The fourth-order time-discrete scheme and split-step direct meshless finite volume method for solving cubic–quintic complex Ginzburg–Landau equations on complicated geometries

Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

“…Similar to above problems, we find the numerical results using the numerical scheme given in Equation (29). The above problem has variable coefficients and the exact solution is not known to us; therefore, we check the accuracy of proposed numerical scheme by means of absolute residual error function, which is a measure of how well the approximation satisfies the original nonlinear fractional differential problem given in Example Section 2.1.…”

“…Liu et al [11] presented and discussed a finite difference/finite element algorithm for casting about for numerical solutions to a time-fractional fourth-order reaction-diffusion problem with a nonlinear reaction term, which is based on a finite difference approximation in time and a finite element method in spatial direction. Moreover, many numerical methods have been developed for solving different classes of fractional differential equations such as spectral methods [12][13][14][15][16][17][18][19], finite difference methods [20][21][22][23][24][25], finite element methods [26,27], finite volume method [28,29], and matrix transfer technique [30,31]. The study of delay differential equations with fractional derivatives is rapidly expanding these days since they are frequently employed in modeling of elastic media and stress-strain behavior for the torsional model, control difficulties, high-speed machining communications, and so on [1].…”

High‐order finite difference/spectral‐Galerkin approximations for the nonlinear time–space fractional Ginzburg–Landau equation

“…The global existence, regularity, asymptotic stability, well-posedness, and the long time behavior of the exact solution were discussed in [3][4][5]. Due to the strong nonlinearity, numerical methods such as spectral method [6,7], finite difference method [8,9], finite volume method [10], and finite element method [11] have been widely considered to analyze the Ginzburg-Landau equation.…”