【摘要】 求解偏微分方程的常用方法包括有限差分法、有限元法等。近年来,小波分析在偏微分方程数值求解中的应用已引起很多学者的关注,例如采用Daubechies小波或shannon小波构造的小波配置方法已经取得较好的结果。钟万勰院士提出的偏微分方程的子域精细积分方法是一种半解析方法,方法简单,精度高。将小波方法和精细积分方法相结合应用于偏微分方程的数值求解中将有利于提高算法的精度和稳定性,为此本文以Burgers方程为例,提出了一种求解一维非线性抛物型偏微分方程的小波精细积分方法。该方法用拟小波配点法对空间域进行离散,建立起对时间的常微分方程组,然后采用精细时程积分方法对该方程组求解。数值计算结果表明,该方法同其它方法相比,具有计算格式简单,数值稳定性和精度较高的优点。更多还原

【Abstract】 To solve the partial differential equations, the finite difference method, finite element methods and other discrete methods in the space domain were used ordinarily. The wavelet analysis methods applied into the numerical solution of partial deference equations have gotten high attention from many authors, for example, the Daubechies scale function and the Shannon wavelet scale function were developed to build wavelet collocation methods and have gotten some good results. The Precise Integration Method in the time domain developed by Zhong is very useful to solve a kinds of differential equations because it possesses very high efficiency and accuracy. In this paper, taking Burgers equation as example, a wavelet precise timeintegration method for the one dimension nonlinear parabolic partial differential equation was proposed. In this method, the spatial domain was discrete by the quasiwavelet collocation method, and so the system of ordinary differential equation to time was built, then the timeintegration method was used to solve the system equations. The computation results show that comparing with other methods the method possesses many merits, such as the simple computational format, higher numerical stability and high precision.更多还原