Fractional generalization of the fermi–Pasta–Ulam–Tsingou media and theoretical analysis of an explicit variational scheme

Abstract

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.