Splines For Two-Dimensional Partial Differential Equations
Di dalam tesis ini, dua kaedah berasaskan splin dibangunkan untuk menyelesaikan persamaan pembezaan separa dua dimensi. Kaedah-kaedah tersebut adalah Kaedah Interpolasi Splin-B Bikubik (KISB) dan Kaedah Interpolasi Splin-B Trigonometri Bikubik (KISTB). Kajian ini adalah kesinambungan daripada per...
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| Format: | Thesis |
| Language: | English |
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2016
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| Online Access: | http://eprints.usm.my/31477/1/NUR_NADIAH_BINTI_ABD_HAMID_24.pdf |
| _version_ | 1825832545402486784 |
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| author | Abd Hamid, Nur Nadiah |
| author_facet | Abd Hamid, Nur Nadiah |
| author_sort | Abd Hamid, Nur Nadiah |
| collection | USM |
| description | Di dalam tesis ini, dua kaedah berasaskan splin dibangunkan untuk menyelesaikan persamaan
pembezaan separa dua dimensi. Kaedah-kaedah tersebut adalah Kaedah Interpolasi
Splin-B Bikubik (KISB) dan Kaedah Interpolasi Splin-B Trigonometri Bikubik (KISTB). Kajian
ini adalah kesinambungan daripada perkembangan terkini di dalam penggunaan kedua-dua
splin terhadap masalah-masalah satu dimensi. Pendekatan KISB dan KISTB adalah serupa kecuali
pada penggunaan fungsi asas splin yang berbeza, iaitu splin-B kubik dan splin-B trigonometri
kubik. Bagi masalah dengan pembolehubah masa, masa tersebut dipecahkan menggunakan
Kaedah Beza Terhingga yang biasa. Pembolehubah ruang pula dipecahkan menggunakan
fungsi permukaan splin bikubik. Dengan menambah syarat-syarat permulaan dan sempadan,
satu sistem persamaan linear yang underdetermined akan terhasil. Sistem ini kemudiannya diselesaikan
menggunakan Kaedah Kuasa Dua Terkecil. Persamaan-persamaan ini diselesaikan
menurut jenis-jenisnya, iaitu persamaan Poisson, persamaan haba, dan persamaan gelombang.
Persamaan-persamaan ini ialah persamaan yang paling mudah masing-masing daripada persamaan
pembezaan separa eliptik, parabolik, dan hiperbolik.
In this thesis, two spline-based methods are developed to solve two-dimensional partial
differential equations. The methods are Bicubic B-spline Interpolation Method (BCBIM) and
Bicubic Trigonometric B-spline Interpolation Method (BCTBIM). This study is a continuation
of recent developments in the application of both splines on the one-dimensional problems.
The approach of BCBIM and BCTBIM are similar except for the use of different spline basis
functions, namely cubic B-spline and cubic trigonometric B-spline, respectively. For problems
with time variable, the time is discretized using the usual Finite Difference Method. The
spatial variables are discretized using the corresponding bicubic spline surface function. By
adding the initial and boundary conditions, an underdetermined system of linear equations results.
This system is then solved using the method of Least Squares. The equations are dealt
according to its types, namely Poisson’s, heat, and wave equations. These equations are the
simplest form of elliptic, parabolic, and hyperbolic partial differential equations, respectively.
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| first_indexed | 2024-03-06T14:54:27Z |
| format | Thesis |
| id | usm.eprints-31477 |
| institution | Universiti Sains Malaysia |
| language | English |
| last_indexed | 2024-03-06T14:54:27Z |
| publishDate | 2016 |
| record_format | dspace |
| spelling | usm.eprints-314772019-04-12T05:25:21Z http://eprints.usm.my/31477/ Splines For Two-Dimensional Partial Differential Equations Abd Hamid, Nur Nadiah QA1 Mathematics (General) Di dalam tesis ini, dua kaedah berasaskan splin dibangunkan untuk menyelesaikan persamaan pembezaan separa dua dimensi. Kaedah-kaedah tersebut adalah Kaedah Interpolasi Splin-B Bikubik (KISB) dan Kaedah Interpolasi Splin-B Trigonometri Bikubik (KISTB). Kajian ini adalah kesinambungan daripada perkembangan terkini di dalam penggunaan kedua-dua splin terhadap masalah-masalah satu dimensi. Pendekatan KISB dan KISTB adalah serupa kecuali pada penggunaan fungsi asas splin yang berbeza, iaitu splin-B kubik dan splin-B trigonometri kubik. Bagi masalah dengan pembolehubah masa, masa tersebut dipecahkan menggunakan Kaedah Beza Terhingga yang biasa. Pembolehubah ruang pula dipecahkan menggunakan fungsi permukaan splin bikubik. Dengan menambah syarat-syarat permulaan dan sempadan, satu sistem persamaan linear yang underdetermined akan terhasil. Sistem ini kemudiannya diselesaikan menggunakan Kaedah Kuasa Dua Terkecil. Persamaan-persamaan ini diselesaikan menurut jenis-jenisnya, iaitu persamaan Poisson, persamaan haba, dan persamaan gelombang. Persamaan-persamaan ini ialah persamaan yang paling mudah masing-masing daripada persamaan pembezaan separa eliptik, parabolik, dan hiperbolik. In this thesis, two spline-based methods are developed to solve two-dimensional partial differential equations. The methods are Bicubic B-spline Interpolation Method (BCBIM) and Bicubic Trigonometric B-spline Interpolation Method (BCTBIM). This study is a continuation of recent developments in the application of both splines on the one-dimensional problems. The approach of BCBIM and BCTBIM are similar except for the use of different spline basis functions, namely cubic B-spline and cubic trigonometric B-spline, respectively. For problems with time variable, the time is discretized using the usual Finite Difference Method. The spatial variables are discretized using the corresponding bicubic spline surface function. By adding the initial and boundary conditions, an underdetermined system of linear equations results. This system is then solved using the method of Least Squares. The equations are dealt according to its types, namely Poisson’s, heat, and wave equations. These equations are the simplest form of elliptic, parabolic, and hyperbolic partial differential equations, respectively. 2016-04 Thesis NonPeerReviewed application/pdf en http://eprints.usm.my/31477/1/NUR_NADIAH_BINTI_ABD_HAMID_24.pdf Abd Hamid, Nur Nadiah (2016) Splines For Two-Dimensional Partial Differential Equations. PhD thesis, Universiti Sains Malaysia. |
| spellingShingle | QA1 Mathematics (General) Abd Hamid, Nur Nadiah Splines For Two-Dimensional Partial Differential Equations |
| title | Splines For Two-Dimensional Partial Differential Equations |
| title_full | Splines For Two-Dimensional Partial Differential Equations |
| title_fullStr | Splines For Two-Dimensional Partial Differential Equations |
| title_full_unstemmed | Splines For Two-Dimensional Partial Differential Equations |
| title_short | Splines For Two-Dimensional Partial Differential Equations |
| title_sort | splines for two dimensional partial differential equations |
| topic | QA1 Mathematics (General) |
| url | http://eprints.usm.my/31477/1/NUR_NADIAH_BINTI_ABD_HAMID_24.pdf |
| work_keys_str_mv | AT abdhamidnurnadiah splinesfortwodimensionalpartialdifferentialequations |