A robust uniform B-spline collocation method for solving the generalized PHI-four equation

In this paper, we develop a numerical solution based on cubic B-spline collocation method. By applying Von-Neumann stability analysis, the proposed technique is shown to be unconditionally stable. The accuracy of the presented method is demonstrated by a test problem. The numerical results are found to be in good agreement with the exact solution

Numerical schemes for general Klein–Gordon equations with Dirichlet and nonlocal boundary conditions

In this work, we address the problem of solving nonlinear general Klein–Gordon equations (nlKGEs). Different fourth- and sixth-order, stable explicit and implicit, finite difference schemes are derived. These new methods can be considered to approximate all type of Klein–Gordon equations (KGEs) including phi-four, forms I, II, and III, sine-Gordon, Liouville, damped Klein–Gordon equations, and many others. These KGEs have a great importance in engineering and theoretical physics.The higher-order methods proposed in this study allow a reduction in the number of nodes, which might also be very interesting when solving multi-dimensional KGEs. We have studied the stability and consistency of the proposed schemes when considering certain smoothness conditions of the solutions. Additionally, both the typical Dirichlet and some nonlocal integral boundary conditions have been studied. Finally, some numerical results are provided to support the theoretical aspects previously considered

Efficient high-order finite difference methods for nonlinear Klein–Gordon equations. I: Variants of the phi-four model and the form-I of the nonlinear Klein–Gordon equation

In this paper, the problem of solving some nonlinear Klein–Gordon equations (KGEs) is considered. Here, we derive different fourth- and sixth-order explicit and implicit algorithms to solve the phi-four equation and the form-I of the nonlinear Klein–Gordon equation. Stability and consistency of the proposed schemes are studied under certain conditions. Numerical results are presented and then compared with others obtained from some methods already existing in the scientific literature to explain the efficiency of the new algorithms. It is also shown that similar schemes can be proposed to solve many classes of nonlinear KGEs