具有Wentzel型边界条件的反源问题解的唯一性

Uniqueness of the Solution to the Inverse Source Problem with a Wentzel Boundary Condition

研究了有界区域内变系数热传导方程的反源问题.与其他模型不同,此模型中的边界条件是Wentzel型边界条件.在关于空间变量积分的附加条件下进行反演,此类附加条件有利于避免单点观测数据下随机性高、误差大的问题.首先,应用变分理论的方法,研究了方程解的正则性;其次,应用H lder不等式、Young不等式以及Friedrichs不等式,证明了仅与时间相关的源项系数的唯一性.结果表明给定的附加条件是充分的,能够反演出未知的源项系数.

In this paper,the inverse source problem of the variable coefficient heat conduction equation in the bounded area is studied.Unlike other models,the boundary conditions in this model are Wentzel type boundary conditions.Inversion is performed under additional conditions regarding the integration of spatial variables,which are beneficial to avoid the problems of high randomness and large errors under single-point observation data.First,the regularity of the solution of the equation is studied by using the method of variational theory;Secondly,the uniqueness of the time-dependent source term coefficients is proved by using Holder inequality,Young inequality,and Friedrichs inequality.The results show that the given additional conditions are sufficient to reverse the unknown source term coefficients.