Energy estimates for two-dimensional space-Riesz fractional wave equation

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Abstract

The fractional wave equation governs the propagation of mechanical diffusive waves in viscoelastic media which exhibits a power-law creep, and consequently provided a physical interpretation of this equation in the framework of dynamic viscoelasticity. In this paper, we first use the energy method to estimate the one-dimensional space-Riesz fractional wave equation. The stiff matrices are proved to be commutative for two-dimensional case, which ensures to carry out of the priori error estimates and the energy method. Then, the unconditional stability and convergence with the global truncation error \(\mathcal {O}(\tau ^{2}+h^{2})\) are theoretically proved with the constant coefficients and numerically verified.

Keywords

Riesz fractional wave equation Nonlocal wave equation Priori error estimates Energy method Numerical stability and convergence 

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Notes

Acknowledgments

We would like to thank the anonymous referees for their input to Example 4.2, and for several suggestions and comments that led to much better results and an improved presentation.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhouPeople’s Republic of China

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