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Fractional generalization of the fermi–Pasta–Ulam–Tsingou media and theoretical analysis of an explicit variational scheme
Communications in Nonlinear Science and Numerical Simulation ( IF 3.4 ) Pub Date : 2020-03-29 , DOI: 10.1016/j.cnsns.2019.105158
J.E. Macías-Díaz

In this work, we consider a partial differential equation that extends the well-known Fermi–Pasta–Ulam–Tsingou chains from nonlinear dynamics. The continuous model under consideration includes the presence of both a damping term and a polynomial function in terms of Riesz space-fractional derivatives. Initial and boundary conditions on a closed and bounded interval are considered in this work. The mathematical model has a fractional Hamiltonian which is conserved when the damping coefficient is equal to zero, and dissipated otherwise. Motivated by these facts, we propose a finite-difference method to approximate the solutions of the continuous model. The method is an explicit scheme which is based on the use of fractional centered differences to approximate the fractional derivatives of the model. A discretized form of the Hamiltonian is also proposed in this work, and we prove analytically that the method is capable of conserving or dissipating the discrete energy under the same conditions that guarantee the conservation or dissipation of energy of the continuous model. We show that solutions of the discrete model exist and are unique under suitable regularity conditions on the reaction function. We establish rigorously the properties of consistency, stability and convergence of the method. To that end, novel technical results are mathematically proved. Computer simulations that assess the capability of the method to preserve the energy are provided for illustration purposes.



中文翻译:

Fermi–Pasta–Ulam–Tssingou介质的分数泛化和显式变分方案的理论分析

在这项工作中,我们考虑了一个偏微分方程,该方程从非线性动力学扩展了著名的费米-帕斯塔-乌拉姆-青果链。所考虑的连续模型包括阻尼项和以Riesz空间分数导数表示的多项式函数。在这项工作中考虑了封闭和有界区间上的初始条件和边界条件。该数学模型具有分数哈密顿量,当阻尼系数等于零时,它是守恒的,否则就消散了。基于这些事实,我们提出了一种有限差分方法来逼近连续模型的解。该方法是一种显式方案,该方案基于使用分数中心差来近似模型的分数导数。在这项工作中还提出了离散形式的哈密顿量,我们通过分析证明了该方法能够在保证连续模型能量守恒或消散的相同条件下保存或消散离散能量。我们证明了离散模型的解存在并且在适当的规则性条件下对反应函数具有唯一性。我们严格建立该方法的一致性,稳定性和收敛性。为此,在数学上证明了新颖的技术成果。为了说明的目的,提供了评估该方法保存能量的能力的计算机仿真。并且我们通过分析证明了该方法能够在保证连续模型能量守恒或消散的相同条件下保存或消散离散能量。我们证明了离散模型的解存在并且在适当的规则性条件下对反应函数具有唯一性。我们严格建立该方法的一致性,稳定性和收敛性。为此,在数学上证明了新颖的技术成果。为了说明的目的,提供了评估该方法保存能量的能力的计算机仿真。并且我们通过分析证明了该方法能够在保证连续模型能量守恒或消散的相同条件下保存或消散离散能量。我们证明了离散模型的解存在并且在适当的规则性条件下对反应函数具有唯一性。我们严格建立该方法的一致性,稳定性和收敛性。为此,在数学上证明了新颖的技术成果。为了说明的目的,提供了评估该方法保存能量的能力的计算机仿真。我们严格建立该方法的一致性,稳定性和收敛性。为此,在数学上证明了新颖的技术成果。为了说明的目的,提供了评估该方法保存能量的能力的计算机仿真。我们严格建立该方法的一致性,稳定性和收敛性。为此,在数学上证明了新颖的技术成果。为了说明的目的,提供了评估该方法保存能量的能力的计算机仿真。

更新日期:2020-03-29
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