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A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations

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Abstract
In this work, we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the Riesz type. The model under consideration generalizes various nonlinear wave equations, including the sine-Gordon and the nonlinear Klein–Gordon models. The system considered in this work is conservative when homogeneous Dirichlet boundary conditions are imposed. Motivated by this fact, we propose a finite-difference method based on fractional centered differences that is capable of preserving the discrete energy of the system. The method under consideration is a nonlinear implicit scheme which has various numerical properties. Among the most interesting numerical features, we show that the methodology is consistent of second order in time and fourth order in space. Moreover, we show that the technique is stable and convergent. Some numerical simulations show that the method is capable of preserving the energy of the discrete system. This characteristic of the technique is in obvious agreement with the properties of its continuous counterpart.
Keywords
Conservative fractional wave equation, Riesz space-fractional equations, Energy-preserving method, Fractional centered differences, High-order approximation, Stability and convergence analyses

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Please use this url to cite or link to this publication:

MLA
Macías-Díaz, Jorge E., Ahmed S. Hendy, and Rob De Staelen. “A Compact Fourth-order in Space Energy-preserving Method for Riesz Space-fractional Nonlinear Wave Equations.” APPLIED MATHEMATICS AND COMPUTATION 325 (2018): 1–14. Print.
APA
Macías-Díaz, J. E., Hendy, A. S., & De Staelen, R. (2018). A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations. APPLIED MATHEMATICS AND COMPUTATION , 325, 1–14.
Chicago author-date
Macías-Díaz, Jorge E., Ahmed S. Hendy, and Rob De Staelen. 2018. “A Compact Fourth-order in Space Energy-preserving Method for Riesz Space-fractional Nonlinear Wave Equations.” Applied Mathematics and Computation 325: 1–14.
Chicago author-date (all authors)
Macías-Díaz, Jorge E., Ahmed S. Hendy, and Rob De Staelen. 2018. “A Compact Fourth-order in Space Energy-preserving Method for Riesz Space-fractional Nonlinear Wave Equations.” Applied Mathematics and Computation 325: 1–14.
Vancouver
1.
Macías-Díaz JE, Hendy AS, De Staelen R. A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations. APPLIED MATHEMATICS AND COMPUTATION . Elsevier; 2018;325:1–14.
IEEE
[1]
J. E. Macías-Díaz, A. S. Hendy, and R. De Staelen, “A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations,” APPLIED MATHEMATICS AND COMPUTATION , vol. 325, pp. 1–14, 2018.
@article{8543443,
  abstract     = {In this work, we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the Riesz type. The model under consideration generalizes various nonlinear wave equations, including the sine-Gordon and the nonlinear Klein–Gordon models. The system considered in this work is conservative when homogeneous Dirichlet boundary conditions are imposed. Motivated by this fact, we propose a finite-difference method based on fractional centered differences that is capable of preserving the discrete energy of the system. The method under consideration is a nonlinear implicit scheme which has various numerical properties. Among the most interesting numerical features, we show that the methodology is consistent of second order in time and fourth order in space. Moreover, we show that the technique is stable and convergent. Some numerical simulations show that the method is capable of preserving the energy of the discrete system. This characteristic of the technique is in obvious agreement with the properties of its continuous counterpart.},
  author       = {Macías-Díaz, Jorge E. and Hendy, Ahmed S. and De Staelen, Rob},
  issn         = {0096-3003},
  journal      = {APPLIED MATHEMATICS AND COMPUTATION },
  keywords     = {Conservative fractional wave equation,Riesz space-fractional equations,Energy-preserving method,Fractional centered differences,High-order approximation,Stability and convergence analyses},
  language     = {eng},
  pages        = {1--14},
  publisher    = {Elsevier},
  title        = {A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations},
  url          = {http://dx.doi.org/10.1016/j.amc.2017.12.002},
  volume       = {325},
  year         = {2018},
}

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