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Numerical investigation of fractional nonlinear sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics

机译：相对论量子力学中出现的分数非线性正弦戈登和Klein-Gordon模型的数值研究

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摘要

This paper presents a method for the approximate solution of the time-fractional nonlinear sine-Gordon and the Klein-Gordon models described in Caputo sense and with the order 1 ＜ γ ＜ 2. This method discretizes the unknown solution in two main steps. First, the temporal discretization of the governing problems is obtained by means of the finite difference scheme. Second, the spatial terms are expanded using local radial basis functions, where each basis function is approximated by a weighted linear summation of function values. The spatial discretization achieved through the local radial basis functions and finite difference (LRBF-FD) is highly accurate, since in local collocation techniques we only consider the nodes located in the subdomain around the collocation node surrounding the local collocation point. In fact, the new approach avoids the ill-conditioning problem resulting from the adoption of the global collocation and leads to sparse systems. The numerical stability and convergence are examined and confirmed numerically to support the theoretical formulation. Numerical experiments assess the effectiveness and capability of the algorithm and demonstrate its good computational performance on irregular domains.
机译：本文提出了一种方法，用于时间 - 分数非线性正弦戈登和Caputo感应中描述的klein-Gordon模型的近似解，并且顺序1 <γ<2。该方法以两个主要步骤离散未知解决方案。首先，通过有限差分方案获得控制问题的时间离散化。其次，使用本地径向基函数扩展空间术语，其中每个基本函数由函数值的加权线性求和近似。通过局部径向基函数和有限差异（LRBF-FD）实现的空间离散化是高度准确的，因为在本地搭配技术中，我们只考虑位于围绕本地焊接点的搭配节点周围的子域内的节点。事实上，新方法避免了通过全球搭配采用并导致稀疏系统产生的不良状态。数值稳定性和收敛性分别检查并证实以支持理论制剂。数值实验评估算法的有效性和能力，并在不规则结构域上展示其良好的计算性能。

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