一类求解抛物型方程侧边值问题的最优滤波方法

An Optimal Filtering Method for Solving a Sideways Parabolic Equation

该文考虑一类特殊的抛物型方程侧边值问题,即一类含有对流项的非标准逆热传导问题.给定在x=1处的温度测量值来确定区间(0,1)上的未知解u(x,t).这是一类不适定问题,即问题的解(如果解存在)不连续依赖于数据.为了求解这一问题,必须采用某些正则化技巧.该文给出了一种最优滤波方法,使得问题的真实解和近似解之间的误差估计达到了Hlder型最优.同时还证明了问题的解在x=0处的收敛性.

The authors consider a sideways parabolic equation in the quarter plane, i.e., a non-standard inverse heat conduction equation with convection term. People want determine the solution μ（x, t） for 0 〈 x 〈 1 from the data along the line x = 1. This is an ill-posed problem in the sense that the solution （if it exists） does not depend continuously on the data. Some special regularization method is needed for solving this problem. This paper considers an optimal filtering method and gives the HSlder optimal error estimate between the exact solution and its regularized approximation solution. Furthermore, the convergence of the regularized approximation solution at x = 0 is also obtained.