A numerically efficient variational algorithm to solve a fractional nonlinear elastic string equation

Abstract

In this work, we propose a fractional extension of the one-dimensional nonlinear vibration problem on an elastic string. The fractional problem is governed by a hyperbolic partial differential equation that considers a nonlinear function of spatial derivatives of the Riesz type and constant damping. Initial and homogeneous Dirichlet boundary conditions are imposed on a bounded interval of the real line. We show that the problem can be expressed in variational form and propose a Hamiltonian function associated to the system. We prove that the total energy of the system is constant in the absence of damping, and it is non-increasing otherwise. Some boundedness properties of the solutions are established mathematically. Motivated by these facts, we design a finite-difference discretization of the continuous model based on the use of fractional-order centered differences. The discrete scheme has also a variational structure, and we propose a discrete form of the Hamiltonian function. As the continuous counterpart, we prove rigorously that the discrete total energy is conserved in the absence of damping, and dissipated when the damping coefficient is positive. The scheme is a second-order consistent discretization of the continuous model. Moreover, we prove the stability and quadratic convergence of the numerical model using a discrete form of the energy method. We provide some computer simulations using an implementation of our scheme to illustrate the validity of the conservative/dissipative properties.