Abstract
A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L2-1σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims.
Highlights
The 2003 Nobel prize winning Ginzburg–Landau model in the field of physics is widely used in superconductors, superfluids, and condensation processes of Bose–Einstein type
The dynamics and well-posedness of the coupled fractional Ginzburg–Landau equation, which describes a class of nonlinear optical fiber materials with active and passive coupled cores, was discussed in [7]
We propose a high order Alikhanov Legendre-Galerkin spectral method for solving the following nonlinear coupled fractional Ginzburg–Landau equations: C β
Summary
The 2003 Nobel prize winning Ginzburg–Landau model in the field of physics is widely used in superconductors, superfluids, and condensation processes of Bose–Einstein type. The dynamics and well-posedness of the coupled fractional Ginzburg–Landau equation, which describes a class of nonlinear optical fiber materials with active and passive coupled cores, was discussed in [7] Motivated by their vast applications, numerical methods dealing with GinzbergLandau problems have gained attention due to the difficulty of obtaining exact solutions for the fractional order form of Ginzberg-Landau models. In [22], a graded mesh finite difference/Galerkin spectral method was used to numerically solve a coupled system of time and space fractional diffusion equations. We propose a high order Alikhanov Legendre-Galerkin spectral method for solving the following nonlinear coupled fractional Ginzburg–Landau equations:. The main concern of this work is to first design a combined numerical scheme for a coupled system (1) of Ginzburg-Landau with the time Caputo fractional derivative and the Riesz space fractional Laplacian operator. Some numerical experiments are done in the penultimate section while the manuscript ends with a section for conclusion and remarks
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