Article
Mathematics
Khalid K. Ali, M. Maneea, Mohamed S. Mohamed
Summary: This study applies the q-homotopy analysis transform method (q-HATM) to solve the Ginzburg-Landau equation and the Ginzburg-Landau coupled system, obtaining analytical solutions in terms of the q-series. The results demonstrate that q-HATM is a reliable and promising approach for solving nonlinear differential equations and provides a valuable tool for researchers in the field of superconductivity.
JOURNAL OF MATHEMATICS
(2023)
Article
Engineering, Civil
Mohamed Drissi, Mohamed Mansouri, Said Mesmoudi, Khalid Saadouni
Summary: This article deals with the resolution of the Ginzburg-Landau envelope equation, which is a nonlinear partial differential equation that requires a robust solver. The study focuses on reducing computational cost by using a high-order solver. The efficiency and robustness of the used algorithm are illustrated by numerical results of a beam resting on a non-linear Winkler foundation.
ENGINEERING STRUCTURES
(2022)
Article
Mathematics, Applied
Xiaolin Li, Shuling Li
Summary: This paper presents an effective linearized element-free Galerkin (EFG) method for solving the complex Ginzburg-Landau (GL) equation, with high precision and convergence rate. A stabilized moving least squares approximation is proposed to enhance stability and performance, and a penalty technique is used to satisfy boundary conditions. Numerical results demonstrate the efficiency of the method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Qifeng Zhang, Jan S. Hesthaven, Zhi-zhong Sun, Yunzhu Ren
Summary: This paper introduces a three-level linearized implicit difference scheme for the two-dimensional spatial fractional nonlinear complex Ginzburg-Landau equation, demonstrating its stability and convergence under certain conditions. By developing a new two-dimensional fractional Sobolev imbedding inequality, energy argument, and careful consideration of the nonlinear term, an optimal convergence order is obtained. Numerical examples validate the theoretical results for different choices of fractional orders alpha and beta.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Shujuan Lu, Tao Xu, Zhaosheng Feng
Summary: In this study, a second-order finite difference scheme is proposed for analyzing a class of space-time variable-order fractional diffusion equation. The scheme is demonstrated to be unconditionally stable and convergent with a convergence order of O(tau(2) + h(2)) under certain conditions, as validated by numerical examples.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics
Mingfa Fei, Wenhao Li, Yulian Yi
Summary: An efficient difference method is proposed to solve one-dimensional and two-dimensional nonlinear time-space fractional Ginzburg-Landau equations, and the convergence of the numerical solution is rigorously studied.
ELECTRONIC RESEARCH ARCHIVE
(2022)
Article
Mathematics, Interdisciplinary Applications
Jincun Liu, Hong Li, Yang Liu
Summary: A fully discrete space-time finite element method is proposed for solving the fractional Ginzburg-Landau equation, utilizing the discontinuous Galerkin finite element scheme in the temporal direction and the Galerkin finite element scheme in the spatial orientation. The well-posedness of the discrete solution is proven by exploiting the valuable properties of Radau numerical integration and Lagrange interpolation polynomials at the Radau points of each time subinterval I-n. Furthermore, the optimal order error estimate in L-8(L-2) is also considered in detail. Numerical examples are provided to assess the validity and effectiveness of the theoretical analysis.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Hengfei Ding, Changpin Li
Summary: In this paper, a new generating function is constructed and used to establish a fourth-order numerical differential formula for approximating the Riesz derivative with order gamma is an element of (1, 2]. The formula is then applied to study the two-dimensional nonlinear spatial fractional complex Ginzburg-Landau equation and a convergence order of O(tau 2 + h4x + h4) is obtained. The unique solvability, unconditional stability, and convergence of the numerical algorithm are proved using discrete energy method and newly derived discrete fractional Sobolev embedding inequalities. Numerical results confirm the theoretical correctness and effectiveness of the proposed scheme. (c) 2023 Elsevier B.V. All rights reserved.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Optics
Salim B. Ivars, Muriel Botey, Ramon Herrero, Kestutis Staliunas
Summary: We propose a method to control turbulence by modifying the excitation cascade. The method is based on the asymmetric coupling between spatiotemporal excitation modes using non-Hermitian potentials. We demonstrate that unidirectional coupling towards larger or smaller wave numbers can increase or reduce the energy flow in turbulent states, thereby influencing the character of turbulence. The study uses the complex Ginzburg-Landau equation, a universal model for pattern formation and turbulence in various systems.
Article
Mathematics, Interdisciplinary Applications
Orazio Descalzi, Carlos Cartes
Summary: This article investigates the formation of localized spatiotemporal chaos in the complex cubic Ginzburg-Landau equation with nonlinear gradient terms and reviews the influence of multiplicative noise on stationary pulses stabilized by nonlinear gradients. Surprising results are obtained through numerical simulations and explained analytically, including the induction of velocity change in propagating dissipative solitons.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
Tiemo Pedergnana, Nicolas Noiray
Summary: This study presents a detailed analysis of the transformation rules for Langevin equations under general nonlinear mappings, and shows how to identify systems with exact potentials by understanding their differential-geometric properties. The results imply a broad class of exactly solvable stochastic models for nonlinear oscillations.
Article
Mathematics, Applied
Pius W. M. Chin
Summary: This article studies the real Ginzburg-Landau equation and proves the existence and uniqueness of the solution in appropriate Sobolev spaces using the Galerkin method and compactness theorem. A reliable nonlinear numerical scheme is designed and shown to be stable, with the optimal rate of convergence determined in some appropriate spaces. The numerical solution preserves all qualitative properties of the exact solution, as demonstrated through numerical experiments with an example to justify the theory.
NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Lu Zhang, Qifeng Zhang, Hai-wei Sun
Summary: This paper presents a fast and high-order finite difference method for two-dimensional space-fractional complex Ginzburg-Landau equations, utilizing innovative time discretization scheme and compact spatial method. The rigorous theoretical analysis using energy argument is conducted, and numerical results demonstrate the performance of the proposed method.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Mingfa Fei, Chengming Huang, Nan Wang, Guoyu Zhang
Summary: This paper presents a linearized Galerkin-Legendre spectral method for solving the one-dimensional nonlinear fractional Ginzburg-Landau equation, with a focus on its unique solvability and boundedness properties. The method is unconditionally convergent in the maximum norm with second-order accuracy in time and spectral accuracy in space. Additionally, a split-step alternating direction implicit Galerkin-Legendre spectral method for two-dimensional problems is introduced without theoretical analysis, and the effectiveness of both proposed schemes is demonstrated through numerical examples.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Jiye Yuan, Tengfei Zhao, Jiqiang Zheng
Summary: This article studies the pointwise convergence for the fractional Schrodinger operator with complex time in one spatial dimension. The results show that the solution converges to the initial data almost everywhere and extend previous research.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
