Analysis of continuous and discrete models for fractional klein-gordon-zakharov systems

Abstract

Resumen En esta tesis, investigamos una extensi´on fraccionaria del sistema Klein–Gordon-Zakharov donde se utilizan diferentes ´ordenes de derivadas espaciales fraccionarias en el sentido de Riesz. Demostramos que la energ´ıa total del sistema se conserva y que las soluciones globales del sistema est´an acotadas. Motivados por estos hechos, proponemos modelos num´ericos para aproximar el sistema continuo subyacente. Para cada uno de los modelos discretizados, introducimos un funcional discreto de energ´ıa adecuado para estimar la energ´ıa total del sistema continuo. Probamos que tal energ´ıa discreta se conserva en todos los casos. La existencia de soluciones de los modelos num´ericos se establece mediante teoremas de punto fijo. Mostramos rigurosamente que los esquemas construidos son capaces de preservar la acotaci´on de las aproximaciones y que producen estimaciones consistentes de la soluci´on real. La estabilidad num´erica y la convergencia tambi´en se prueban te´oricamente. Como una de las consecuencias, se muestra rigurosamente la unicidad de las soluciones num´ericas para todos modelos discretizados. Finalmente, se proporcionan comparaciones de las soluciones num´ericas para evaluar las capacidades de estos m´etodos discretos para preservar la energ´ıa discreta de sus sistemas.

Abstract In this thesis, we investigate a fractional extension of the Klein–Gordon–Zakharov system where different orders of fractional spatial derivatives are utilized in the Riesz sense. We show that the total energy of the system is conserved, and that the global solutions of the system are bounded. Motivated by these facts, we propose numerical models to approximate the underlying continuous system. For each of the discretized models, we introduce a proper discrete energy functional to estimate the total energy of the continuous system. We prove that such a discrete energy is conserved in all cases. The existence of solutions of the numerical models is established via fixed-point theorems. We show rigorously that the schemes constructed are capable of preserving the boundedness of the approximations and that they yield consistent estimates of the true solution. Numerical stability and convergence are likewise proved theoretically. As one of the consequences, the uniqueness of numerical solutions is shown rigorously for all discretized models. Finally, comparisons of the numerical solutions are provided, in order to evaluate the capabilities of these discrete methods to preserve the discrete energy of their systems.