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Register a new userSimultaneous determination of coefficients, internal sources and an obstacle of a diffusion equation from a single measurement
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Yavar Kian
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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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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.
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 deals with the uniqueness in identifying multiple parameters simultaneously in the one-dimensional time-fractional diffusion-wave equation of fractional time-derivative order $in (0,2)$ with the zero Robin boundary condition. Using the Laplace transform and a transformation formula, we prove the uniqueness in determining an order of the fractional derivative, a spatially varying potential, initial values and Robin coefficients simultaneously by boundary measurement data, provided that all the eigenmodes of an initial value do not vanish. Furthermore, for another formulation of inverse problem with input source term in place of initial value, by the uniqueness in the case of non-zero initial value and a Duhamel principle, we prove the simultaneous uniqueness in determining multiple parameters for a time-fractional diffusion-wave equation.
We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form $mathbb{R}^Nsetminus K$, where $Ksubsetmathbb{R}^N$ is a compact obstacle. The model we study is motivated by applications in biology and takes into account long range dispersal events that may be anisotropic depending on how a given population perceives the environment. To formulate this in a meaningful manner, we introduce a new theoretical framework which is of both mathematical and biological interest. The main goal of this paper is to construct an entire solution that behaves like a planar travelling wave as $tto-infty$ and to study how this solution propagates depending on the shape of the obstacle. We show that whether the solution recovers the shape of a planar front in the large time limit is equivalent to whether a certain Liouville type property is satisfied. We study the validity of this Liouville type property and we extend some previous results of Hamel, Valdinoci and the authors. Lastly, we show that the entire solution is a generalised transition front.
Let $0le u_0(x)in L^1(R^2)cap L^{infty}(R^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|ge r_1$ and is monotone decreasing for all $|x|ge r_1$ for some constant $r_1>0$ and ${ess}inf_{2{B}_{r_1}(0)}u_0ge{ess} sup_{R^2setminus B_{r_2}(0)}u_0$ for some constant $r_2>r_1$. Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum cite{DP2}, cite{DS}, and prove the collapsing behaviour of the maximal solution of the equation $u_t=Deltalog u$ in $R^2times (0,T)$, $u(x,0)=u_0(x)$ in $R^2$, near its extinction time $T=int_{R^2}u_0dx/4pi$.
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