Purpose

The purpose of this paper is to provide an insight and to solve numerically the identification of timewise terms and free boundaries coefficient appearing in the heat equation from over-determination conditions.

Design/methodology/approach

The formulated coefficient identification problem is inverse and ill-posed, and therefore, to obtain a stable solution, a nonlinear Tikhonov regularization least-squares approach is used. For the numerical discretization, the finite difference method combined with a regularized nonlinear minimization is performed using the MATLAB subroutine lsqnonlin.

Findings

The numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.

Research limitations/implications

The mathematical formulation is restricted to identify coefficients in unknown components dependent on time, and this may be considered as a research limitation. However, there is no research implication to overcome this, as the known input data is also limited to single temperature in heat equation with Stefan conditions, and the first- and second-order heat moments measurements at a particular time location.

Practical implications

As noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise.

Social implications

The identification of the timewise terms and free boundaries will be of great interest in the heat transfer community and related fluid flow applications.

Originality/value

The current investigation advances previous studies, which assumed that the coefficient multiplying the lower order temperature term depends on time. The knowledge of this physical property coefficient is very important in heat transfer and fluid flow. The originality lies in the insight gained by performing for the numerical simulations of inversion to find the timewise terms and free boundaries coefficient dependent on time in the heat equation from noisy measurements.

中文翻译：

---

同时识别热方程的时间项和自由边界

目的

本文的目的是提供洞察力，并从数值上解决由超确定条件引起的热方程中出现的时间项和自由边界系数的识别问题。

设计/方法/方法

公式化的系数识别问题是逆的且不适定的，因此，为了获得稳定的解，使用了非线性Tikhonov正则化最小二乘法。对于数值离散化，使用MATLAB子程序lsqnonlin进行了有限差分方法和正则化非线性最小化的组合。

发现

为两个示例提供的数值结果显示了计算方法的效率以及数值解决方案的准确性和稳定性，即使在输入数据中存在噪声的情况下也是如此。

研究局限/意义

数学公式被限制为取决于时间来识别未知分量中的系数，并且这可以被认为是研究限制。但是，没有研究意义可以克服这一问题，因为已知的输入数据也仅限于具有Stefan条件的热方程式中的单个温度，以及在特定时间位置的一阶和二阶热矩测量。

实际影响

由于嘈杂的数据被倒置，因此该研究模拟了实际情况，在这些情况下，实际测量结果固有地被噪声污染。

社会影响

在传热界和相关的流体流动应用中，时间项和自由边界的识别将引起极大的兴趣。

创意/价值

当前的研究推进了先前的研究，这些研究假设系数乘以低阶温度项取决于时间。该物理特性系数的知识在传热和流体流动中非常重要。独创性在于通过进行反演的数值模拟而获得的见解，以便从热测量中找到热方程中随时间变化的时间项和自由边界系数。