A compact difference scheme for a two dimensional nonlinear fractional Klein–Gordon equation in polar coordinates

Wang, Zhen; Vong, Seakweng

doi:10.1016/j.camwa.2016.04.005

Cited by 11 publications

(2 citation statements)

References 34 publications

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“…Based on the piecewise linear interpolation, Sun and Wu proposed an ℒ1 formula in , with the truncation error

O (τ^{2 - α})

, where 0 < α < 1. One could refer to (the fractional Cattaneo equation , the fractional convection‐diffusion equation , the fractional Klein‐Gordon equation , the nonlinear time‐fractional Schrödinger equations , the fractional integro‐differential equation and the multiterm time–space variable‐order fractional advection–diffusion equations for more details and applications. In , a formula called ℒ1‐2 was proposed to approximate the Caputo fractional derivative by Gao, Sun, and Zhang, where they utilized the quadratic interpolation for the integrand f ( t ) on the interval [ t j ‐1 , t j ] for j ≥ 2 and the linear interpolation for that on the interval [ t 0 , t 1 ].…”

Section: Introductionmentioning

confidence: 99%

Fast high order difference schemes for the time fractional telegraph equation

Liang

Yao

Wang

2019

Numerical Methods Partial

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In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the ℱℒ2‐1σ formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order Oτ2+h2 and Oτ2+h4, respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the ℒ2‐1σ method.

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“…Based on the piecewise linear interpolation, Sun and Wu proposed an ℒ1 formula in , with the truncation error

O (τ^{2 - α})

Section: Introductionmentioning

confidence: 99%

Fast high order difference schemes for the time fractional telegraph equation

Liang

Yao

Wang

2019

Numerical Methods Partial

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“…[30][31][32][33][34] Simultaneously, some numerical methods were developed to solve the fractional Klein-Gordon equations. [35][36][37][38] Wang et al investigated an efficient conservative difference scheme for the nonlinear coupled space fractional Klein-Gordon-Schrödinger (NCSFKGS) equations in [39]. Over the past 30 years, Fourier spectral method has been actively applied to solve integer partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

A Fourier spectral method for the nonlinear coupled space fractional Klein‐Gordon‐Schrödinger equations

et al. 2019

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In this paper, we give an efficient numerical method for the nonlinear coupled space fractional Klein-Gordon-Schrödinger (NCSFKGS) equations, based on the Crank-Nicolson method, the central difference method and the Fourier spectral method. As far as we know, no one has studied the Equations (5)-(6) in our paper, these equations are different from those in [39] which only considers space fractional Schrödinger equation while the Klein-Gordon equation is classical, here, we consider the two equations which are both space fractional. In this paper, the Crank-Nicolson method and the central difference method are used to discretize the space fractional Schrödinger equation and the space fractional Klein-Gordon equation in time direction, respectively. The Fourier spectral method is used to discretize the NCSFKGS equations in space direction. This numerical method conserves the mass and energy in the discrete level. The convergence of the numerical method is of second order accuracy in time and spectral accuracy in space. Rigorous proofs are developed here for the conservation laws and the convergence of the numerical method. Numerical experiments are presented to confirm the theoretical results and they proved that our numerical method is very efficient (it only takes a few seconds to get the numerical solutions). K E Y W O R D Sconvergence analysis, fast Fourier transform, Fourier spectral method, mass and energy conservation laws, nonlinear coupled space fractional Klein-Gordon-Schrödinger equations

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