Direct and inverse source problems for degenerate parabolic equations
-
M. S. Hussein
,
Vitaly L. Kamynin
und
Andrey B. Kostin
Abstract
Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problems for which sufficient conditions guaranteeing the unique solvability are fulfilled, and present the results of numerical simulations. It is hoped that the analysis initiated in this study will open up new avenues for research in the field of direct and inverse problems for degenerate parabolic equations with applications.
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: EP/P005985
Funding statement: V. L. Kamynin and A. B. Kostin were partially supported by the Programme of Competitiveness Increase of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute); contract no 02.a03.21.0005, 27.08.2013. D. Lesnic would like to acknowledge some small financial support received from the EPSRC funded research network on inverse problems EP/P005985/1 for a week research visit of Professor V. L. Kamynin to Leeds in July 2017.
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© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Identification of the interface between acoustic and elastic waves from internal measurements
- A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem
- Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues
- The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data
- Identifying space-time dependent force on the vibrating Euler–Bernoulli beam by a boundary functional method
- A stability result for the determination of order in time-fractional diffusion equations
- Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system
- An inverse problem of triple-thickness parameters determination for thermal protective clothing with Stephan–Boltzmann interface conditions
- Direct and inverse source problems for degenerate parabolic equations
- Partial inverse problems for quadratic differential pencils on a graph with a loop
Artikel in diesem Heft
- Frontmatter
- Identification of the interface between acoustic and elastic waves from internal measurements
- A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem
- Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues
- The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data
- Identifying space-time dependent force on the vibrating Euler–Bernoulli beam by a boundary functional method
- A stability result for the determination of order in time-fractional diffusion equations
- Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system
- An inverse problem of triple-thickness parameters determination for thermal protective clothing with Stephan–Boltzmann interface conditions
- Direct and inverse source problems for degenerate parabolic equations
- Partial inverse problems for quadratic differential pencils on a graph with a loop
