Reproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equation
The Swift–Hohenberg equation is a nonlinear partial differential equation of fourth order that models the formation and evolution of patterns in a wide range of physical systems. We study the 1D Swift–Hohenberg equation in order to demonstrate the utility of the reproducing kernel method. The soluti...
| প্রধান লেখক: | , , |
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| বিন্যাস: | Journal article |
| প্রকাশিত: |
Elsevier
2018
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| _version_ | 1826292965674319872 |
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| author | Bakhtiari, P Abbasbandy, S Van Gorder, R |
| author_facet | Bakhtiari, P Abbasbandy, S Van Gorder, R |
| author_sort | Bakhtiari, P |
| collection | OXFORD |
| description | The Swift–Hohenberg equation is a nonlinear partial differential equation of fourth order that models the formation and evolution of patterns in a wide range of physical systems. We study the 1D Swift–Hohenberg equation in order to demonstrate the utility of the reproducing kernel method. The solution is represented in the form of a series in the reproducing kernel space, and truncating this series representation we obtain the n-term approximate solution. In the first approach, we aim to explain how to construct a reproducing kernel method without using Gram-Schmidt orthogonalization, as orthogonalization is computationally expensive. This approach will therefore be most practical for obtaining numerical solutions. Gram-Schmidt orthogonalization is later applied in the second approach, despite the increased computational time, as this approach will prove theoretically useful when we perform a formal convergence analysis of the reproducing kernel method for the Swift–Hohenberg equation. We demonstrate the applicability of the method through various test problems for a variety of initial data and parameter values. |
| first_indexed | 2024-03-07T03:22:49Z |
| format | Journal article |
| id | oxford-uuid:b8022cb8-736b-44bb-ac32-28ca4e63d39f |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:22:49Z |
| publishDate | 2018 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:b8022cb8-736b-44bb-ac32-28ca4e63d39f2022-03-27T04:52:51ZReproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b8022cb8-736b-44bb-ac32-28ca4e63d39fSymplectic Elements at OxfordElsevier2018Bakhtiari, PAbbasbandy, SVan Gorder, RThe Swift–Hohenberg equation is a nonlinear partial differential equation of fourth order that models the formation and evolution of patterns in a wide range of physical systems. We study the 1D Swift–Hohenberg equation in order to demonstrate the utility of the reproducing kernel method. The solution is represented in the form of a series in the reproducing kernel space, and truncating this series representation we obtain the n-term approximate solution. In the first approach, we aim to explain how to construct a reproducing kernel method without using Gram-Schmidt orthogonalization, as orthogonalization is computationally expensive. This approach will therefore be most practical for obtaining numerical solutions. Gram-Schmidt orthogonalization is later applied in the second approach, despite the increased computational time, as this approach will prove theoretically useful when we perform a formal convergence analysis of the reproducing kernel method for the Swift–Hohenberg equation. We demonstrate the applicability of the method through various test problems for a variety of initial data and parameter values. |
| spellingShingle | Bakhtiari, P Abbasbandy, S Van Gorder, R Reproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equation |
| title | Reproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equation |
| title_full | Reproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equation |
| title_fullStr | Reproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equation |
| title_full_unstemmed | Reproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equation |
| title_short | Reproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equation |
| title_sort | reproducing kernel method for the numerical solution of the 1d swift hohenberg equation |
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