Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals

The retrospective inverse problem for evolution equations is formulated as the reconstruction of unknown initial data by a given solution at the final time. We consider the inverse retrospective problem for a one-dimensional parabolic equation in two disconnected intervals with weak solutions in wei...

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Bibliographic Details
Published in:Computation Vol. 11; no. 10; p. 204
Main Authors: Koleva, Miglena N., Vulkov, Lubin G.
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.10.2023
Subjects:
Boundary conditions
Boundary value problems
Conjugate gradient method
Difference equations
Differential equations
Discretization
finite-difference scheme
Heat
Ill posed problems
Intervals
Inverse problems
iterative conjugate gradient method
Iterative methods
Numerical analysis
parabolic equation on partitioned domain
Partial differential equations
Regularization methods
retrospective inverse problem
Sobolev space
Temperature
weighted Sobolev spaces
Germany
ISSN:2079-3197, 2079-3197
Online Access:Get full text
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Summary:The retrospective inverse problem for evolution equations is formulated as the reconstruction of unknown initial data by a given solution at the final time. We consider the inverse retrospective problem for a one-dimensional parabolic equation in two disconnected intervals with weak solutions in weighted Sobolev spaces. The two solutions are connected with nonstandard interface conditions, and thus this problem is solved in the whole spatial region. Such a problem, as with other inverse problems, is ill-posed, and for its numerical solution, specific techniques have to be used. The direct problem is first discretized by a difference scheme which provides a second order of approximation in space. For the resulting ordinary differential equation system, the positive coerciveness is established. Next, we develop an iterative conjugate gradient method to solve the ill-posed systems of the difference equations, which are obtained after weighted time discretization, of the inverse problem. Test examples with noisy input data are discussed.
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ISSN:2079-3197
2079-3197
DOI:10.3390/computation11100204