偏微分方程解存在性的计算机辅助证明

A Computer-assisted Proof Method to Existence of Solution to Partial Differential Equations

针对具有初值的非线性的偏微分方程,首先,采用牛顿法求出方程的近似解。接着,利用计算机软件Matlab的帮助,证明在这个近似解的附近存在精确解。主要的方法是将偏微分方程的解转化为一个紧算子的不动点,然后在计算机中构造一个候补的集合,验证该算子在这个集合中存在一个不动点。这个过程是通过将不动点定理转化为可以计算的条件,然后在计算机软件中进行验算。最后,将理论应用在两类偏微分方程上,得出相应方程的近似解,以及相应的数值验算结果。

For nonlinear partial differential equations with initial value,firstly,we use Newton ＇s method to obtain an approximate solution. Then,by Matlab,we prove that there exists an exact solution near the approximate solution. This can be achieved by letting the solution of the equation be a fixed-point of a compact operator,and then constructing a candidate set so that we prove there exists a solution in the set. We create some computable criteria so that the proof can be applicable in a computer.Lastly we apply our theory to the Emden equation and get some numerical verification results.