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 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

Splines For Linear Two-Point Boundary Value Problems

Linear two-point boundary value problems of order two are solved using cubic trigonometric B-spline, cubic Beta-spline and extended cubic B-spline interpolation methods. Cubic Beta-spline has two shape parameters, b1 and b2 while extended cubic B-spline has one, l . In this method, the parameters were varied and the corresponding approximations were compared to the exact solution to obtain the best values of b1, b2 and l . The methods were tested on four problems and the obtained approximated solutions were compared to that of cubic B-spline interpolation method. Trigonometric B-spline produced better approximation for problems with trigonometric form whereas Beta-spline and extended cubic B-spline produced more accurate approximation for some values of b1, b2 and l . All in all, extended cubic B-spline interpolation produced the most accurate solution out of the three splines. However, the method of finding l cannot be applied in the real world because there is no exact solution provided. That method was implemented in order to test whether values of l that produce better approximation do exist. Thus, an approach of finding optimized l is developed and Newton’s method was applied to it. This approach was found to approximate the solution much better than cubic B-spline interpolation method

Reforming the Malaysian tax system: the effects of Special Voluntary Disclosure Programme (SVDP), role of Inland Revenue Board Malaysia (IRBM), and demographics of taxpayers on Malaysian tax awareness

A sound tax system improves government revenue by encouraging voluntary tax compliance to ensure good tax awareness among citizens. However, many issues arise with regard to tax collection. As these issues remain unresolved, the government needs to frequently implement a tax amnesty programme which is not sustainable or cost-effective in the longer run. Tax awareness becomes a significant factor in managing tax compliance behaviour within contemporary literature. Hence, this paper aims at ascertaining the effects of Special Voluntary Disclosure Programme (SVDP), the role of Inland Revenue Board Malaysia (IRBM), and demographic factors on tax awareness among Malaysian taxpayers. This paper used Smart PLS version 3 to run the data. The results confirmed that SVDP, IRBM’s role, and public/ private sectors’ employment are crucial in influencing tax awareness. These findings have implications for tax policies, such as developing tax knowledge programmes and raising public awareness among individual taxpayers in Malaysia

Extended cubic B-spline method for linear two-point boundary value problems

Second order linear two-point boundary value problems were solved using extended cubic B-spline interpolation method. Extended cubic B-spline is an extension of cubic B-spline consisting of one shape parameter, called λ. The resulting approximated analytical solution for the problems would be a function of λ. Optimization of λ was carried out to find the best value of λ that generates the closest fit to the differential equations in the problems. This method approximated the solutions for the problems much more accurately compared to finite difference, finite element, finite volume and cubic B-spline interpolation methods

Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method

This paper is concerned with the numerical solution of the nonlinear Schrödinger (NLS) equation with Neumann boundary conditions by quintic B-spline Galerkin finite element method as the shape and weight functions over the finite domain. The Galerkin B-spline method is more efficient and simpler than the general Galerkin finite element method. For the Galerkin B-spline method, the Crank Nicolson and finite difference schemes are applied for nodal parameters and for time integration. Two numerical problems are discussed to demonstrate the accuracy and feasibility of the proposed method. The error norms L 2 , L ∞ and conservation laws I 1 ,   I 2 are calculated to check the accuracy and feasibility of the method. The results of the scheme are compared with previously obtained approximate solutions and are found to be in good agreement

Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method

The non-linear Schrödinger (NLS) equation has often been used as a model equation in the study of quantum states of physical systems. Numerical solution of NLS equation is obtained using cubic B-spline Galerkin method. We have applied the Crank–Nicolson scheme for time discretization and the cubic B-spline basis function for space discretization. Three numerical problems, including single soliton, interaction of two solitons and birth of standing soliton, are demonstrated to evaluate to the performance and accuracy of the method. The error norms and conservation laws are determined and found to be in good agreement with the published results. The obtained results show that the approach is feasible and accurate. The proposed method has almost second order convergence. The linear stability of the method is performed using the Von Neumann method