The uniqueness of inverse problems for a fractional equation with a single measurement

This article is concerned with an inverse problem on simultaneously determining some unknown coefficients and/or an order of derivative in a multidimensional time-fractional evolution equation either in a Euclidean domain or on a Riemannian manifold. Based on a special choice of the Dirichlet boundary input, we prove the unique recovery of at most two out of four x-dependent coefficients (possibly with an extra unknown fractional order) by a single measurement of the partial Neumann boundary output. Especially, both a vector-valued velocity field of a convection term and a density can also be uniquely determined. The key ingredient turns out to be the time-analyticity of the decomposed solution, which enables the construction of Dirichlet-to-Neumann maps in the frequency domain and thus the application of inverse spectral results. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

Kian Y.1 , Li Z.2 , Liu Y.3 , Yamamoto M. 4, 5, 6
  • 1 Aix-Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France
  • 2 School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong, 255049, China
  • 3 Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, N12W7, Kita-Ward, Sapporo, 060-0812, Japan
  • 4 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
  • 5 Honorary Member of Academy of Romanian Scientists, Splaiul Independentei Street, No. 54, Bucharest, 050094, Romania
  • 6 Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow, 117198, Russian Federation
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