On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps
Background. A scalar three-dimensional boundary value problem of wave diffraction on an inhomogeneous scatterer for the Helmholtz equation with a nonlinear dependence of the scattering wavenumber on the field is considered. Materials and methods. The boundary value problem is reduced to the volume n...
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| Format: | Article |
| Language: | English |
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Penza State University Publishing House
2023-10-01
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| Series: | Известия высших учебных заведений. Поволжский регион: Физико-математические науки |
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| _version_ | 1797672547724034048 |
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| author | Yuriy G. Smirnov Dar'ya A. Labutkina |
| author_facet | Yuriy G. Smirnov Dar'ya A. Labutkina |
| author_sort | Yuriy G. Smirnov |
| collection | DOAJ |
| description | Background. A scalar three-dimensional boundary value problem of wave diffraction on an inhomogeneous scatterer for the Helmholtz equation with a nonlinear dependence of the scattering wavenumber on the field is considered. Materials and methods. The boundary value problem is reduced to the volume nonlinear Lippmann – Schwinger integral equation on the scatterer. The method of contracting maps is used to study the integral equation. Results. The existence and uniqueness of the solution in the space of continuous functions under certain conditions on the parameters of the problem are proved. Conclusions. The convergence of the iterative process in the method of contracting maps is proved and an estimate of the convergence rate is presented. |
| first_indexed | 2024-03-11T21:31:40Z |
| format | Article |
| id | doaj.art-39780eead7ee489eaabd3f6cb0c93181 |
| institution | Directory Open Access Journal |
| issn | 2072-3040 |
| language | English |
| last_indexed | 2024-03-11T21:31:40Z |
| publishDate | 2023-10-01 |
| publisher | Penza State University Publishing House |
| record_format | Article |
| series | Известия высших учебных заведений. Поволжский регион: Физико-математические науки |
| spelling | doaj.art-39780eead7ee489eaabd3f6cb0c931812023-09-27T08:18:22ZengPenza State University Publishing HouseИзвестия высших учебных заведений. Поволжский регион: Физико-математические науки2072-30402023-10-01310.21685/2072-3040-2023-3-1On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps Yuriy G. Smirnov0Dar'ya A. Labutkina1Penza State UniversityPenza State UniversityBackground. A scalar three-dimensional boundary value problem of wave diffraction on an inhomogeneous scatterer for the Helmholtz equation with a nonlinear dependence of the scattering wavenumber on the field is considered. Materials and methods. The boundary value problem is reduced to the volume nonlinear Lippmann – Schwinger integral equation on the scatterer. The method of contracting maps is used to study the integral equation. Results. The existence and uniqueness of the solution in the space of continuous functions under certain conditions on the parameters of the problem are proved. Conclusions. The convergence of the iterative process in the method of contracting maps is proved and an estimate of the convergence rate is presented. helmholtz equationintegral equationsmethod of contracting mapssolvability of boundary value problemnumerical method |
| spellingShingle | Yuriy G. Smirnov Dar'ya A. Labutkina On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps Известия высших учебных заведений. Поволжский регион: Физико-математические науки helmholtz equation integral equations method of contracting maps solvability of boundary value problem numerical method |
| title | On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps |
| title_full | On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps |
| title_fullStr | On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps |
| title_full_unstemmed | On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps |
| title_short | On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps |
| title_sort | on the solution of the nonlinear lippmann schwinger integral equation by the method of contracting maps |
| topic | helmholtz equation integral equations method of contracting maps solvability of boundary value problem numerical method |
| work_keys_str_mv | AT yuriygsmirnov onthesolutionofthenonlinearlippmannschwingerintegralequationbythemethodofcontractingmaps AT daryaalabutkina onthesolutionofthenonlinearlippmannschwingerintegralequationbythemethodofcontractingmaps |