Inverse problems for heat equation and space–time fractional diffusion equation with one measurement
Inverse problems for heat equation and space–time fractional diffusion equation with one measurement
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DOI：
10.1016/j.jde.2020.05.022
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发表时间：
2019-03
影响因子：
2.4
通讯作者：
T. Helin;M. Lassas;L. Ylinen;Zhidong Zhang
中科院分区：
文献类型：
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作者：
T. Helin;M. Lassas;L. Ylinen;Zhidong Zhang
给定一个无边界的连通紧致黎曼流形 (M, g)，dim M≥ 2，我们考虑一个时空分数扩散方程，其内部源由流形的开子集 V 支持。方程的时间分数部分由阶次为 αε(0, 1] 的 Caputo 导数给出，空间分数部分由 (− Δ g) β 给出，其中 βε(0, 1] 和 Δ g 是流形上的 Laplace–Beltrami 算子。α= β= 1 的情况对应于流形上的标准热方程，是一个重要的特例。我们构建了一个特定的源，以便测量V 确定黎曼等距的流形。
Given a connected compact Riemannian manifold (M, g) without boundary, dim M≥ 2, we consider a space–time fractional diffusion equation with an interior source that is supported on an open subset V of the manifold. The time-fractional part of the equation is given by the Caputo derivative of order α∈(0, 1], and the space fractional part by (− Δ g) β, where β∈(0, 1] and Δ g is the Laplace–Beltrami operator on the manifold. The case α= β= 1, which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on V determines the manifold up to a Riemannian isometry.