Source Identification for a Two-Dimensional Parabolic Equation with an Integral Constraint

Miglena N. Koleva () and Lubin G. Vulkov
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Miglena N. Koleva: Department of Mathematics, Faculty of Natural Sciences and Education, “Angel Kanchev” University of Ruse, 8 Studentska Str., 7017 Ruse, Bulgaria
Lubin G. Vulkov: Department of Applied Mathematics and Statistics, Faculty of Natural Sciences and Education, “Angel Kanchev” University of Ruse, 8 Studentska Str., 7017 Ruse, Bulgaria

Mathematics, 2025, vol. 13, issue 11, 1-19

Abstract: We consider a two-dimensional parabolic problem subject to both Neumann and Dirichlet boundary conditions, along with an integral constraint. Based on the integral observation, we solve the inverse problem of a recovering time-dependent right-hand side. By exploiting the structure of the boundary conditions, we reduce the original inverse problem to a one-dimensional formulation. We conduct a detailed analysis of the existence and uniqueness of the solution to the resulting one-dimensional loaded initial-boundary value problem. Furthermore, we derive estimates for both the solution and the unknown function. The direct and inverse problems are numerically solved by finite difference schemes. Numerical verification of the theoretical results is provided.

Keywords: parabolic equation; inverse problem; source identification; integral measurement; loaded equation; existence and uniqueness; stability estimates (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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