Purpose

The purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.

Design/methodology

For the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.

Findings

The numerical results demonstrate that accurate and stable solutions are obtained.

Originality/value

The inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.

中文翻译：

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二维自由边界问题中瞬态对流系数的确定

目的

研究的目的是数值求解确定瞬态对流系数和自由边界的反问题，以及二维对流扩散方程中的温度，初始条件和边界条件辅以非局部积分观测。 . 从文献中已经知道，这个逆问题有一个唯一的解。然而，由于对输入数据中的噪声不稳定，问题仍然存在。

设计/方法论

对于数值离散化，本文采用交替方向显式有限差分法和 Tikhonov 正则化方法，寻找稳定准确的数值解。使用 MATLAB 例程 lsqnonlin 以计算方式解决由此产生的非线性最小化问题。精确的和数值模拟的噪声输入数据都被反转。

发现

数值结果表明得到了准确和稳定的解。

原创性/价值

本文提出的逆问题已经被证明是局部唯一可解的，但目前还没有实现数值解；因此，这项工作的主要独创性是尝试这项任务。