Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation
Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation
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DOI：
10.1088/1751-8121/aa763a
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发表时间：
2017-04
期刊：
影响因子：
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通讯作者：
Yikan Liu;Zhidong Zhang
中科院分区：
文献类型：
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作者：
Yikan Liu;Zhidong Zhang
本文考虑（时间分数阶）扩散方程（α tα−△）u（x，t）=ρ（t）g（x）（0<α <$1）中ρ（t）的重构问题.我们主要关注x 0 suppg的情况，它具有实际重要性，但文献中尚未得到很好的研究。假设ρ的符号变化有限，并有一个额外的观测区间，基于一个反卷积不等式和正解的一个下估计，建立了该问题的多重对数稳定性.同时，我们发展了一个不动点迭代的数值重建，并证明了其收敛性。所提出的方法的性能说明了几个数值例子。
In this article, we consider the reconstruction of ρ(t) in the (time-fractional) diffusion equation (∂tα−△)u(x,t)=ρ(t)g(x) (0<α⩽1) by observation at a single point x0. We are mainly concerned with the situation of x0∉suppg, which is practically important but has not been well investigated in literature. Assuming finite sign changes of ρ and an extra observation interval, we establish the multiple logarithmic stability for the problem based on a reverse convolution inequality and a lower estimate for positive solutions. Meanwhile, we develop a fixed-point iteration for the numerical reconstruction and prove its convergence. The performance of the proposed method is illustrated by several numerical examples.