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

• Published in: Nonlinear analysis (Vilnius, Lithuania) Vol. 23; no. 1; pp. 50 - 62
• Main Authors: Martín-Vaquero, Jesús, Hernández Encinas, Ascensión, Queiruga-Dios, Araceli, Martínez, Víctor, Del Rey, Ángel Martín Martín
• Format: Journal Article
• Language: English
• Published: Vilnius University Press 01-01-2018
• Subjects: consistency; finite difference methods; Klein–Gordon equations; nonlocal boundary conditions; stability
• ISSN: 1392-5113; 2335-8963
• 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.