ABSTRACT This paper concerns a one-phase inverse Stefan problem in one-dimensional space. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We also consider noisy data for this problem. As such, we first regularize the proposed problem by the discrete mollification method. We apply the integration matrix using Bernstein basis polynomials for the discrete mollification method. Through this method, the execution time was gradually decreased. We then extend the space marching algorithm for solving our problem. Moreover, proofs of stability and convergence of the process are given. Finally, the results of this paper have been illustrated and examined by some numerical examples. Numerical examples confirm the efficiency of the proposed method.

中文翻译：

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一种基于离散混合方法的数值方案，使用伯恩斯坦基多项式求解逆一维 Stefan 问题

摘要 本文涉及一维空间中的一相逆Stefan 问题。从解不连续依赖于数据的意义上说，这个问题是不适定的。我们还考虑了这个问题的噪声数据。因此，我们首先通过离散混合方法对所提出的问题进行正则化。我们将使用 Bernstein 基多项式的积分矩阵应用于离散混合方法。通过这种方法，执行时间逐渐减少。然后我们扩展空间行进算法来解决我们的问题。此外，还给出了过程的稳定性和收敛性的证明。最后，通过一些数值例子对本文的结果进行了说明和检验。数值例子证实了所提出方法的效率。