热传导方程初边值问题的三次样条解

Cubic Spline Solution to Initial Boundary Value Problems of Heat Transfer Equations

摘要: 本文利用Legendre-Gauss-Lobatto节点为配置点，构造含参数的三次样条函数来数值求解带Dirichlet边界条件的热传导方程。其中，空间方向用含参数的三次样条函数离散，选取适当的参数值以提高数值误差的精度；时间方向用Crank-Nicolson格式离散，构造算法格式。并给出相应的算法格式和数值算例，表明该算法格式的有效性和高精度。

Abstract: In order to investigate the numerical solution approximated to the exact solution of the initial boundary value problems of the heat transfer equations, the cubic spline solution is given here. Using the parametric cubic spline function based on Legendre-Gauss-Lobatto node for the numerical solution of Dirichlet boundary value problems of the heat transfer equations, the spatial direction is discretized by cubic spline function with parameters, and the parameters are appropriately chosen to improve accuracy of numerical errors and use the Crank-Nicolson format to construct the algorithm scheme in the time direction. Finally, the algorithm and numerical examples are given to demonstrate the high efficiency and accuracy of the proposed algorithm.