Numerical schemes for general Klein–Gordon equations with Dirichlet and nonlocal boundary conditions
Jesús Martín-Vaquero
University of Salamanca, Spain
Ascensión Hernández Encinas
University of Salamanca, Spain
Araceli Queiruga-Dios
University of Salamanca, Spain
Víctor Martínez
Spanish National Research Council, Spain
Ángel Martín del Rey
University of Salamanca, Spain
Published 2018-02-20


Klein–Gordon equations
nonlocal boundary conditions
finite difference methods

How to Cite

Martín-Vaquero J., Hernández Encinas A., Queiruga-Dios A., Martínez V. and del Rey Ángel M. (2018) “Numerical schemes for general Klein–Gordon equations with Dirichlet and nonlocal boundary conditions”, Nonlinear Analysis: Modelling and Control, 23(1), pp. 50-62. doi: 10.15388/NA.2018.1.5.


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.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Please read the Copyright Notice in Journal Policy