Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior Degeneracy
A simultaneous reconstruction of the initial condition and the space-dependent reaction coefficient in a multidimensional hyperbolic partial differential equation with interior degeneracy is of concern. A temporal integral observation is utilized to achieve that purpose. The well-posedness, existenc...
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| Format: | Article |
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MDPI AG
2023-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/14/3186 |
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| author | Hamed Ould Sidi Mahmoud A. Zaky Rob H. De Staelen Ahmed S. Hendy |
| author_facet | Hamed Ould Sidi Mahmoud A. Zaky Rob H. De Staelen Ahmed S. Hendy |
| author_sort | Hamed Ould Sidi |
| collection | DOAJ |
| description | A simultaneous reconstruction of the initial condition and the space-dependent reaction coefficient in a multidimensional hyperbolic partial differential equation with interior degeneracy is of concern. A temporal integral observation is utilized to achieve that purpose. The well-posedness, existence, and uniqueness of the inverse problem under consideration are discussed. The inverse problem can be reformulated as a least squares minimization and the Fréchet gradients are determined, using the adjoint and sensitivity problems. Finally, an iterative construction procedure is developed by employing the conjugate gradient algorithm while invoking the discrepancy principle as a stopping criterion. Some numerical experiments are given to ensure the performance of the reconstruction scheme in one and two dimensions. |
| first_indexed | 2024-03-11T00:52:03Z |
| format | Article |
| id | doaj.art-b361ca1198c2485eb634e821ae0faf48 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T00:52:03Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b361ca1198c2485eb634e821ae0faf482023-11-18T20:21:53ZengMDPI AGMathematics2227-73902023-07-011114318610.3390/math11143186Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior DegeneracyHamed Ould Sidi0Mahmoud A. Zaky1Rob H. De Staelen2Ahmed S. Hendy3Department of Mathematics, Faculty of Sciences, University of Nouakchott Al Aasriya, Nouakchott Bp 6093, MauritaniaDepartment of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi ArabiaBeheer en Algemene Directie, Ghent University Hospital, Corneel Heymanslaan 10, B-9000 Ghent, BelgiumDepartment of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, RussiaA simultaneous reconstruction of the initial condition and the space-dependent reaction coefficient in a multidimensional hyperbolic partial differential equation with interior degeneracy is of concern. A temporal integral observation is utilized to achieve that purpose. The well-posedness, existence, and uniqueness of the inverse problem under consideration are discussed. The inverse problem can be reformulated as a least squares minimization and the Fréchet gradients are determined, using the adjoint and sensitivity problems. Finally, an iterative construction procedure is developed by employing the conjugate gradient algorithm while invoking the discrepancy principle as a stopping criterion. Some numerical experiments are given to ensure the performance of the reconstruction scheme in one and two dimensions.https://www.mdpi.com/2227-7390/11/14/3186wave probleminterior degeneracycoefficient and initial function reconstructionCGM |
| spellingShingle | Hamed Ould Sidi Mahmoud A. Zaky Rob H. De Staelen Ahmed S. Hendy Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior Degeneracy Mathematics wave problem interior degeneracy coefficient and initial function reconstruction CGM |
| title | Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior Degeneracy |
| title_full | Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior Degeneracy |
| title_fullStr | Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior Degeneracy |
| title_full_unstemmed | Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior Degeneracy |
| title_short | Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior Degeneracy |
| title_sort | numerical reconstruction of a space dependent reaction coefficient and initial condition for a multidimensional wave equation with interior degeneracy |
| topic | wave problem interior degeneracy coefficient and initial function reconstruction CGM |
| url | https://www.mdpi.com/2227-7390/11/14/3186 |
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