Fourier spectral method for the modified Swift-Hohenberg equation

Abstract

In this paper, we consider the Fourier spectral method for numerically solving the modified Swift-Hohenberg equation. The semi-discrete and fully discrete schemes are established. Moreover, the existence, uniqueness and the optimal error bound are also considered.

Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation

“…There are some wonderful works about the modified Swift-Hohenberg including the bifurcation analysis [31] and the proof of the existence of attractors. For example, [27] and [32] prove the existence of the global attractor, [23] shows the existence of the pullback attractor, [21] presents the existence of the uniform attractors, and [34] gives the numerical solution for this equation by Fourier spectral method. In 2013, Deng [3,6] investigated the steady Swift-Hohenberg equation for its homoclinic solutions.…”

Novel Patterns in Fractional-in-Space Nonlinear Coupled FitzHugh–Nagumo Models with Riesz Fractional Derivative

“…Therefore, the development of time-saving, accurate, and stable numerical methods of FDEs is a focus. Some numerical methods of the FDEs have been announced, such as the finite difference method [13], finite difference predictor-corrector method [14,15], reproducing kernel method [16,17], matrix approach method [18], spectral method [19][20][21][22], and so on [23][24][25][26][27][28][29].…”

“…The Fourier spectral method [19][20][21][22]30,31] is a good method for studying the fractional diffusion model. Some scholars used the Fourier spectral method to solve the space fractional Klein-Gordon-Schrödinger equations [19], modified Swift-Hohenberg equation [20], fractional variable-coefficient KdV-modified KdV equation [30], and 2D space fractional Gray-Scott model [31], and so on [21,22].…”

“…The Fourier spectral method [19][20][21][22]30,31] is a good method for studying the fractional diffusion model. Some scholars used the Fourier spectral method to solve the space fractional Klein-Gordon-Schrödinger equations [19], modified Swift-Hohenberg equation [20], fractional variable-coefficient KdV-modified KdV equation [30], and 2D space fractional Gray-Scott model [31], and so on [21,22]. In this manuscript, we use the Fourier spectral method for spatial discretization and the Runge-Kutta method for time discretization to solve the FitzHugh-Nagumo model with the Riesz fractional derivative.…”

The Fourier spectral approximation for Kolmogorov-Spiegel-Sivashinsky equation

“…They have the natural advantage in keeping the physical properties of primitive problems. During the past years, many papers have already been published to study Fourier spectral method, for example [3,6,16,17].…”