Direct and inverse source problems for degenerate parabolic equations

Direct and inverse source problems for degenerate parabolic equations AbstractDegenerate 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Inverse and III-posed Problems de Gruyter

Direct and inverse source problems for degenerate parabolic equations

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Publisher
de Gruyter
Copyright
© 2020 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1569-3945
eISSN
1569-3945
DOI
10.1515/jiip-2019-0046
Publisher site
See Article on Publisher Site

Abstract

AbstractDegenerate 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.

Journal

Journal of Inverse and III-posed Problemsde Gruyter

Published: Jun 1, 2020

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