In this article, an efficient difference scheme for the coupled fractional Ginzburg–Landau equations with the fractional Laplacian is studied. We construct the discrete scheme based on the implicit midpoint method in time and a weighted and shifted Grünwald difference method in space. Then, we prove that the scheme is uniquely solvable, and the numerical solutions are bounded and unconditionally convergent in the lh2 norm. Finally, numerical tests are given to confirm the theoretical results and show the effectiveness of the scheme.

An efficient difference scheme for the coupled nonlinear fractional Ginzburg–Landau equations with the fractional Laplacian

Exact solutions of the stochastic fractional long-short wave interaction system with multiplicative noise in generalized elastic medium

(2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p [282]. B−L [427]. α [216, 483]. α− z [322]. N = 2 [507]. D [222]. ẍ+ f(x)ẋ + g(x) = 0 [112, 111, 8, 5, 6]. Eτ,ηgl3 [148]. g [300]. κ [244]. L [205, 117]. L [164]. L∞ [368]. M [539]. P [27]. R [147]. Z2 [565]. Z n 2 [131]. Z2 × Z2 [25]. D(X) [166]. S(N) [110]. ∫l2 [154]. SU(2) [210]. N [196, 242]. O [386]. osp(1|2) [565]. p [113, 468]. p(x) [17]. q [437, 220, 92, 183]. R, d = 1, 2, 3 [279]. SDiff(S) [32]. σ [526]. SLq(2) [185]. SU(N) [490]. τ [440]. U(1) N [507]. Uq(sl 2) [185]. φ 2k [283]. φ [553]. φ4 [365]. ∨ [466]. VOA[M4] [33]. Z [550].

/pdf/a-complete-bibliography-of-publications-in-the-journal-of-3brkwhz8a7.pdf

A Complete Bibliography of Publications in the Journal of Mathematical Physics: 2005{2009

In this work, we study numerically two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equations. A centered finite difference method is exploited to discretize the spatial variables and leads to a system of the ordinary differential equation, in which the resulting coefficient matrix is complex symmetric and possesses the block Toeplitz structure. An exponential Runge–Kutta method is employed to solve such a system of the ordinary differential equation. Theoretically, the proposed method is second-order accuracy in space and fourth-order accuracy in time, respectively. In the practical implementation, the product of a block Toeplitz matrix exponential and a vector is calculated by the shift-invert Lanczos method. Meanwhile, the sectorial operator (the coefficient matrix) guarantees the fast approximation by the shift-invert Lanczos method. Numerical experiments are carried out to testify the theoretical results and demonstrate that the proposed method enjoys the excellent computational advantage.

Exponential Runge–Kutta Method for Two-Dimensional Nonlinear Fractional Complex Ginzburg–Landau Equations

This paper deals with the asymptotic behaviour of solutions for non-autonomous stochastic fractional Ginzburg–Landau equations driven by multiplicative noise with α∈(0,1). We first present some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Then we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equations in L2(Rn). At last, we prove the finiteness of fractal dimension of random attractors.

Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg–Landau equations with multiplicative noise

This paper is concerned with the well-posedness as well as long-term dynamics of stochastic Ginzburg–Landau equations driven by nonlinear noise. We will apply a specific method to solve stochastic Ginzburg–Landau equations, known as the variational approach. We prove the existence and uniqueness of the solutions by assuming that the coefficients satisfy certain monotonicity assumptions. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. At the same time, the existence of invariant measures for the stochastic Ginzburg–Landau equations is also established.

Dynamics of stochastic Ginzburg–Landau equations driven by nonlinear noise

This paper deals with the well-posedness of the solutions of the fractional complex Ginzburg-Landauequation driven by locally Lipschitz nonlinear diffusion terms defined on $ R^n $. We first give the pathwise uniform estimates and uniform estimates on average. Then we prove the existence, uniqueness and measurability of solutions for the equation. 

https://www.aimsciences.org/data/article/export-pdf?id=64182fe36565307f274b0c33

Well-posedness of fractional stochastic complex Ginzburg-Landau equations driven by regular additive noise

Abstract This article deals with approximations of center manifolds for delay stochastic differential equations with additive noise. We first prove the existence and smoothness of random center manifolds for these approximation equations. Then we show that the C k {C}^{k} invariant center manifolds of the system with colored noise approximate that of the original system.

/pdf/approximations-of-center-manifolds-for-delay-stochastic-13faunld.pdf

Approximations of center manifolds for delay stochastic differential equations with additive noise

Nowadays, more and more people have already realised that optimisation issues constrained by Differential and Algebraic Equations (DAEs) or Partial Differential and Algebraic Equations (PDAEs) are extremely important to our daily lives, as applications of this type of problem are very widely spread. Unfortunately, it is hard to solve these problems and therefore standard optimisation tools are needed to make them be convenient to deal with. It is clear that there are way too many algebraic modeling tools built into high-level programming languages, such as Python and MATLAB. Consequently, additional freedom was allowed by these tools in incorporating new elements, language, and workflows. The purpose of this article is to encourage model solvers to be able to use the most suitable tools in different situations, which including the advantages and disadvantages of existing tools for solving differential equations, to give them a comprehensive summary of performance of those tools. According to the research result, Pyomo.dae is the stable and flexible feature9, which is useful for modal transformations. Besides, Pyomo.dae is also suitable for three different differential equations. In addition, the Pymanopt can be used for automated differentiation when being applied in optimaisation on manifolds. The PyDEns is so flexible for convenient experimentation that it is famous for its applications for neural networks. At last, Pandapower is good at handling parameters.

Comparing existing tools in Python for modelling real-life optimisation problems with differential equations

Dynamics of The Stochastic g-Navier-Stokes Equations Driven by Nonlinear Noise

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