一类求解抛物型方程侧边值问题的最优滤波方法

数学物理学报 ›› 2009, Vol. 29 ›› Issue (2): 245-252.

一类求解抛物型方程侧边值问题的最优滤波方法

李洪芳, 傅初黎, 熊向团, 南楠

(1. 兰州大学数学与统计学院 |甘肃兰州 730000|2. 中国石油大学(华东)数学与计算科学学院 |山东东营 257061;
3. 复旦大学数学科学学院上海 200433; 4. 艺洛软件技术(上海)有限公司上海 201203)

收稿日期:2007-07-30 修回日期:2008-06-05 出版日期:2009-04-25 发布日期:2009-04-25
基金资助:
国家自然科学基金(10671085)、兰州大学理论物理与数学纯基础科学基金(Lzu05005)和中国石油大学(华东)引进人才(博士)科研基金(Y080807)资助.

An Optimal Filtering Method for Solving a |Sideways Parabolic Equation

LI Hong-Fang, FU Chu-Li, XIONG Xiang-Qiu, NA Nan

(1. School of Mathematics and Statistics, Lanzhou University, Gansu Lanzhou 730000|2. College of Mathematics and Computational Science, China University of Petroleum, Shandong Dongying 257061; 3. School of Mathematical Sciences, Fudan University, Shanghai 200433|4. ILOG Software Technology (Shanghai) Co., Ltd., Shanghai 201203)

Received:2007-07-30 Revised:2008-06-05 Online:2009-04-25 Published:2009-04-25
Supported by:
国家自然科学基金(10671085)、兰州大学理论物理与数学纯基础科学基金(Lzu05005)和中国石油大学(华东)引进人才(博士)科研基金(Y080807)资助.

摘要/Abstract

摘要：

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Hölder型最优. 同时还证明了问题的解在x=0处的收敛性.

关键词: 不适定问题, 抛物型方程侧边值问题, 最优滤波方法, 误差估计

Abstract:

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 u(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 H\"{o}lder 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.

Key words: Ill-posed problem, Sideways parabolic equation, Optimal filter , method, Error estimate

中图分类号:

35R25

李洪芳, 傅初黎, 熊向团, 南楠. 一类求解抛物型方程侧边值问题的最优滤波方法[J]. 数学物理学报, 2009, 29(2): 245-252.

LI Hong-Fang, FU Chu-Li, XIONG Xiang-Qiu, NA Nan. An Optimal Filtering Method for Solving a |Sideways Parabolic Equation[J]. Acta mathematica scientia,Series A, 2009, 29(2): 245-252.

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