An energy-preserving computational approach for the semilinear space fractional damped Klein–Gordon equation with a generalized scalar potential

Hendy, Ahmed S.; Taha, Thiab R.; Сураган, Дурвудхан; Zaky, Mahmoud A.

doi:10.1016/j.apm.2022.04.009

Cited by 8 publications

(2 citation statements)

References 48 publications

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“…The applicability and accuracy of the solution method were demonstrated by two numerical examples for solving the nonlinear Schrödinger equation and the nonlinear Ginzburg-Landau equation. A future extension of our work will focus on the physical properties of the scheme such as the energy-preservation following the results in Hendy et al 62 and Zaky and Hendy. 63 In addition, we will try to provide a rigorous convergence analysis for the proposed numerical scheme.…”

Section: Discussionmentioning

confidence: 99%

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

Mostafa

Zaky

Hafez

et al. 2022

Math Methods in App Sciences

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We present a class of orthogonal functions on infinite domain based on Jacobi polynomials. These functions are generated by applying a tanh transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo‐derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced tanh Jacobi functions, we develop an efficient spectral tanh Jacobi collocation method for the numerical simulation of nonlinear Schrödinger equations on the infinite domain without using artificial boundary conditions. The applicability and accuracy of the solution method are demonstrated by two numerical examples for solving the nonlinear Schrödinger equation and the nonlinear Ginzburg–Landau equation.

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Section: Discussionmentioning

confidence: 99%

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

Mostafa

Zaky

Hafez

et al. 2022

Math Methods in App Sciences

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“…Fan and Jiang [10] consider an unstructured mesh finite element method for the two-dimensional time-space fractional Schrödinger equation. High-order numerical schemes for time-space fractional differential equations with variable-order fractional derivatives or delay can be found in the work of Zaky et al [11][12][13][14][15]. Furthermore, the L1 Galerkin methods are proposed for the case with nonsmooth solution [16,17].…”

Section: Introductionmentioning

confidence: 99%

L1/LDG algorithm for time‐Caputo space‐Riesz fractional convection equation in two dimensions

Cai,

2023

Math Methods in App Sciences

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This paper proposes an L1/LDG algorithm for a time–space fractional convection equation on a bounded rectangular domain . Here, the temporal partial derivative is in the sense of Caputo with order , and the spatial partial derivatives are of Riesz type with orders in ‐ and ‐directions, respectively. L1 approximation is utilized to evaluate the temporal Caputo derivative, while the local discontinuous Galerkin (LDG) method is adopted to deal with the spatial Riesz derivative. The rigorous numerical stability analysis and error estimate are presented which are supported by illustrative numerical examples.

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Improved Uniform Error Bounds of Exponential Wave Integrator Method for Long-Time Dynamics of the Space Fractional Klein-Gordon Equation with Weak Nonlinearity

Jia,

Jiang

2023

J Sci Comput

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In this paper, we derive the improved uniform error bounds for the long-time dynamics of the d-dimensional (d = 2, 3) nonlinear space fractional sine-Gordon equation (NSFSGE). The nonlinearity strength of the NSFSGE is characterized by ε 2 where 0 < ε ≤ 1 is a dimensionless parameter. The second-order time-splitting method is applied to the temporal discretization and the Fourier pseudo-spectral method is used for the spatial discretization. To obtain the explicit relation between the numerical errors and the parameter ε, we introduce the regularity compensation oscillation technique to the convergence analysis of fractional models. Then we establish the improved uniform error bounds O ε 2 τ 2 for the semi-discretization scheme and O h m + ε 2 τ 2 for the full-discretization scheme up to the long time at O(1/ε 2 ). Further, we extend the time-splitting Fourier pseudo-spectral method to the complex NSFSGE as well as the oscillatory complex NSFSGE, and the improved uniform error bounds for them are also given. Finally, extensive numerical examples in two-dimension or three-dimension are provided to support the theoretical analysis. The differences in dynamic behaviors between the fractional sine-Gordon equation and classical sine-Gordon equation are also discussed.

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An energy-preserving computational approach for the semilinear space fractional damped Klein–Gordon equation with a generalized scalar potential

Cited by 8 publications

References 48 publications

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

L1/LDG algorithm for time‐Caputo space‐Riesz fractional convection equation in two dimensions

Improved Uniform Error Bounds of Exponential Wave Integrator Method for Long-Time Dynamics of the Space Fractional Klein-Gordon Equation with Weak Nonlinearity

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