Mathematical Sciences: Inversion and Observability of Parabolic Initial Value Problems

在许多工程和科学应用中，通过对系统在空间域内部的不同点的不完全观测来确定系统在空间和时间上演化的初始边界条件的反问题具有相当重要的意义。本课题研究某些抛物型偏微分方程解这些问题的一系列理论和数值方面。我们将研究如何根据噪声数据下稳定的有限组内部温度测量来确定表面温度和热流密度。对初始数据的本征函数展开和初始边界数据的惠特克基数展开的方法进行了理论和数值分析。

The inverse problem of determining the initial boundary conditions of a system evolving in space and time from incomplete observations of the system at various points interior to the spatial domain, is of considerable importance in many engineering and scientific applications. This project deals with a selection of theoretical and numerical aspects of these problems for certain classes of parabolic partial differential equations. Determination of surface temperature and heat flux from a finite set of interior temperature measurements stable under noisy data will be investigated. Theoretical and numerical analysis of methods involving eigenfunction expansions of initial data and Whittaker Cardinal expansions of initial boundary data will be studied.