【作者】 黎丽梅；

【导师】 徐大；

【作者基本信息】 湖南师范大学 ， 计算数学， 2013， 博士

【摘要】 分数阶偏微分方程是一类很重要的微分方程,它源于许多科学领域,例如带记忆的热传导、多孔粘弹性介质的压缩、原子反应、动力学、生命科学、药理学、病理学、牛顿流体学等.在这些实际问题中,很多数学家以及应用领域中的研究者正在尝试着用分数阶微分方程来建模.由于分数阶导数是拟微分算子,它具有保记忆性(非局部性),因此在对实际问题进行优美刻画的同时,也给数值计算带来了相当大的困难,特别是对于高维问题的数值计算.交替方向有限元方法是一类计算高维偏微分方程的有效数值方法,可以把多维问题降维,分解成连续解几个一维问题.因此它既具备了交替方向存储量少、计算量低又有有限元高精度的特点.本文旨在用交替方向有限元方法研究二维的分数阶偏微分方程的数值解,设计了这类方程稳定的有效的数值格式,建立了所给格式的误差估计.本文的工作主要包括以下三个部分：第一部分利用交替方向隐式有限差分法研究二维的分数阶发展方程的数值解.时间方向采用向后欧拉方法,空间方向使用二阶差商,积分项用一阶卷积求积公式逼近,得到全离散格式.然后用离散的能量方法证明了全离散格式的稳定性和收敛性.最后用数值结果和数值模拟验证了理论分析的收敛阶和所给格式的有效性.该方法能够有效地减少由全局时间依赖性所引起的对存储量的要求,从而可以计算长时间的解.第二部分讨论了利用交替方向隐式Galerkin有限元方法研究二维的分数阶发展方程的数值解,空间方向采用有限元方法,时间方向使用向后欧拉格式、Grank-Nicolson格式,积分项运用相应的卷积求积公式逼近,得到全离散格式.严格证明了全离散格式的稳定性和误差估计,并用数值例子检验理论分析的正确性.该方法在空间方向上能达到任意阶精度,也能够有效地减少由全局时间依赖性所引起的对存储量的要求.第三部分利用交替方向隐式Galerkin有限元方法研究二维的时间分数阶扩散-波动方程的数值解.时间方向使用Grank-Nicolson格式,空间方向采用Galerkin有限元,得到全离散格式.证明了全离散格式的稳定性和收敛性,数值例子验证其结论的正确性.更多还原

【Abstract】 The fractional partial differential equation is a class of very important differential equations, which originated from many science fields, such as, heat conduction with memory, compression of porous viscoelastic media, atomic reaction, dynamics, life sciences, pharmacology, pathology, Newton fluidics, etc. So many mathematicians and researchers in the application fields are trying to use fractional differential equations to model. Fractional derivative is pseudo-differential operator and has the character of memory (nonlocal). Although it can describe practical problems beautifully, it can also bring con-siderable difficulties in numerical computation, especially for high-dimensional problems.Alternating direction implicit (ADI) finite element method is an effec-tive numerical methods to calculate high-dimensional partial differential equa-tions, which reduce a multidimensional problem to sets of independent one-dimensional problems. So ADI finite element method has less storage, low computation and the feature of high accuracy. The main objective of this thesis is to use ADI method to investigate the numerical solutions of the two-dimensional time fractional partial differential equations. Stable and efficient numerical schemes are proposed for these equations and the error estimates of our proposed numerical schemes are also established. The main work of this thesis contains the following three parts:In the first part, we use ADI finite difference method to solve the nu-merical solutions of the two dimensional fractional evolution equation. We first use the second-order difference quotient for the spatial discretization and the backward Euler for the time stepping combined with order one convolu-tion quadrature approximating the integral term and obtain the full discrete scheme. Then, applying the method of discrete energy method, we prove that the full discrete scheme is unconditionally stable and convergent. Finally, some numerical results and numerical simulations are presented to confirm the rates of convergence and the robustness of the numerical schemes. This method can effectively reduce the storage capacity which is caused by overall time-dependent and can calculate the long time solution.In the second part, ADI Galerkin schemes are formulated and analyzed for the two-dimensional time fractional evolution equation. We first use the Galerkin finite element for the spatial discretization and the backward Eu-ler, Grank-Nicolson for the time stepping combined with order one and two convolution quadrature and obtain the full discrete scheme. Then we rigorously prove the stability and error estimates of the full discrete scheme. Numerical examples is examined to the accuracy of the theoretical analysis. The method achieve arbitrary order accuracy in the spatial direction and effectively reduce the storage volume requirements caused by the overall time dependence.In the third part, we apply ADI finite element method to investigate the numerical solutions of the two-dimensional fractional diffusion-wave equation. We use Galerkin finite element method in space and Crank-Nicolson method in time and obtain the full discrete scheme. We prove the stability and error estimates of the scheme. Numerical examples verify the correctness of the conclusions.更多还原