等离子体中分数阶微分方程求解的有限元方法研究

Fractional differential equations are of great importance in plasma, because the physical processes in the plasma can be accurately described by the fractional differential equations when the media possess the nonlocal properties. By using the computational mathematics, plasma physics, and fluid mechanics, this project will investigate the finite element methods for fractional differential equations in plasma. The project is composed of the following parts: For the plasma models with the coupling of high frequency-low frequency waves and nonlocal properties, it will establish the finite element methods for fractional coupled nonlinear Schrödinger equations, make the numerical simulations for the vector nonlinear waves, and reveal the effect of fractional derivative on the coupling of high frequency—low frequency waves; For the plasma models with the influence of magnetic fields and nonlocal properties, it will establish multiscale finite element methods for fractional MHD equations, make the numerical simulations for the magnetohydrodynamics problems, and explore the interactions between the fractional derivative and multiphysics coupling. This project will promote the cross development of computational mathematics, plasma physics, and fluid mechanics, improve the numerical methods for fractional differential equations, extend the applications of fractional differential equations, and provide new mathematical theories for investigating the plasma with nonlocal properties.

分数阶微分方程在等离子体中占有十分重要的地位，这是因为当介质具有非局部性质时，等离子体中的物理过程需要通过分数阶微分方程进行准确的描述。本项目综合运用计算数学、等离子体物理学和流体力学的知识，研究等离子体中分数阶微分方程求解的有限元方法，具体内容包括：针对高频—低频波发生耦合作用且具有非局部性质的等离子体模型，本项目将建立分数阶耦合非线性Schrödinger方程的有限元方法，对矢量非线性波动现象进行数值模拟，揭示分数阶导数对高频—低频波耦合作用的物理机理的影响；针对受到磁场作用且具有非局部性质的等离子体模型，本项目将建立分数阶MHD方程的多尺度有限元方法，对磁流体力学问题进行数值模拟，探索分数阶导数对多物理场耦合的作用规律。本项目将推动计算数学、等离子体物理学与流体力学的交叉发展，完善分数阶微分方程的数值算法，扩大分数阶微分方程的应用范围，为研究具有非局部性质的等离子体提供新的数学理论。

本项目综合运用了计算数学、等离子体物理学和流体力学的知识，研究了等离子体中非线性分数阶微分方程的数值解法。具体研究内容包括：根据具体的物理环境建立等离子体的流体力学方程组，如Euler-Possion方程组、MHD方程组等；在小振幅的假设下，本项目通过约化摄动理论、变分原理和分数阶微积分理论从流体力学方程组中推导出了分数阶Schamel-KdV方程、非线性Schrödinger方程等非线性偏微分方程，构造了求解分数阶微分方程的数值算法，建立了算法的收敛性和稳定性分析，并通过数值算例验证了理论分析，编写了新的程序；本项目对等离子体中的非线性物理过程开展了数值工作，分析了Riesz分数阶导数和Caputo分数阶导数对等离子体中的孤立波、激波、怪波等非线性波动现象的影响，探讨了分数阶导数的阶数与非线性波动的空间—时间分布、振幅、速度、非线性叠加等物理性质之间的关系，揭示了非局部性质对等离子体的作用规律。.. 本项目已发表多篇SCI论文，相关研究成果在理论方面能够丰富分数阶微分方程的数值算法，在应用方面能够指导等离子体物理实验，并且推动了计算数学、等离子体物理学与流体力学的交叉发展。