带奇异逆平方势的时间分数阶扩散波方程的逆源问题
Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential
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中文总结 AI 辅助
本文针对带奇异逆平方势的时间分数阶扩散波方程逆源问题，利用Hardy型不等式等建立正问题适定性，引入Tikhonov正则化结合伴随共轭梯度法实现源的数值重构，经数值实验验证方法有效稳定。
中文摘要 AI 辅助
本文研究带奇异逆平方势的时间分数阶扩散波方程的逆源问题。假设源项由已知的时间因子和待恢复的未知空间分量组成，需从终端状态测量中恢复该未知空间分量。通过利用Hardy型不等式及相关奇异椭圆算子的谱性质，在合适的能量框架内建立了正问题的适定性与正则性。随后证明终端观测算子是紧的，且在时间因子满足合适的非退化条件下，空间源具有唯一性。为稳定所得的不适定逆问题，引入Tikhonov正则化方法；通过含右侧分数阶导数的伴随问题推导正则化泛函的梯度，进而得到基于伴随的共轭梯度法，并结合精确线搜索实现未知源的数值重构。在一维和二维空间域上分别使用精确及含噪声的终端数据开展数值实验，以验证所提源重构方法的有效性与稳定性。
英文摘要
This paper investigates an inverse source problem for a time-fractional diffusion-wave equation with a singular inverse-square potential. The source term is assumed to consist of a known temporal factor and an unknown spatial component, which is to be recovered from terminal-state measurements. The well-posedness and regularity of the forward problem are established within an appropriate energy framework by exploiting Hardy-type inequalities and the spectral properties of the associated singular elliptic operator. The terminal observation operator is then shown to be compact, and uniqueness of the spatial source is established under a suitable nondegeneracy condition on the temporal factor. To stabilize the resulting ill-posed inverse problem, a Tikhonov regularization approach is introduced. The gradient of the regularized functional is derived through an adjoint problem involving a right-sided fractional derivative, leading to an adjoint-based conjugate gradient method with an exact line search for the numerical reconstruction of the unknown source. Numerical experiments are conducted on both one-and two-dimensional spatial domains, using both exact and noisy terminal data, to demonstrate the effectiveness and stability of the proposed source reconstruction method.
