Abstract In the article, we solve the inverse problems to recover unknown space-time dependent functions of heat conductivity and heat source for a nonlinear convective-diffusive equation, without needing of initial temperature, final time temperature, and internal temperature data. After adopting a homogenization technique, a set of spatial boundary functions are derived, which satisfy the homogeneous boundary conditions. The homogeneous boundary functions and zero element constitute a linear space, and then a new energetic functional is derived in the linear space, which preserves the time-dependent energy. The linear systems and iterative algorithms to recover the unknown parameters with energetic boundary functions as the bases are developed, which are convergent fast at each time marching step. The data required for the recovery of unknown functions are parsimonious, including the boundary data of temperatures and heat fluxes and the boundary data of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the exact solutions with the identified results, which are obtained under large noisy disturbance.

中文翻译：

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通过能量边界函数方法识别非线性对流扩散方程的热导率和源函数

摘要 在本文中，我们解决了非线性对流扩散方程的热导率和热源的未知时空相关函数的反问题，而无需初始温度、最终时间温度和内部温度数据。采用齐次化技术后，推导出一组满足齐次边界条件的空间边界函数。齐次边界函数和零元构成一个线性空间，然后在线性空间中推导出一个新的能量函数，它保留了与时间相关的能量。开发了以能量边界函数为基础的线性系统和迭代算法来恢复未知参数，它们在每个时间行进步骤中都快速收敛。恢复未知函数所需的数据是简约的，包括温度和热通量的边界数据和待恢复的未知函数的边界数据。通过将精确解与在大噪声干扰下获得的识别结果进行比较，证实了当前方法的准确性和鲁棒性。