含两个未知边界的抛物型方程反问题稳定数值算法

在物理学中模拟均匀的多孔介质流时会遇到一类一维抛物型方程反问题,该问题由一个含两未知边界条件的抛物型方程以及在某指定内点上测量得到的特定数据条件所构成。为了能够更好地求解该类反问题,本文首先证明解的唯一性,然后给出其离散后的有限差分格式以及该格式下的数值解的稳定性条件,并通过切比雪夫多项式逼近未知函数,利用最小二乘法解出未知项的系数,最后给出数值试验。

A Stable Numerical Algorithm for an Inverse Parabolic Problem with Two Unknown Boundaries

A one-dimensional inverse parabolic problem can be encounter when we research simulation of homogeneous porous medium flow in physics. The problem consists of a parabolic e- quation with two conditions which are unknown at the boundaries and a condition which is deter- mined from an over-specified data measured at an interior point. In order to solve this problem, uniqueness of the solution should be proved first, then the problem is discretized and a finite dif- ference scheme is given. Stability conditions for numerical solution to inverse problem are stated. A set of Chebyshev polynomials are approximate to the unknown function and the unknown set of expansion coefficients in unknown function are determined from the Least-squares method. In the last section the paper gives some numerical examples.