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A numerically efficient Hamiltonian method for fractional wave equations

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  • Macías-Díaz, J.E.

Abstract

In this work, we consider a partial differential equation that extends the well known wave equation. The model under consideration is a multidimensional equation which includes the presence of both a damping term and a fractional Laplacian of the Riesz type. Homogeneous Dirichlet boundary conditions on a closed and bounded spatial interval are considered in this work. The mathematical model has a fractional Hamiltonian which is conserved when the damping coefficient is equal to zero, and dissipated otherwise. Motivated by these facts, we propose a finite-difference method to approximate the solutions of the continuous model. The method is an implicit scheme which is based on the use of fractional centered differences to approximate the spatial fractional derivatives of the model. A discretized form of the Hamiltoninan is also proposed in this work, and we prove analytically that the method is capable of preserving/dissipating the discrete energy when the continuous model preserves/dissipates the energy. We establish rigorously the properties of consistency, stability and convergence of the method, and provide some a priori bounds for the numerical solutions. Moreover, we prove the existence and the uniqueness of the numerical solutions as well as the unconditional stability of the method in the linear regime. Some computer simulations that assess the capability of the method to preserve/dissipate the energy are carried out for illustration purposes.

Suggested Citation

  • Macías-Díaz, J.E., 2018. "A numerically efficient Hamiltonian method for fractional wave equations," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 231-248.
    • Handle: RePEc:eee:apmaco:v:338:y:2018:i:c:p:231-248
    DOI: 10.1016/j.amc.2018.06.003
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    References listed on IDEAS

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    1. Gafiychuk, V.V. & Datsko, B.Yo., 2006. "Pattern formation in a fractional reaction–diffusion system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(2), pages 300-306.
    2. Qianqian Yang & Fawang Liu & Ian Turner, 2010. "Stability and Convergence of an Effective Numerical Method for the Time-Space Fractional Fokker-Planck Equation with a Nonlinear Source Term," International Journal of Differential Equations, Hindawi, vol. 2010, pages 1-22, January.
    3. Macías-Díaz, J.E. & Hendy, A.S. & De Staelen, R.H., 2018. "A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 1-14.
    4. Qin, Wendi & Ding, Deqiong & Ding, Xiaohua, 2015. "Two boundedness and monotonicity preserving methods for a generalized Fisher-KPP equation," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 552-567.
    5. Manuel Duarte Ortigueira, 2006. "Riesz potential operators and inverses via fractional centred derivatives," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2006, pages 1-12, August.
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