分数阶扩散波方程数值解的QTT方法研究
发布时间：2018-10-05 12:57
【摘要】：分数阶微分方程在科学计算领域有广泛的应用,如分数阶微分方程可以描述物理中的许多现象,特别的,分数阶扩散波方程可以准确地描述反常扩散现象.基于有限差分的微分方程的QTT数值方法已经有了一些研究,但都是整数阶微分方程,分数阶微分方程的QTT数值方法目前还没有相关研究,解决这个问题的关键在于Caputo分数阶导数算子的QTT分解的构造以及空间紧致格式算子的QTT分解的构造,本文主要研究分数阶扩散波方程的QTT数值算法.首先,在Toeplitz矩阵QTT分解的基础上,得到了 Hankel矩阵的QTT分解,以及Toeplitz矩阵和Hankel矩阵的逆矩阵的QTT分解,并且指出了Toeplitz矩阵和Hankel矩阵的QTT分解的核之间的联系,二者的QTT核可以较为容易的相互转化.其次,分数阶扩散波方程离散格式主要有Caputo分数阶导数算子、拉普拉斯算子和空间紧格式算子.本文得到了 Caputo分数阶导数算子和空间紧格式算子的低秩QTT显示表示,并基于分数阶扩散波方程矩阵形式在整体结构上的特殊性,得到了方程的低秩QTT显示表示.最后,运用DMRG方法求解分数阶扩散波方程.得到了该问题比较精确的快速数值算法,数值实验表明QTT分解方法是解决这类方程的有力工具.
[Abstract]:Fractional differential equations are widely used in the field of scientific computation, such as fractional differential equations can describe many phenomena in physics, especially, fractional diffusion wave equations can accurately describe anomalous diffusion phenomena. The QTT numerical methods for differential equations based on finite difference have been studied, but they are all integer-order differential equations. The QTT numerical methods of fractional differential equations have not been studied yet. The key to solve this problem lies in the construction of the QTT decomposition of the Caputo fractional derivative operator and the construction of the QTT decomposition of the spatial compact format operator. This paper mainly studies the QTT numerical algorithm for the fractional order diffusion wave equation. Firstly, on the basis of QTT factorization of Toeplitz matrix, the QTT decomposition of Hankel matrix and the QTT factorization of inverse matrix of Toeplitz matrix and Hankel matrix are obtained, and the relation between the kernel of Toeplitz matrix and QTT factorization of Hankel matrix is pointed out. Their QTT cores can be easily transformed into each other. Secondly, the discrete schemes of fractional diffusion wave equations mainly include Caputo fractional derivative operator, Laplace operator and spatial compact format operator. In this paper, the low rank QTT representation of Caputo fractional derivative operator and space compact format operator is obtained. Based on the particularity of the matrix form of fractional diffusion wave equation in the global structure, the low rank QTT representation of the equation is obtained. Finally, the fractional diffusion wave equation is solved by DMRG method. A fast numerical algorithm for solving the problem is obtained. Numerical experiments show that the QTT decomposition method is a powerful tool for solving this kind of equations.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2253469
[Abstract]:Fractional differential equations are widely used in the field of scientific computation, such as fractional differential equations can describe many phenomena in physics, especially, fractional diffusion wave equations can accurately describe anomalous diffusion phenomena. The QTT numerical methods for differential equations based on finite difference have been studied, but they are all integer-order differential equations. The QTT numerical methods of fractional differential equations have not been studied yet. The key to solve this problem lies in the construction of the QTT decomposition of the Caputo fractional derivative operator and the construction of the QTT decomposition of the spatial compact format operator. This paper mainly studies the QTT numerical algorithm for the fractional order diffusion wave equation. Firstly, on the basis of QTT factorization of Toeplitz matrix, the QTT decomposition of Hankel matrix and the QTT factorization of inverse matrix of Toeplitz matrix and Hankel matrix are obtained, and the relation between the kernel of Toeplitz matrix and QTT factorization of Hankel matrix is pointed out. Their QTT cores can be easily transformed into each other. Secondly, the discrete schemes of fractional diffusion wave equations mainly include Caputo fractional derivative operator, Laplace operator and spatial compact format operator. In this paper, the low rank QTT representation of Caputo fractional derivative operator and space compact format operator is obtained. Based on the particularity of the matrix form of fractional diffusion wave equation in the global structure, the low rank QTT representation of the equation is obtained. Finally, the fractional diffusion wave equation is solved by DMRG method. A fast numerical algorithm for solving the problem is obtained. Numerical experiments show that the QTT decomposition method is a powerful tool for solving this kind of equations.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
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