一维热传导方程移动边界识别的计算方法

构造了一种正则化的积分方程方法来由Cauchy数据确定一维热传导方程的移动边界．在将区域延拓至规则区域后，通过Fourier方法将问题转化为一个第一类Volterra积分方程．然后分别用Lavrentiev正则化方法以及Tikhonov正则化方法将不稳定的第一类Volterra积分方程转化为适定的第二类积分方程，并分别将积分方程转化为常微分方程组，并用Runge—Kutta方法数值求解，以及直接离散来求解．最后通过自由边界上的条件得到数值的移动边界．通过一些数值试验表明此方法是有效可行的，并且给出的方法无需迭代，数值计算较简单．

Computational methods for determining a moving boundary in one-dimensional heat equation

In this paper,we use a regularized integral equation method(RIEM) to determine a moving boundary from Cauchy data in a one-dimensional heat equation.By the Fourier method,the problem can be transformed to a Volterra integral equation of the first kind,and then we can use the Lavrentiev regularization and the Tikhonov regularization to solve this integral equation.In the Lavrentiev regularization,we change the integral equation into differential equations and then solve them by the Runge-Kutta method.In the Tikhonov method,we give the convergence analysis and solve the integral equation directly.Then,the boundary geometry p(t) is got by th boundary condition.Principally,the RIEM possesses the following advantages:it does not need any iteration,and a regularized numerical solution can be obtained.