Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation
In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynam...
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| Format: | Article |
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MDPI AG
2024-01-01
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| Online Access: | https://www.mdpi.com/2075-1680/13/1/64 |
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| author | Miglena N. Koleva Lubin G. Vulkov |
| author_facet | Miglena N. Koleva Lubin G. Vulkov |
| author_sort | Miglena N. Koleva |
| collection | DOAJ |
| description | In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green’s second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations. |
| first_indexed | 2024-03-08T11:05:44Z |
| format | Article |
| id | doaj.art-caf959d1caad4bd3bdd20633c58fed86 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-08T11:05:44Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj.art-caf959d1caad4bd3bdd20633c58fed862024-01-26T15:04:07ZengMDPI AGAxioms2075-16802024-01-011316410.3390/axioms13010064Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s EquationMiglena N. Koleva0Lubin G. Vulkov1Department of Mathematics, Faculty of Natural Sciences and Education, University of Ruse “Angel Kanchev”, 8 Studentska Str., 7017 Ruse, BulgariaDepartment of Applied Mathematics and Statistics, Faculty of Natural Sciences and Education, University of Ruse “Angel Kanchev”, 8 Studentska Str., 7017 Ruse, BulgariaIn this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green’s second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations.https://www.mdpi.com/2075-1680/13/1/64Laplace’s equationdynamical boundary conditionsinverse problemsGreen’s identitymeshless methodconjugate gradient method |
| spellingShingle | Miglena N. Koleva Lubin G. Vulkov Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation Axioms Laplace’s equation dynamical boundary conditions inverse problems Green’s identity meshless method conjugate gradient method |
| title | Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation |
| title_full | Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation |
| title_fullStr | Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation |
| title_full_unstemmed | Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation |
| title_short | Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation |
| title_sort | numerical reconstruction of the source in dynamical boundary condition of laplace s equation |
| topic | Laplace’s equation dynamical boundary conditions inverse problems Green’s identity meshless method conjugate gradient method |
| url | https://www.mdpi.com/2075-1680/13/1/64 |
| work_keys_str_mv | AT miglenankoleva numericalreconstructionofthesourceindynamicalboundaryconditionoflaplacesequation AT lubingvulkov numericalreconstructionofthesourceindynamicalboundaryconditionoflaplacesequation |