【摘要】：分?jǐn)?shù)階微分方程在科學(xué)計(jì)算領(lǐng)域有廣泛的應(yīng)用,如分?jǐn)?shù)階微分方程可以描述物理中的許多現(xiàn)象,特別的,分?jǐn)?shù)階擴(kuò)散波方程可以準(zhǔn)確地描述反常擴(kuò)散現(xiàn)象.基于有限差分的微分方程的QTT數(shù)值方法已經(jīng)有了一些研究,但都是整數(shù)階微分方程,分?jǐn)?shù)階微分方程的QTT數(shù)值方法目前還沒(méi)有相關(guān)研究,解決這個(gè)問(wèn)題的關(guān)鍵在于Caputo分?jǐn)?shù)階導(dǎo)數(shù)算子的QTT分解的構(gòu)造以及空間緊致格式算子的QTT分解的構(gòu)造,本文主要研究分?jǐn)?shù)階擴(kuò)散波方程的QTT數(shù)值算法.首先,在Toeplitz矩陣QTT分解的基礎(chǔ)上,得到了 Hankel矩陣的QTT分解,以及Toeplitz矩陣和Hankel矩陣的逆矩陣的QTT分解,并且指出了Toeplitz矩陣和Hankel矩陣的QTT分解的核之間的聯(lián)系,二者的QTT核可以較為容易的相互轉(zhuǎn)化.其次,分?jǐn)?shù)階擴(kuò)散波方程離散格式主要有Caputo分?jǐn)?shù)階導(dǎo)數(shù)算子、拉普拉斯算子和空間緊格式算子.本文得到了 Caputo分?jǐn)?shù)階導(dǎo)數(shù)算子和空間緊格式算子的低秩QTT顯示表示,并基于分?jǐn)?shù)階擴(kuò)散波方程矩陣形式在整體結(jié)構(gòu)上的特殊性,得到了方程的低秩QTT顯示表示.最后,運(yùn)用DMRG方法求解分?jǐn)?shù)階擴(kuò)散波方程.得到了該問(wèn)題比較精確的快速數(shù)值算法,數(shù)值實(shí)驗(yàn)表明QTT分解方法是解決這類(lèi)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.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.8

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