A linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg–Landau equation

A Space-Time Finite Element Method for the Fractional Ginzburg–Landau Equation

“…Let λ n be sufficiently small and satisfy µ − ( Ĉ1 + Ĉ2 )λ n > 0. By combining Equations (16)(17)(18)(19)(20) and setting δ = Re G(V), V ≥ δC 3 U n > 0.…”

“…Recently, some numerical methods for the fractional Ginzburg-Landau equation have been proposed to analyze the behavior of the solution of the equation. For solving this model, some efficient numerical methods, including finite difference methods [6][7][8][9][10][11][12][13][14][15][16] and finite element methods [17,18], were developed. Since the Galerkin finite element method can solve differential equations with more complex geometries and has high-order accuracy, numerous researchers considered the finite element method along the spacial direction to solve space fractional differential equations (see [19,20] and the references therein).…”

The construction of several new inequalities for definite integral

“…For instance, the Cauchy-Schwarz inequality is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, for details we refer to [1,2]. The discrete and continuous Gronwall-Bellman inequalities are often used in the analysis of existence, boundedness, stability of numerical solutions of differential equations and integral equations [3][4][5][6]. In 2005, Yang et al [7] established an extension on Hardy-Hilbert integral inequality by introducing a power exponent function, and show the best possible coefficient.…”

A low-rank Lie-Trotter splitting approach for nonlinear fractional complex Ginzburg-Landau equations

“…In [19], the authors used an exponential Runge-Kutta method to solve the two-dimensional (2D) FGLE. Other related work can be found in [20][21][22][23][24] and references therein.…”