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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Main Author:	Abd Hamid, Nur Nadiah
Format:	Thesis
Language:	English
Published:	2016
Subjects:	QA1 Mathematics (General)
Online Access:	http://eprints.usm.my/31477/1/NUR_NADIAH_BINTI_ABD_HAMID_24.pdf
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id	my-usm-ep.31477
record_format	uketd_dc
spelling	my-usm-ep.314772019-04-12T05:25:21Z Splines For Two-Dimensional Partial Differential Equations 2016-04 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 http://eprints.usm.my/31477/ http://eprints.usm.my/31477/1/NUR_NADIAH_BINTI_ABD_HAMID_24.pdf application/pdf en public phd doctoral Universiti Sains Malaysia Pusat Pengajian Sains Matematik (School of Mathematical Sciences)
institution	Universiti Sains Malaysia
collection	USM Institutional Repository
language	English
topic	QA1 Mathematics (General)
spellingShingle	QA1 Mathematics (General) Abd Hamid, Nur Nadiah Splines For Two-Dimensional Partial Differential Equations
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.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Abd Hamid, Nur Nadiah
author_facet	Abd Hamid, Nur Nadiah
author_sort	Abd Hamid, Nur Nadiah
title	Splines For Two-Dimensional Partial Differential Equations
title_short	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_sort	splines for two-dimensional partial differential equations
granting_institution	Universiti Sains Malaysia
granting_department	Pusat Pengajian Sains Matematik (School of Mathematical Sciences)
publishDate	2016
url	http://eprints.usm.my/31477/1/NUR_NADIAH_BINTI_ABD_HAMID_24.pdf
_version_	1747820433258840064

Splines For Two-Dimensional Partial Differential Equations

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