初始场与边界热流同时识别反问题的正则化方法

考虑含Neumann边界条件的一维热方程的初始场与边界热流同时识别反问题. 首先, 给定右边界热流密度及温度值和终止时刻温度值, 建立Fredholm积分方程组模型, 证明其解的唯一性. 其次, 构造左边界热流密度与初始温度场同时反演的正则化算法. 数值实例结果表明, 采取的数据预处理算法及改进正则化方法, 双函数反演的精度更高.

Regularization Method for Simultaneous Identification of Initial Field and Boundary Heat Flow

We considered an inverse problem of simultaneous identification of the initial field and boundary heat flux for one-dimensional heat equation with Neumann boundary conditions. Firstly, the heat flux density and temperature at the right boundary and the terminal temperature were given, a model of Fredholm integral equation was established and the uniqueness of its solution was proved. Secondly, a regularization method for  simultaneous inversion of the heat flux density at the left boundary and initial temperature field was constructed. The results of numerical examples show that the data preprocessing algorithm and improved regularization method achieve higher accuracy of double-function inversion.