二维偏微分方程的小波配点法

A Wavelet Collocation Method for Solving the Two-Dimensional Heat Equation

根据多分辨分析,使用任意连续的尺度函数,在边界处结合外尺度函数,构造了区间上的插值基函数,并结合二元张量积小波分析将此方法推广到了二维。同时,给出了边值条件的积分处理方法,形成了求解二维偏微分方程的小波配点法。以二维热传导方程定解问题为例,选择Shannon函数进行了数值计算。结果表明,数值解达到了较高的精度,表明该方法适用于高维情形。

Based on multi-resolution analysis,the interpolation base functions in interval is proposed,by using arbitrary continuous scaling function and external scaling function in the boundry.And then this method is extended to two-dimensional function in combination with two-tensor product wavelet analysis.At the same time,an integral approach of dealing with boundary condition is suggested,whereby forming a wavelet collocation method for solving the two-dimensional differential equation.At last,taking the two-dimensional heat equation for example,and using Shannon scaling function to carry out numerical calculation,the results indicate that when the numerical solution reaches higher accuracy,this method can be adaptable to high-dimensional case.