A mixed initial boundary value problem for the heat conduction equation is investigated and solved. The problem statement includes three intervals: the first one (0 → T₁) describes heating of an internal combustion chamber and the second one (T₁ → T₂), the chamber cooling and slower cooling of its wall. The third interval describes natural cooling of the chamber wall when the chamber temperature coincides with that of the environment. The applicability of the Fourier transform in time to this problem is proved. This makes it possible to transform the governing equation to an ordinary differential equation. By using the resulting equation, an inverse boundary value problem for the heat conduction equation is solved by a nonlinear method of projection regularization, and an error estimate of the approximate solution is obtained.

中文翻译：

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导热方程反边值问题的研究

研究并求解了热传导方程的混合初边值问题。问题陈述包括三个间隔：第一个间隔（0→ T ₁）描述了内燃机的加热，第二个间隔（T ₁ → T ₂），腔室的冷却和壁的冷却较慢。第三间隔描述了当腔室温度与环境温度一致时腔室壁的自然冷却。证明了傅里叶变换及时适用于该问题。这使得可以将控制方程式转换为常微分方程式。通过使用所得方程，通过非线性投影正则化方法解决了导热方程的逆边值问题，并获得了近似解的误差估计。