Abstract
The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline is presented for the approximate solution of this equation with Dirichlet boundary conditions. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Several examples are discussed to exhibit the feasibility and capability of the approach. The absolute errors and error norms are also computed at different times to assess the performance of the proposed approach and the results were found to be in good agreement with known solutions and with existing schemes in literature.
Highlights
The generalized nonlinear Klien-Gordon (KG) equation arises in various problems in science and engineering
This paper focuses on the analysis and numerical solution of the generalized nonlinear KG equation, which is given in the following form [14]: L2
The KG equation is important in mathematical physics especially in quantum mechanics and it is well known as a soliton equation
Summary
The generalized nonlinear Klien-Gordon (KG) equation arises in various problems in science and engineering. Sirendaoreji solved the nonlinear KG equation using the auxiliary equation method to construct new exact traveling wave solutions with quadratic and cubic nonlinerity [8]. Caglar et al [11] has introduced a cubic B-spline interpolation method to solve the two-point boundary value problem. Hamid et al [12] has introduced an alternative cubic trigonometric B-spline interpolation method to solve the same problem. A new three-time level implicit approach which combines a finite difference approach and cubic trigonometric Bspline collocation method (CTBCM) is proposed to solve generalized nonlinear KG equation. The finite difference approach is proposed to discretize time derivative and cubic trigonometric collocation method is applied to interpolate the solutions at timet. 3h k~ sin ( ) sin (h) sin ( ): Due to local support properties of B-spline basis function, there are only three non-zero basis functions T4,j{3(xj), T4,j{2(xj) and T4,j{1(xj) are included over subinterval 1⁄2xj,xjz1: the approximation ukj and its derivatives with respect to x can be simplified as:
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