The Inverse Source Problem for a Class of Stochastic Convection-Diffusion Equations
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摘要:
考虑了一类由分数阶Brown运动驱动的随机对流扩散方程的源项反演问题。正问题部分首先利用分离变量法，得出了方程的温和解，进一步在期望的意义下，讨论了正问题的适定性。反问题部分研究了由终止时刻的随机数据来反演随机源项的部分统计量，并证明了相应的唯一性和不稳定性。最后进行了一些数值模拟，验证了相应的理论结果。
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关键词:
- 随机对流扩散方程 /
- 反源问题 /
- 分数阶Brown运动 /
- 唯一性 /
- 不适定性
Abstract:The inverse source problem for a class of stochastic convection-diffusion equations driven by the fractional Brownian motion with the Hurst index, was considered. The direct problem is to study the solution to the stochastic convection-diffusion equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. The direct problem is well-posed. The uniqueness and instability of the inverse source problem was proved. Some numerical simulation examples verify the theoretical analysis.
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图 1
$ H=0.9, \delta=0.04 $ 时$ f $ 和$ \sigma^2 $ 的相对误差Figure 1. The relative errors of the reconstruction for
$ f\, $ and$ \sigma^2\, $ with respect to$ H=0.9 $ and$ \delta=0.04 $ 
图 2
$ H=0.4, N=6 $ 时的$ f $ 和$ \sigma^2 $ Figure 2. The reconstruction of
$ f\,$ and$ \sigma^2\, $ for the inverse problem with$ H=0.4 $ and$ N=6 $ 
图 3
$ H=0.5, N=6 $ 时的$ f $ 和$ \sigma^2 $ Figure 3. The reconstruction of
$ f\,$ and$ \sigma^2\, $ for the inverse problem with$ H=0.5 $ and$ N=6 $ -
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