On the solution of a Riesz space-fractional nonlinear wave equation through an efficient and energy-invariant scheme

ABSTRACT

In this work, we consider a damped hyperbolic partial differential equation in multiple spatial dimensions with spatial partial derivatives of non-integer order. The equation under investigation is a fractional extension of the well-known sine-Gordon and Klein–Gordon equations from relativistic quantum mechanics. The system has associated an energy functional which is conserved in the undamped regime, and dissipated in the damped case. In this manuscript, we restrict our study to a bounded spatial domain and propose an explicit finite-difference discretization of the problem using fractional centred differences. Together with the scheme, we propose an approximation for the energy functional and show that the energy of the discrete system is conserved/dissipated when the energy of the continuous model is conserved/dissipated. The method guarantees that the energy functionals are positive, in agreement with the continuous counterparts. We show in this work that the method is a uniquely solvable, consistent, stable and convergent technique.

KEYWORDS:

• Space-fractional wave equation
• positive energy invariants
• Riesz fractional derivatives
• dissipation-preserving scheme
• stability and convergence analysis

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 35R11

Acknowledgments

The author wishes to thank A. J. Serna-Reyes for pointing out some typos in a preliminary version of this paper and for improving the computer code used to produce the simulations. The author also wishes to thank the anonymous reviewers and the editor in charge of handling this paper for all their criticisms and comments. All of their suggestions contributed significantly to improve the quality of this work.

Disclosure statement

The author has no conflict of interest to declare.

ORCID

J. E. Macías-Díaz http://orcid.org/0000-0002-7580-7533