带记忆项的热传导方程反源问题
Inverse Source Problem for Heat Conduction Equation with Memory Term
摘要: 本文研究含记忆项热传导过程中时间相关未知源项的反演问题，反演输入数据为空间内部一点数 据。 针对该问题，我们建立了关于重建未知源项的Tikhonov正则化泛函的迭代算法。 在数值实现 上，将反问题转化为最优化问题，证明了目标泛函关于源项是Freche´t可导的。 通过构造伴随问题 可以得到泛函关于源项的梯度，采用共轭梯度法(CGM)反演源项。 最后，通过数值算例验证了所 提方法的有效性。
Abstract: In this work, we study the inverse problem of retrieving the time-dependent unknown source term in heat conduction processes with memory term, where the input data for inversion is the data at an internal point in space. For this problem, we establish an iterative algorithm based on the Tikhonov regularization functional for reconstructing the unknown source term. In numerical implementation, the inverse problem is trans- formed into an optimization problem, and it is proved that the objective functional is Freche´t differentiable with respect to the source term. The gradient of the functional with respect to the source term can be obtained by constructing an adjoint problem, and the conjugate gradient method (CGM) is used to invert the source term. Finally, numerical examples verify the effectiveness of the proposed method.
文章引用：张亚欣, 张萌萌. 带记忆项的热传导方程反源问题[J]. 应用数学进展, 2025, 14(11): 233-247. https://doi.org/10.12677/AAM.2025.1411479

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