含一个未知边界的抛物型方程反问题稳定数值算法

A Stable Numerical Algorithm of an Inverse Problem for a Parabolic Equation with One Unknown Boundary

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反问题,该问题由一个含一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。为了能够更好的求解该类反问题,首先证明解的唯一性,然后给出其离散后的有限差分格式求解该反问题,并讨论了该格式的稳定性条件,最后给出数值试验表明该方法的有效性和可行性。

A one-dimensional inverse parabolic problem can be encountered in simulation of ho- mogeneous porous medium flow in physics. It consists of a parabolic equation with one un- known boundary and specific data measured at an interior point. The uniqueness of the solution is shown at first, then the problem is discretized and a finite difference scheme is given. Stabili- ty conditions for numerical solution to the inverse problem are argued. Numerical examples are cited to show the feasibility and effectiveness of the solution.