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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.

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Acknowledgments

The author wishes to thank the anonymous reviewers for their comments and criticisms. All of their comments were taken into account in the revised version of the paper, resulting in a substantial improvement with respect to the original submission. The present work reports on a set of final results of the research project “Conservative methods for fractional hyperbolic systems: analysis and applications”, funded by the National Council for Science and Technology of Mexico (CONACYT) through grant A1-S-45928.

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Correspondence to Jorge E. Macías-Díaz.

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Appendix: Matlab code

Appendix: Matlab code

The purpose of this appendix is to provide a Matlab computer implementation of the finite-difference scheme (28)–(29). Beforehand, it is worth pointing out that the implementation can be applied to solve systems like that described in Example 1 of Section 4, in which the initial conditions are given by the algebraic sum of the kink and the anti-kink of the Toda lattice, and F = 0. However, a suitable modification of this program may be used to solve different general scenarios.

The modifiable variables of the code are the following:

  • The value of L, which corresponds to the value of b in B = (0, L).

  • The value of T, which corresponds to the value of T.

  • The value of alpha, which corresponds to the value of of the fractional order of differentiation α.

  • The value of damp, which corresponds to the value of the damping coefficient γ.

  • The value of h, which corresponds to the value of h.

  • The value of tau, which corresponds to the value of τ.

As outcomes, the algorithm obtains the graphs of the numerical solutions for U and \(\mathcal {H}\) versus x and t, as well as the graph of \(\mathcal {E}\) versus t.

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Macías-Díaz, J.E. A numerically efficient variational algorithm to solve a fractional nonlinear elastic string equation. Numer Algor (2020). https://doi.org/10.1007/s11075-020-00880-2

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Keywords

  • Nonlinear vibration problem
  • Fractional elastic string
  • Discrete variational derivative method
  • Numerical efficiency analysis

Mathematics Subject Classification (2010)

  • 65Mxx
  • 65Qxx