The objective of this article is to introduce a novel technique for computing numerical solutions to the nonlinear inverse heat conduction problem. This involves solving nonlinear parabolic equations with Cauchy data provided on one side $\Gamma$ of the boundary of the computational domain $\Omega$. The key step of our proposed method is the truncation of the Fourier series of the solution to the governing equation. The truncation technique enables us to derive a system of 1D ordinary differential equations. Then, we employ the well-known Runge-Kutta method to solve this system, which aids in addressing the nonlinearity and the lack of data on $\partial \Omega \setmunus \Gamma$. This new approach is called the dimensional reduction method. By converting the high-dimensional problem into a 1D problem, we achieve exceptional computational speed. Numerical results are provided to support the effectiveness of our approach.

翻译：本文旨在介绍一种求解非线性逆热传导问题数值解的新技术。该方法涉及求解带有柯西数据的非线性抛物方程，其中数据仅在计算域 $\Omega$ 的边界一侧 $\Gamma$ 上提供。我们提出的方法的关键步骤是对控制方程的解的傅里叶级数进行截断。截断技术使我们能够推导出一个一维常微分方程组。然后，我们采用经典的龙格-库塔方法求解该方程组，这有助于处理非线性以及 $\partial \Omega \setminus \Gamma$ 上数据缺失的问题。这种新方法被称为降维方法。通过将高维问题转化为一维问题，我们实现了极高的计算速度。文中提供了数值结果以支持该方法的有效性。