Fourth-Order Spline Methods For Solving Nonlinear Schrödinger Equation
The Nonlinear Schrödinger (NLS) equation is an important and fundamental equation in Mathematical Physics. In this thesis, fourth-order cubic B-spline collocation method and fourth-order cubic Exponential B-spline collocation method are developed in order to solve problems involving the NLS equat...
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| Format: | Thesis |
| Language: | English |
| Published: |
2021
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| Subjects: | |
| Online Access: | http://eprints.usm.my/55075/1/AZHAR%20BIN%20AHMAD%20-Thesis.pdf |
| _version_ | 1825907222245277696 |
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| author | Ahmad, Azhar |
| author_facet | Ahmad, Azhar |
| author_sort | Ahmad, Azhar |
| collection | USM |
| description | The Nonlinear Schrödinger (NLS) equation is an important and fundamental equation
in Mathematical Physics. In this thesis, fourth-order cubic B-spline collocation
method and fourth-order cubic Exponential B-spline collocation method are developed
in order to solve problems involving the NLS equation. The established Cubic
B-spline Collocation Method and Cubic Exponential B-spline Collocation Method are
of second-order accuracy. The methods developed in this thesis are of fourth-order
accuracy. The time dimension of the NLS equation is discretized using the Finite Difference
Method and the space dimension is discretized based on the particular B-spline
methods used. The Taylor series approach and Besse approaches are used to handle
the nonlinear term of the NLS equation. Since the methods result in an underdetermined
system, the supplementary initial and boundary conditions are used to solve the
system. The developed methods are tested for stability and are found to be unconditionally
stable. Error analysis and convergence analysis are also carried out. The
efficiency of the methods are assessed on three test problems involving solitons and
the approximations are found to be very accurate. Besides that, the numerical order
of convergence is calculated and associated theoretical statements are proved. In conclusion,
the proposed methods in this study worked well and give accurate numerical
results for the NLS equation. |
| first_indexed | 2024-03-06T16:00:58Z |
| format | Thesis |
| id | usm.eprints-55075 |
| institution | Universiti Sains Malaysia |
| language | English |
| last_indexed | 2024-03-06T16:00:58Z |
| publishDate | 2021 |
| record_format | dspace |
| spelling | usm.eprints-550752022-10-03T03:53:48Z http://eprints.usm.my/55075/ Fourth-Order Spline Methods For Solving Nonlinear Schrödinger Equation Ahmad, Azhar QA101-145 Elementary Mathematics, Arithmetic The Nonlinear Schrödinger (NLS) equation is an important and fundamental equation in Mathematical Physics. In this thesis, fourth-order cubic B-spline collocation method and fourth-order cubic Exponential B-spline collocation method are developed in order to solve problems involving the NLS equation. The established Cubic B-spline Collocation Method and Cubic Exponential B-spline Collocation Method are of second-order accuracy. The methods developed in this thesis are of fourth-order accuracy. The time dimension of the NLS equation is discretized using the Finite Difference Method and the space dimension is discretized based on the particular B-spline methods used. The Taylor series approach and Besse approaches are used to handle the nonlinear term of the NLS equation. Since the methods result in an underdetermined system, the supplementary initial and boundary conditions are used to solve the system. The developed methods are tested for stability and are found to be unconditionally stable. Error analysis and convergence analysis are also carried out. The efficiency of the methods are assessed on three test problems involving solitons and the approximations are found to be very accurate. Besides that, the numerical order of convergence is calculated and associated theoretical statements are proved. In conclusion, the proposed methods in this study worked well and give accurate numerical results for the NLS equation. 2021-06 Thesis NonPeerReviewed application/pdf en http://eprints.usm.my/55075/1/AZHAR%20BIN%20AHMAD%20-Thesis.pdf Ahmad, Azhar (2021) Fourth-Order Spline Methods For Solving Nonlinear Schrödinger Equation. PhD thesis, Perpustakaan Hamzah Sendut. |
| spellingShingle | QA101-145 Elementary Mathematics, Arithmetic Ahmad, Azhar Fourth-Order Spline Methods For Solving Nonlinear Schrödinger Equation |
| title | Fourth-Order Spline Methods For
Solving Nonlinear Schrödinger
Equation |
| title_full | Fourth-Order Spline Methods For
Solving Nonlinear Schrödinger
Equation |
| title_fullStr | Fourth-Order Spline Methods For
Solving Nonlinear Schrödinger
Equation |
| title_full_unstemmed | Fourth-Order Spline Methods For
Solving Nonlinear Schrödinger
Equation |
| title_short | Fourth-Order Spline Methods For
Solving Nonlinear Schrödinger
Equation |
| title_sort | fourth order spline methods for solving nonlinear schrodinger equation |
| topic | QA101-145 Elementary Mathematics, Arithmetic |
| url | http://eprints.usm.my/55075/1/AZHAR%20BIN%20AHMAD%20-Thesis.pdf |
| work_keys_str_mv | AT ahmadazhar fourthordersplinemethodsforsolvingnonlinearschrodingerequation |