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Register a new userUnique determination of several coefficients in a fractional diffusion(-wave) equation by a single measurement
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We consider the inverse problem of determining different type of information about a diffusion process, described by ordinary or fractional diffusion equations stated on a bounded domain, like the density of the medium or the velocity field associated with the moving quantities from a single boundary measurement. This properties will be associated with some general class of time independent coefficients that we recover from a single Neumann boundary measurement, on some parts of the boundary, of the solution of our diffusion equation with a suitable boundary input, located on some parts of the boundary.
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In this article, for a two dimensional fractional diffusion equation, we study an inverse problem for simultaneous restoration of the fractional order and the source term from the sparse boundary measurements. By the adjoint system corresponding to our diffusion equation, we construct useful quantitative relation between unknowns and measurements. From Laplace transform and the knowledge in complex analysis, the uniqueness theorem is proved.
This article is devoted to the simultaneous resolution of three inverse problems, among the most important formulation of inverse problems for partial differential equations, stated for some class of diffusion equations from a single boundary measurement. Namely, we consider the simultaneous unique determination of several class of coefficients, some internal sources (a source term and an initial condition) and an obstacle appearing in a diffusion equation from a single boundary measurement. Our problem can be formulated as the simultaneous determination of information about a diffusion process (velocity field, density of the medium), an obstacle and of the source of diffusion. We consider this problems in the context of a classical diffusion process described by a convection-diffusion equation as well as an anomalous diffusion phenomena described by a time fractional diffusion equation.
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This paper is concerned with the inverse problem on determining an orbit of the moving source in a fractional diffusion(-wave) equations in a connected bounded domain of $mathbb R^d$ or in the whole space $mathbb R^d$. Based on a newly established fractional Duhamels principle, we derive a Lipschitz stability estimate in the case of a localized moving source by the observation data at $d$ interior points. The uniqueness for the general non-localized moving source is verified with additional data of more interior observations.
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