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Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme

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  • Jorge E. Macías-Díaz

    (Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes Ags. 20131, Mexico)

Abstract

In this work, we investigate numerically a one-dimensional wave equation in generalized form. The system considers the presence of constant damping and functional anomalous diffusion of the Riesz type. Reaction terms are also considered, in such way that the mathematical model can be presented in variational form when damping is not present. As opposed to previous efforts available in the literature, the reaction terms are not only functions of the solution. Instead, we consider the presence of smooth functions that depend on fractional derivatives of the solution function. Using a finite-difference approach, we propose a numerical scheme to approximate the solutions of the fractional wave equation. Along with this integrator, we propose discrete forms of the local and the total energy operators. In a first stage, we show rigorously that the energy properties of the continuous system are mimicked by our discrete methodology. In particular, we prove that the discrete system is dissipative (respectively, conservative) when damping is present (respectively, absent), in agreement with the continuous model. The theoretical numerical analysis of this system is more complicated in light of the presence of the functional form of the anomalous diffusion. To solve this problem, some novel technical lemmas are proved and used to establish the stability and the quadratic convergence of the scheme. Finally, we provide some computer simulations to show the capability of the scheme to conserve/dissipate the energy. Various fractional problems with functional forms of the anomalous diffusion of the solution are considered to that effect.

Suggested Citation

  • Jorge E. Macías-Díaz, 2019. "Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme," Mathematics, MDPI, vol. 7(11), pages 1-27, November.
  • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1095-:d:286331
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    References listed on IDEAS

    as
    1. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
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    3. Hassan Khan & Rasool Shah & Poom Kumam & Dumitru Baleanu & Muhammad Arif, 2019. "An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations," Mathematics, MDPI, vol. 7(5), pages 1-19, May.
    4. Pundikala Veeresha & Doddabhadrappla Gowda Prakasha & Dumitru Baleanu, 2019. "An Efficient Numerical Technique for the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equation," Mathematics, MDPI, vol. 7(3), pages 1-18, March.
    5. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    6. X. Wang & F. Liu & X. Chen, 2015. "Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations," Advances in Mathematical Physics, Hindawi, vol. 2015, pages 1-14, November.
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