M. Rageh Taha Radwan |
| author_sort | Adel R. Hadhoud |
| collection | DOAJ |
| description | This paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo sense. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>−</mo></mrow></semantics></math></inline-formula>approximation method is used for time-fractional derivative discretization. Using Von Neumann stability analysis, the proposed technique is shown to be conditionally stable. Numerical examples are solved to verify the accuracy and effectiveness of this method. The error norms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>2</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula> are also calculated at different values of <i>N</i> and <i>t</i> to evaluate the performance of the suggested method. |
| first_indexed | 2024-03-09T19:49:19Z |
| format | Article |
| id | doaj.art-812aef2313cd4cbd9069964de16428fb |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T19:49:19Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-812aef2313cd4cbd9069964de16428fb2023-11-24T01:14:05ZengMDPI AGFractal and Fractional2504-31102022-02-016312710.3390/fractalfract6030127Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation MethodAdel R. Hadhoud0Abdulqawi A. M. Rageh1Taha Radwan2Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebeen El-Kom 13829, EgyptDepartment of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebeen El-Kom 13829, EgyptDepartment of Mathematics, College of Science and Arts, Qassim University, Ar Rass 51452, Saudi ArabiaThis paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo sense. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>−</mo></mrow></semantics></math></inline-formula>approximation method is used for time-fractional derivative discretization. Using Von Neumann stability analysis, the proposed technique is shown to be conditionally stable. Numerical examples are solved to verify the accuracy and effectiveness of this method. The error norms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>2</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula> are also calculated at different values of <i>N</i> and <i>t</i> to evaluate the performance of the suggested method.https://www.mdpi.com/2504-3110/6/3/127fractional Schrödinger equationstrigonometric B-splines methodCaputo derivativevon neumann methodstability analysis |
| spellingShingle | Adel R. Hadhoud Abdulqawi A. M. Rageh Taha Radwan Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method Fractal and Fractional fractional Schrödinger equations trigonometric B-splines method Caputo derivative von neumann method stability analysis |
| title | Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method |
| title_full | Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method |
| title_fullStr | Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method |
| title_full_unstemmed | Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method |
| title_short | Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method |
| title_sort | computational solution of the time fractional schrodinger equation by using trigonometric b spline collocation method |
| topic | fractional Schrödinger equations trigonometric B-splines method Caputo derivative von neumann method stability analysis |
| url | https://www.mdpi.com/2504-3110/6/3/127 |
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