基于控制的分数阶热方程空间源重构
A control-based spatial source reconstruction in fractional heat equations
AI总结：
针对分数阶热方程的反源问题，利用零能控性及谱分析推导傅里叶系数重构公式，从部分观测数据恢复空间源，数值实验验证了方法的准确性与稳定性。
摘要：
本文研究了涉及分数阶拉普拉斯算子的非局部热方程的反源问题。主要目标是从系统状态及其时间导数在域的一个子集上的部分观测中重构源项的空间分量。利用分数阶热方程在分数阶s∈(1/2,1)区间内的零能控性，推导了未知源傅里叶系数的重构公式。该方法基于谱分析和Volterra积分方程，为在有限测量数据下恢复空间源提供了稳健框架。数值实验证实了所提方法的准确性和稳定性。
英文摘要：
This article addresses the inverse source problem for a nonlocal heat equation involving the fractional Laplacian. The primary goal is to reconstruct the spatial component of the source term from partial observations of the system's state and its time derivative over a subset of the domain. A reconstruction formula for the Fourier coefficients of the unknown source is derived, leveraging the null controllability property of the fractional heat equation when the fractional order lies in the interval $s\in(1/2,1)$. The methodology builds on spectral analysis and Volterra integral equations, providing a robust framework for recovering spatial sources under limited measurement data. Numerical experiments confirm the accuracy and stability of the proposed approach.
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