两类时间分数阶偏微分方程的数值算法

发布时间：2018-06-14 10:26

本文选题：分数阶微分 + 再生核理论　；参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：分数阶微分算子因其非局部性,更适合用来描述实际生活中一些复杂的动态行为。近年来,分数阶微分方程的应用已遍及众多科学领域,但是方程的解析解较难得到。因此,探究方程的数值算法成为定性地研究此类方程的一个重要手段,而如何构造高精度的数值算法需要进一步地探究。基于样条思想与再生核理论,本文系统地研究了两类时间分数阶偏微分方程的数值算法,即变时间分数阶Mobile-Immobile对流扩散方程和非线性时间分数阶薛定谔方程。主要结果如下:首先,本文讨论了变时间分数阶Mobile-Immobile对流扩散方程解的光滑性,由此可以用样条函数来逼近方程的解。基于再生核函数和样条多项式,本文构造了此类方程的一个级数形式的近似解表达式。同时,构建了一个简便的数值算法来得到ε-近似解。为了说明算法的有效性,本文给出了算法的收敛性及稳定性分析。其次,本文研究了一类非线性时间分数阶薛定谔方程的数值算法。此类方程的解通常具有弱奇性,本文引入分数阶积分算子作用原来微分方程的两端,得到积分方程再进行求解。与一般的分数阶微分方程不同,此类方程的解属于复数域。本文将方程解的实虚部拆分,在相应的再生核空间中分别导出了实虚部易于计算的解表达式。同时,讨论了方程ε-近似解的存在性。数值结果表明,该算法具有很好的收敛性和较高的数值精度。
[Abstract]:Fractional differential operators are more suitable to describe some complex dynamic behaviors in real life because of their nonlocality. In recent years, fractional differential equations have been widely used in many scientific fields, but the analytical solutions of the equations are difficult to obtain. Therefore, exploring the numerical algorithm of equations becomes an important means to qualitatively study such equations, and how to construct high-precision numerical algorithms needs to be further explored. Based on spline theory and reproducing kernel theory, two kinds of numerical algorithms for fractional partial differential equations of time, namely, variable time fractional Mobile-Immobile convection-diffusion equation and nonlinear time fractional Schrodinger equation, are studied systematically in this paper. The main results are as follows: firstly, we discuss the smoothness of the solution of the Mobile-Immobile convection-diffusion equation with variable time fractional order, so that the solution of the equation can be approximated by spline function. Based on the reproducing kernel function and spline polynomial, an approximate solution expression of a series form of this kind of equation is constructed in this paper. At the same time, a simple numerical algorithm is constructed to obtain the 蔚-approximate solution. In order to illustrate the validity of the algorithm, the convergence and stability analysis of the algorithm are given in this paper. Secondly, the numerical algorithm for a class of nonlinear time fractional Schrodinger equations is studied. The solution of this kind of equation is usually weak singularity. In this paper, the fractional integral operator is introduced to act on the two ends of the original differential equation, and then the integral equation is obtained and solved. Unlike ordinary fractional differential equations, the solutions of such equations belong to the complex field. In this paper, the real imaginary part of the solution of the equation is split, and the expressions of the real imaginary part are derived in the corresponding reproducing kernel space. At the same time, the existence of 蔚-approximate solution of the equation is discussed. Numerical results show that the algorithm has good convergence and high numerical accuracy.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

【参考文献】