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

Abstract

[EN]In this work, we address the problem of solving nonlinear general Klein–Gordonequations (nlKGEs). Different fourth- and sixth-order, stable explicit and implicit, finite differenceschemes 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, dampedKlein–Gordon equations, and many others. These KGEs have a great importance in engineeringand theoretical physics.The higher-order methods proposed in this study allow a reduction in the number of nodes, whichmight also be very interesting when solving multi-dimensional KGEs. We have studied the stabilityand consistency of the proposed schemes when considering certain smoothness conditions of thesolutions. Additionally, both the typical Dirichlet and some nonlocal integral boundary conditionshave been studied. Finally, some numerical results are provided to support the theoretical aspectspreviously considered