非线性分数阶Allen-Cahn方程的数值算法研究

分数阶Allen-Cahn模型是研究相变理论和界面问题的基本相场模型，目前在计算材料学和分子动力学方面有着广泛的应用。鉴于此类方程中的非线性势函数项以及分数阶导算子导致的复杂性，寻求高效的数值方法就成为一个迫切需要解决的问题。本项目将基于再生核函数空间理论，结合拟牛顿法、有限差分法和同伦法，研究非线性分数阶Allen-Cahn方程的有效数值方法。由于近似函数的分数阶导数具有一致收敛性，正与再生核数值算法的一致收敛性相吻合，因此再生核函数空间是数值逼近分数阶方程的理想空间框架。我们采用建立新分数阶再生核空间来减少由于分数阶导数的历史记忆性所引起的存储量。同时为了便于处理分数阶Allen-Cahn方程中的非线性势函数项和不同的边界条件,我们选取勒让德多项式作为分数阶再生核空间的基函数。我们将进一步利用再生核函数理论的优势，证明提出的分数阶再生核数值算法的一致收敛性和能量稳定性。

The fractional Allen-Cahn models are basic phase field models for studying phase transition theory and interface problems. Recently, the fractional Allen-Cahn equations have been widely used in the fields of computational materials science and molecular dynamics. Due to the complexity of the nonlinear potential function term and the fractional derivative operator in the fractional Allen-Cahn models, it is very important to search for efficient numerical methods. In this project, based on the theory of reproducing kernel functions, we will study the effective numerical methods of fractionial Allen-Cahn equations by combining the quasi-Newton method, the finite difference method and the homotopy perturbation method. Because the uniform convergence of fractional derivative of the approximate function is consistent with the uniform convergence of the reproducing kernel numerical algorithm, the reproducing kernel function space is an ideal space framework for numerical approximation to the fractional equations. A new fractional reproducing kernel space is established to reduce the storage caused by the historical memory of fractional derivatives. At the same time, in order to deal with the nonlinear potential term and different boundary value conditions in the fractional Allen-Cahn equation, we choose Legendre polynomials as the basis function of the fractional reproducing kernel space. Furthermore, by using advantages of reproducing kernel function theory, we will prove the uniform convergence and energy stability of the proposed fractional reproducing kernel numerical algorithm.