时—空分数阶扩散方程的快速算法以及MT-TSCR-FDE的快速数值解法

发布时间：2018-07-26 20:47

【摘要】：如今,分数阶微积分已成为流行在社会科学与工程的重要工具。特别是时空分数阶扩散方程正越来越多地应用于研究许多领域的反常扩散现象。由于分数阶导数的非局部性,其数值求解会生成满系数矩阵,并且此矩阵的求解需要消耗大量的计算成本及存储量。所以我们展开研究快速数值方法来解决这一问题。第一步：我们给出时-空分数阶双边扩散方程的一般形式：对于时间分数阶,我们采用Caputo分数阶导数；对于空间分数阶,采用左和右的Ricmann-Liouville空间分数阶导数,并利用修正的Grunwald-Letnikov近似。给出相应的有限差分格式及矩阵格式。第二步：分析有限差分格式的满系数矩阵,它可以分解成Toeplitz矩阵与向量乘积之和。根据Toeplitz矩阵与循环矩阵的关系,以及循环矩阵的性质,提出用傅里叶变换法求解矩阵向量的乘积,开发一个基于快速傅里叶变换(FFT)的快速解决方案。第三步：基于FFT的快速解决方案,我们开发了两种快速数值方法：一种是O(N log N)的最小剩余共轭梯度平方快速迭代方法,一种是O(N log2 N)的快速有限差分法。与常规有限差分法相比能够极大地减少计算成本和存储空间,同时保持相同的精度。第四步：对于多项分数阶时-空Caputo-Resz方程(MT-TSCR-FDE), P(Dt)u(x,t)=p(x)Rxβ+q(x)Rxγ-h(x)u(x,t)+F(x,t)首先利用预估-校正法对此多项分数阶方程进行数值逼近,然后应用上面提出的方法和步骤,分析研究数值求解的快速方法,以极大地减少计算成本和存储空间。第五步：数值实验。分别针对时-空Caputo-Riesz分数阶扩散方程,一个有解析解的时-空分数阶扩散方程,以及最后的MT-TSCR-FDE给出实例数值模拟,给出相应的误差分析,以及CPU时间分析,通过优良的数值结果证明本文提出的几个快速方法的有效性。
[Abstract]:Today, fractional calculus has become an important tool in social science and engineering. In particular, fractional diffusion equations are increasingly used to study anomalous diffusion phenomena in many fields. Because of the nonlocality of fractional derivative, the numerical solution of the matrix will generate a full coefficient matrix, and the solution of this matrix needs a lot of computation cost and storage. So we study the fast numerical method to solve this problem. The first step: we give the general form of the time-space fractional two-sided diffusion equation: for the time fractional order, we use the Caputo fractional derivative; for the space fractional order, we use the left and right Ricmann-Liouville space fractional derivative. The modified Grunwald-Letnikov approximation is used. The corresponding finite difference scheme and matrix scheme are given. The second step is to analyze the full coefficient matrix of finite difference scheme, which can be decomposed into the sum of the product of Toeplitz matrix and vector. According to the relationship between Toeplitz matrix and cyclic matrix, and the properties of cyclic matrix, a fast solution based on fast Fourier transform (FFT) is developed by using Fourier transform method to solve the product of matrix vector. Step 3: based on the fast solution of FFT, we develop two fast numerical methods: one is the fast iterative method of minimum residual conjugate gradient square of O (N log N) and the other is the fast finite difference method of O (N log2 N). Compared with the conventional finite difference method, the computational cost and storage space can be greatly reduced, while maintaining the same accuracy. The fourth step: for the multiterm fractional space-time Caputo-Resz equation (MT-TSCR-FDE), P (Dt) u (XT) p (x) Rx 尾 q (x) Rx 纬 -h (x) u (XT) F (XT, we first use the predictor-correction method to approximate the multivariate fractional order equation, and then apply the methods and steps mentioned above to analyze and study the fast method of numerical solution. To greatly reduce computational costs and storage space. Step 5: numerical experiment. For time-space Caputo-Riesz fractional order diffusion equation, a time-space fractional diffusion equation with analytic solution, and the final MT-TSCR-FDE numerical simulation, the corresponding error analysis and CPU time analysis are given. The validity of several fast methods proposed in this paper is proved by good numerical results.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

【相似文献】