缓增分数阶扩散方程的循环与反循环分裂迭代算法

发布时间：2018-08-29 20:23

【摘要】：缓增分数阶扩散方程是通过乘以缓增指数因子,修正了分数阶扩散方程,更好的描述自然界中生命力有限的微粒或有限空间中的反常扩散现象。缓增幂律跳跃分布产生了缓增空间分数阶导数,缓增幂律等待时间得到了缓增时间分数阶导数,从而改善了分数阶扩散方程的缺陷。近年来,缓增分数阶扩散方程的数值算法引起人们研究兴趣。本文主要研究了两类不同参数的循环与反循环分裂迭代法,快速求解缓增分数阶扩散方程的数值解。本文的主要工作如下:(1)对于缓增分数阶两点边值问题,利用缓增加权移位的Grünwald差分算子(tempered-WSGD)近似缓增Riemann-Liouville分数阶导数,得到的线性系统中的系数矩阵是一个稠密、非对称、具有Toeplitz结构的矩阵。用循环与反循环分裂迭代法求解该Toeplitz系统,在每次迭代时,通过使用快速傅里叶变换求解线性系统,计算量仅需要O(Nlog N),N表示空间网格的节点个数。并详细证明循环与反循环分裂迭代法是无条件收敛的,数值算例表明该算法是可行有效的快速算法。(2)对于扩散系数相等的缓增分数阶两点边值问题,用tempered-WSGD对缓增Riemann-Liouville分数阶的左右导数进行逼近,得到了一个对称的、正定的,具有Toeplitz结构的线性系统。利用双参数循环与反循环分裂迭代法求解Toeplitz系统,并对收敛性进行了证明,且分析了双参数的选取。数值算例表明快速算法的收敛速度快。(3)对于缓增分数阶扩散方程,用隐式的二阶有限差分格式离散之后得到一个对称的、正定的,具有Toeplitz结构的线性系统,双参数的循环与反循环分裂迭代法应用到求解Toeplitz系统,并证明了该方法无条件收敛于线性系统的唯一解,数值实例也验证了双参数的循环与反循环分裂迭代法的收敛速度是快速的。
[Abstract]:Slowly increasing fractional diffusion equation is a modified fractional diffusion equation by multiplying slowly increasing exponent factor to better describe the phenomenon of anomalous diffusion in nature with limited vitality or limited space. The jump distribution of the slowly increasing power law produces the fractional derivative of the slowly increasing space, and the waiting time of the slowly increasing power law obtains the fractional derivative of the slowly increasing time, which improves the defect of the fractional diffusion equation. In recent years, numerical algorithms for slowly increasing fractional diffusion equations have attracted much attention. In this paper, two kinds of cyclic and inverse cyclic splitting iterative methods with different parameters are studied to solve the numerical solutions of slowly increasing fractional diffusion equations. The main work of this paper is as follows: (1) for the two-point boundary value problem of slowly increasing fractional order, the coefficient matrix in linear system is dense and asymmetric by using Gr 眉 nwald difference operator (tempered-WSGD) with slowly increasing weight shift. A matrix with a Toeplitz structure. The Toeplitz system is solved by cyclic and inverse cyclic splitting iteration method. In each iteration, the linear system is solved by using the fast Fourier transform. The computation only needs O (Nlog N) N to represent the number of nodes in the spatial grid. It is proved in detail that the iterative method of cyclic and inverse cyclic splitting is unconditionally convergent. Numerical examples show that the algorithm is a feasible and efficient fast algorithm. (2) for a slowly increasing fractional two-point boundary value problem with equal diffusion coefficient, By using tempered-WSGD to approximate the left and right derivatives of slowly increasing Riemann-Liouville fractional order, a symmetric, positive definite linear system with Toeplitz structure is obtained. The two-parameter cyclic and anti-cyclic splitting iteration method is used to solve the Toeplitz system. The convergence of the system is proved and the selection of two parameters is analyzed. Numerical examples show that the convergence rate of the fast algorithm is fast. (3) for slowly increasing fractional diffusion equations, a symmetric, positive definite linear system with Toeplitz structure is obtained after discretization by implicit second-order finite difference scheme. The two-parameter cyclic and anti-cyclic splitting iteration method is applied to the solution of Toeplitz system, and it is proved that the method converges unconditionally to the unique solution of the linear system. Numerical examples also show that the convergence rate of the iterative method with two parameters is fast.
【学位授予单位】：西安理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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