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This paper applies the Improved Order of Convergent of the Cubic B-Spline collocation method (IOCuBSM) to obtain the numerical solution of the nonlinear Schrödinger Equation with a single soliton as the test problem. The approximate solution is based on the Crank-Nicolson scheme with the finite-difference method applied to time discretization and the improved order of convergent cubic B-Splines as the interpolation function in the space dimension. The error between numerical and analytical solutions is determined by discrete maximum norm and Euclidean norm and the method is proved to be unconditionally stable. Furthermore, the computational time for the methods is also studied. [ABSTRACT FROM AUTHOR]

In this study, a numerical technique for solving the generalized time-fractional Navier-Stokes equation in cylindrical coordinates via cubic B-spline (CBS) basis functions is proposed. The suggested technique for solving the problem is derived from the finite difference method and the collocation method. Caputo's derivative of order 0 < α ≤ 1 and CBS functions are employed for the time and spatial discretization, respectively. The Fourier stability analysis, derived from the von Neumann method was performed and indicate unconditionally stable. The convergence analysis of the presented numerical technique is of order O(h2 + Δt2−α). [ABSTRACT FROM AUTHOR]

In this research, the Benjamin-Bona-Mahony equation is solved numerically using cubic B-spline (CBS) collocation method. Forward difference approximation is used to discretize the time derivative while the CBS function is used to discretize the space dimension. To deal with nonlinearity, the nonlinear term is replaced with a known parameter. At every time level, calculations are repeated two or three times to increase the accuracy of the results. Applying von-Neumann stability analysis, the proposed technique is shown to be unconditionally stable under Crank-Nicolson scheme. Comparison of the numerical results is done with the results from some recent schemes. [ABSTRACT FROM AUTHOR]

Finite difference method (FDM) and Cubic Exponential B-Spline collocation method (CuEBCM) from different background are used as the numerical solution of the Nonlinear Schrödinger equation. The approach is based on the Crank-Nicolson scheme where FDM is applied on the time and space discretization, while the Cubic Exponential B-Spline is utilized as an interpolation function in the space dimension. The error between the numerical and the analytical solutions is measured by using the discrete maximum norm and Euclidean norm when the analytical solutions exist. The results from CuEBCM are found to be more accurate compared to the FDM. [ABSTRACT FROM AUTHOR]

Coupled Nonlinear Schrödinger (CNLS) equation is a second order nonlinear partial differential equation commonly related to nonlinear optical fiber. In this paper, CNLS equation is solved using Finite Difference Method (FDM) and Hybrid Cubic B-Spline collocation method (HCBM) with appropriate initial and boundary conditions. Theta-weighted scheme is applied to the equations and the nonlinear terms are linearized using Taylor series expansion. The temporal space is discretized by forward difference and for the spatial dimensions, central difference is applied for FDM while B-Spline functions are applied for HCBM. The HCBM is shown to be unconditionally stable using von Neumann stability analysis. To test the accuracy, a numerical example is discussed and the error norms are computed. The results obtained show that FDM and HCBM are reliable and easy to implement. [ABSTRACT FROM AUTHOR]

In this research, the Benjamin-Bona-Mahony (BBM) equation is solved numerically using cubic B-spline (CBS) collocation method. The BBM equation is transformed into a system of two equations and Besse’s formula is applied to the nonlinear term. Then, Forward Difference Approximation is used to discretize the time derivative while the CBS function is used to discretize the space dimension. Two numerical examples are discussed and compared with the exact solutions. [ABSTRACT FROM AUTHOR]

The Degasperis-Procesi (DP) equation is solved numerically using the Quartic B-Spline Collocation Method (QBCM). The quartic B-spline function is used to set up the collocation method to solve the nonlinear DP equation. The quartic B-spline is applied as an interpolating function in the spatial dimension while the Finite Difference method (FDM) is used to discretize the time derivative. The nonlinear DP equation is linearized using Taylor's expansion and then the equation is discretized using the θ-weighted scheme. Crank-Nicolson and Implicit schemes are used. In order to demonstrate the capability of the schemes, two problems are solved and the solutions obtained are compared with the analytical solutions. The proposed numerical schemes are found to be in line with the analytical solutions [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]