求解电报方程的重心Lagrange插值配点法

发布时间：2017-12-28 13:12

本文关键词：求解电报方程的重心Lagrange插值配点法　出处：《宁夏大学》2016年硕士论文　论文类型：学位论文

【摘要】：电报方程,又名传输线方程。最初来自于学者对电流,电压信号在传输线上传播的研究。由于电报方程是一类特殊的偏微分方程,很难求得解析解,数值解是其主要的求解方法。因此,探究快速、准确求解电报方程的数值算法具有十分重要的意义。本文采用重心Lagrange插值配点法求解了一、二维常系数与变系数电报方程。它是一种新型的无网格方法,利用重心Lagrange插值建立近似函数,由配点法离散系统方程。该方法无需划分任何形式的网格,也不需要求解积分,因此具有运算简单、计算效率高的特点。论文的主要研究内容包括：(1)重心Lagrange插值配点法的计算精度依赖于插值节点的选取。在空间域和时间域上,选取Chebyhev-Gauss-Lobatto节点,利用重心Lagrange插值构造了包含时间和空间变量的近似函数。(2)将变量的重心Lagrange插值公式代入电报方程,利用Kronecker积离散微分算子。掌握微分方程的偏导数与重心Lagrange插值微分矩阵间的对应关系,可以直接写出其他偏微分方程问题的重心Lagrange插值离散公式。(3)利用矩阵的Kronecker积符号将离散代数方程组化为简单的矩阵形式。对于变系数电报方程的离散系统方程组的矩阵形式,系数矩阵是和微分矩阵的阶数相同的对角矩阵,其他与常系数方程作同样处理。本文采用置换法施加边界条件,同时列出一维、二维问题边界条件的具体施加过程。(4)数值算例中,求解方程的计算程序均通过MATLAB软件编写完成。了解计算节点的编号次序,为编写初始条件和边界条件相关程序带来许多便利。重心Lagrange插值配点法的计算结果与其他数值算法的结果相比,充分体现了重心Lagrange插值配点法在程序实现和提高运算精度方面的优势。
[Abstract]:The telegraph equation, also known as the transmission line equation. It was originally derived from the study of the propagation of current and voltage signals on the transmission line. Because the telegraph equation is a special kind of partial differential equation, it is difficult to obtain analytical solution. Numerical solution is the main solution. Therefore, it is of great significance to explore the fast and accurate numerical algorithm for solving the telegraph equation. In this paper, we use the center of gravity Lagrange interpolation collocation method to solve the one - and two - dimensional constant coefficient and variable coefficient telegraph equations. It is a new type of meshless method, which uses the center of gravity Lagrange interpolation to establish an approximate function and discrete system equations by the method of distribution point. The method does not need to divide any form of grid, and does not need to solve the integral, so it has the characteristics of simple operation and high calculation efficiency. The main contents of this paper are as follows: (1) the calculation accuracy of the centroid Lagrange interpolation point method depends on the selection of interpolated nodes. In space and time domain, the Chebyhev-Gauss-Lobatto nodes are selected and the approximate functions containing time and space variables are constructed by using the center of gravity Lagrange interpolation. (2) the Lagrange interpolation formula of the center of gravity of the variable is replaced by the telegraph equation, and the discrete differential operator of the Kronecker product is used. Mastering the corresponding relation between the partial derivative of differential equation and the barycentric Lagrange interpolation differential matrix, we can directly write out the barycentric Lagrange interpolation discrete formula of other partial differential equations. (3) the discrete algebraic equation is transformed into a simple matrix form by using the Kronecker product symbol of the matrix. For the variable coefficient telegraph equation, the matrix form of the discrete system equations is the same diagonal matrix as the order of the differential matrix, and the other is the same as the constant coefficient equation. In this paper, the boundary conditions are applied by the substitution method, and the specific application process of the boundary conditions of the one-dimensional and two-dimensional problems is listed. (4) in numerical examples, the calculation program for solving the equation is written by MATLAB software. The understanding of the numbering of computing nodes brings many conveniences for the preparation of initial conditions and boundary conditions related programs. The computed results of barycenter Lagrange interpolation collocation method are compared with the results of other numerical algorithms, which fully embody the advantage of Lagrange interpolation collocation method in program implementation and operation accuracy improvement.
【学位授予单位】：宁夏大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82

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