Inverse problems for heat equation and space-time fractional diffusion equation with one measurement

2019/03/11 by Tapio Helin, Matti Lassas, Helin, Tapio +5 · 4 citations
Mathematics · Computer Science · #Fractional Differential Equations Solutions #Numerical methods in inverse problems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1903.04348

Abstract

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.

Citations

• Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen

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• Numerical identification of the time-dependent coefficient in the heat equation with fractional Laplacian
• Anisotropic Calderón problem for a logarithmic Schrödinger operator of order 2+ on closed Riemannian manifolds
• A control-based spatial source reconstruction in fractional heat equations
• A Hyperbolic Inverse Problem for lower order terms on a closed manifold with disjoint data

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