Nur Nadiah Abd Hamid

Solving Boussinesq equation using quintic B-spline and quintic trigonometric B-spline interpolation methods

The quintic B-spline (QBS) and quintic trigonometric B-spline (QTBS) functions are used to set up the collocation methods in finding solutions for the Boussinesq equation. The QBS and QTBS are applied as interpolating functions in the spatial dimension while the finite difference method (FDM) is used to discretize the time derivative. The nonlinear Boussinesq equation is linearized using Taylor’s expansion. The von Neumann stability analysis is used to analyze the schemes and they are shown to be conditionally stable. In order to demonstrate the capability of the schemes, some problems are solved and compared with the analytical solutions and generated results from the FDM. The proposed numerical schemes are found to be accurate.

Solving Dym equation using quartic B-spline and quartic trigonometric B-spline collocation methods

The nonlinear Dym equation is solved numerically using the quartic B-spline (QuBS) and quartic trigonometric B-spline (QuTBS) collocation methods. The QuBS and QuTBS are utilized as interpolating functions in the spatial dimension while the finite difference method (FDM) is applied to discretize the temporal space with the help of theta-weighted method. The nonlinear term in the Dym equation is linearized using Taylor’s expansion. Two schemes are performed on both methods which are Crank-Nicolson and fully implicit. Applying the Von-Neumann stability analysis, these schemes are found to be conditionally stable. Several numerical examples of different forms are discussed and compared in term of errors with exact solutions and results from the FDM.

Bicubic B-spline interpolation method for two-dimensional Laplace's equations

Two-dimensional Laplaces equation is solved using bicubic B-spline interpolation method. An arbitrary surface with some unknown coefficients is generated using bicubic B-spline surfaces formula. This surface is presumed to be the solution for the equation. The values of the coefficients are calculated by spline interpolation technique using the corresponding differential equations and boundary conditions. This method produces approximated analytical solution for the equation. A numerical example will be presented along with a comparison of the results with finite element and isogeometrical methods.

Solving nonlinear Benjamin-Bona-Mahony equation using cubic B-spline and cubic trigonometric B-spline collocation methods

In this research, the nonlinear Benjamin-Bona-Mahony (BBM) equation is solved numerically using the cubic B-spline (CuBS) and cubic trigonometric B-spline (CuTBS) collocation methods. The CuBS and CuTBS are utilized as interpolating functions in the spatial dimension while the standard finite difference method (FDM) is applied to discretize the temporal space. In order to solve the nonlinear problem, the BBM equation is linearized using Taylor’s expansion. Applying the von-Neumann stability analysis, the proposed techniques are shown to be unconditionally stable under the Crank-Nicolson scheme. Several numerical examples are discussed and compared with exact solutions and results from the FDM.

Solving the nonlinear Schrödinger equation using cubic B-spline interpolation and finite difference methods on dual solitons

In this paper, Nonlinear Schrodinger (NLS) equation with Neumann boundary conditions is solved using cubic B-spline interpolation method (CuBSIM) and finite difference method (FDM). Firstly, FDM is applied on the time discretization and cubic B-spline is utilized as an interpolation function in the space dimension with the help of theta-weighted method. The second approach is based on the FDM applied on the time and space discretization with the help of theta-weighted method. The CuBSIM is shown to be stable by using von Neumann stability analysis. The proposed method is tested on the interaction of the dual solitons of the NLS equation. The accuracy of the numerical results is measured by the Euclidean-norm and infinity-norm. CuBSIM is found to produce more accurate results than the FDM.

Solving Buckmaster equation using cubic B-spline and cubic trigonometric B-spline collocation methods

The cubic B-spline and cubic trigonometric B-spline functions are used to set up the collocation in finding solutions for the Buckmaster equation. These splines are applied as interpolating functions in the spatial dimension while the finite difference method (FDM) is used to discretize the time derivative. The Buckmaster equation is linearized using Taylor’s expansion and solved using two schemes, namely Crank-Nicolson and fully implicit. The von Neumann stability analysis is carried out on the two schemes and they are shown to be conditionally stable. In order to demonstrate the capability of the schemes, some problems are solved and compared with analytical and FDM solutions. The proposed methods are found to generate more accurate results than the FDM.

Extended cubic B-spline method for solving a linear system of second-order boundary value problems

A method based on extended cubic B-spline is proposed to solve a linear system of second-order boundary value problems. In this method, two free parameters,

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

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