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Jorge Eduardo Macías-Díaz

Bio: Jorge Eduardo Macías-Díaz is an academic researcher from Autonomous University of Aguascalientes. The author has contributed to research in topics: Nonlinear system & Bounded function. The author has an hindex of 26, co-authored 192 publications receiving 1692 citations. Previous affiliations of Jorge Eduardo Macías-Díaz include University of New Orleans & Tallinn University.


Papers
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Journal ArticleDOI
TL;DR: The governing partial differential equation generalizes the Hodgkin–Huxley, the Allen–Cahn and the Fisher–Kolmogorov–Petrovskii–Piscounov equations, and guarantees the unconditional stability.
Abstract: For the first time in literature, semi-implicit spectral approximations for nonlinear Caputo time- and Riesz space-fractional diffusion equations with both smooth and non-smooth solutions are proposed. More precisely, the governing partial differential equation generalizes the Hodgkin–Huxley, the Allen–Cahn and the Fisher–Kolmogorov–Petrovskii–Piscounov equations. The schemes employ a Legendre-based Galerkin spectral method for the Riesz space-fractional derivative, and L1-type approximations with both uniform and graded meshes for the Caputo time-fractional derivative. More importantly, by using fractional Gronwall inequalities and their associated discrete forms, sharp error estimates are proved which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes. This analysis encompasses both uniform meshes as well as meshes that are graded in time, and guarantees the unconditional stability. The numerical results that accompany our analysis confirm our theoretical error estimates, and give significant insights into the convergence behavior of our schemes for problems with smooth and non-smooth solutions.

78 citations

Journal ArticleDOI
TL;DR: It is shown here that the undamped regime has an associated positive energy functional, and it is preserved throughout time under suitable boundary conditions, and the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative.

77 citations

Journal ArticleDOI
TL;DR: This work establishes that the discrete energy is conserved in the undamped regime, and that it dissipates in the damped scenario, and proposes an explicit finite-difference discretization of the authors' fractional model based on the use of fractional centered differences.

63 citations

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TL;DR: Numerical evidence on the presence of the phenomenon of nonlinear supratransmission in sine-Gordon systems of Riesz space-fractional order is found.

50 citations

Journal ArticleDOI
TL;DR: A general nonlinear wave equation with Riesz space-fractional derivatives that generalizes various classical hyperbolic models, including the sine-Gordon and the Klein–Gordon equations from relativistic quantum mechanics is investigated.

50 citations


Cited by
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01 Jan 2012

3,692 citations

Journal Article
TL;DR: Explicit formulas and graphs of few special functions are derived in this article on the basis of various definitions of various fractional derivatives and their applications are also reviewed in the paper, where the authors also review their applications.
Abstract: Explicit formula and graphs of few special functions are derived in the paper on the basis of various definitions of various fractional derivatives and various fractional integrals. Their applications are also reviewed in the paper.

140 citations