关于Riesz分数阶复Ginzburg-Landau-Schrödinger方程两种可靠技术的比较在超导体建模中的应用
On Comparison of Two Reliable Techniques for the Riesz Fractional Complex Ginzburg- Landau-Schrödinger Equation in Modelling Superconductivity
AI总结：
本文比较了两种可靠方法用于Riesz分数阶复Ginzburg-Landau-Schrödinger方程的求解，分析了其稳定性及精度，展示了不同分数阶次下的绝对误差和L^2、L^infinity范数结果。
摘要：
本文研究了带有Riesz分数阶导数的复Ginzburg-Landau-Schrödinger（CGLS）方程，采用二阶隐式有限差分法（IFDM）进行求解，并进一步利用时间分裂傅里叶谱（TSFS）技术进行比较研究和精度检验。在有限差分离散化中，Riesz分数阶导数通过分数中心差分方法进行近似。进一步分析了所提出方法的稳定性，证明TSFS技术具有无条件稳定性。此外，还列出了不同分数阶次下所获解的|χ(x, t)|^2的绝对误差，并展示了L^2和L^infinity误差范数，结果也通过图形进行展示。
英文摘要：
In the present paper, the Complex Ginzburg-Landau-Schrödinger (CGLS) equation with the Riesz fractional derivative has been treated by a reliable implicit finite difference method (IFDM) of second order and furthermore for the purpose of a comparative study, and also for the investigation of the accuracy of the resulting solutions another effective spectral technique viz. time-splitting Fourier spectral (TSFS) technique has been utilized. In the case of the finite difference discretization, the Riesz fractional derivative is approximated by the fractional centered difference approach. Further the stability of the proposed methods has been analysed thoroughly and the TSFS technique is proved to be unconditionally stable. Moreover the absolute errors for the solutions of |χ(x, t)|^2 obtained from both the techniques for various fractional order have been tabulated. Further the L^2 and L^infinity error norms has been displayed for |χ(x, t)|^2 and the results are also graphically depicted.
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