Splines For Two-Dimensional Partial Differential Equations
| dc.contributor.author | Abd Hamid, Nur Nadiah | |
| dc.date.accessioned | 2017-01-09T04:17:21Z | |
| dc.date.available | 2017-01-09T04:17:21Z | |
| dc.date.issued | 2016-04 | |
| dc.description.abstract | 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. For Poisson’s equations, BCBIM is found to produce comparable results with that of Finite Element Method. BCBIM generates slightly more accurate results than BCTBIM except for problems with trigonometric exact solutions. BCBIM scheme is proved to be consistent and unconditionally stable whereas BCTBIM conditionally stable.The numerical results of BCBIM and BCTBIM are found to be sublinearly convergent in directions x and y. For the heat equation, BCBIM is found to produce more accurate results than BCTBIM for both examples with trigonometric and non-trigonometric exact solutions. Otherwise, for the wave equation, BCTBIM is found to produce better results than BCBIM. Therefore, for the heat and wave equations, BCBIM and BCTBIM do not necessarily produce more accurate results than each other. Similarly, BCBIM schemes for the heat and wave equations are proved to be consistent and unconditionaly stable whereas BCTBIM schemes are shown to be conditionally stable. The numerical results from both BCBIM and BCTBIM are found to be sublinearly convergent in directions x, y, and t. These methods are then tested to more general and well-known partial differential equations with promising results. The equations are the advection-diffusion, Burgers’ equations, and linear hyperbolic equation that have many applications in the fields of fluid mechanics and wave phenomena. | en_US |
| dc.identifier.uri | http://hdl.handle.net/123456789/3352 | |
| dc.language.iso | en | en_US |
| dc.publisher | Universiti Sains Malaysia | en_US |
| dc.subject | Splines | en_US |
| dc.subject | Two-dimensional partial | en_US |
| dc.title | Splines For Two-Dimensional Partial Differential Equations | en_US |
| dc.type | Thesis | en_US |
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