时间分数阶微分方程的弱奇异解的高精度数值逼近方法的研究

Time-fractional differential equations have recently been widely used to descirbe the phenomenon of anomalous diffusion in physics. In this project, we intend to consider the numerical approximations for a class of time-fractional initial and boundary problems with negative semi-definite spatial operators. First, we plan to construct the semi-discrete schemes of the differential equations such that the spatial discrete operators preserve negative semi-definiteness. We intend to establish the priori error estimate in space concerning the smooth and non-smooth inital conditions and source terms. Second, We intend to propose a class of novel convolution quadrature formulae for approximating the semi-discrete schemes with respect to time such that the errors between the solutions of the fully discrete schemes and the weakly singular solutions of the differential equations can preserve high order of accuray. We intend to analyze the unconditionally stability of the fully discrete schemes and establish the global error estimate concerning the symmetric and non-symmetric spatial discrete operators. Third, to overcome the Dahlquist restriction on the A-stability of linear multistep methods, using the proposed convolution quadrature formulae, we intend to construct a class of boundary value methods with A-stability for approximating the semi-descritized schemes in time. Furthermore, we analyze the unconditionally stability of the boundary value methods and establish the priori error estimate on the weakly singular solutions of the time-fractional differential equations.

时间分数阶微分方程近来被广泛应用于描述具有反常扩散的物理现象。在本项目中，我们拟考虑一类空间算子满足半负定性的时间分数阶初边值问题的数值离散。第一，拟建立方程的半离散格式，使得空间离散算子保持半负定性质。拟建立当初始条件及已知源项充分光滑和非光滑情形下空间离散方法的先验误差估计。第二，拟提出一类新的离散卷积求积公式用于半离散格式时间方向的逼近，使得到的全离散格式的解逼近方程的弱奇异解能够保持高阶精度。拟分别研究当空间离散算子为对称和非对称时，所得全离散格式的无条件稳定性分析和全局误差估计。第三，为了克服对于线性多步法的A稳定性的Dahlqiust限制条件，基于所构造的离散卷积公式，我们拟构造一类具有A稳定性的边值方法用于半离散格式时间方向的数值逼近。进一步地，拟分析全离散格式的无条件稳定性，建立当方程的解具有弱奇异性时所构造的边值方法的先验误差估计。

时间分数阶微分方程目前主要被用于描述在实际具有反常扩散现象的一类问题。亚扩散方程的解的理论性质和数值模拟在最近几十年获得了很多学者的关注，取得了一系列研究结果。结合现有结论，本项目主要工作在于：第一，讨论非光滑初值和源项条件下，常系数线性亚扩散方程的解的适定性质和光滑性质。对于一类具有奇异性的源项，证明了相应的方程的解具有更弱的正则性。第二，应用Galerkin有限元和Lumped mass有限元构造半离散数值格式用于对空间方向的数值逼近，并给出严格的误差估计，使其适用于非光滑初值和源项的情形。第三，在时间方向上，基于卷积向后差分格式，Crank-Nicolson格式构造新的全离散格式，分别命名为GLBE，FBDF22，CN-I和CN-II格式，并利用Laplace变换方法提出新的全离散格式误差分析方法，对于所提出的格式提供严格的收敛精度的理论证明。大量的数值例子也验证了所提出的新的格式对于亚扩散方程解的弱奇异性，甚至是奇异性都能够保持最优精度的逼近。