Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle
In this paper, we use a hybrid method based on a variant of Trefftz’s method (TM), in combination with the usual Boundary Collocation Method (BCM) to find the approximate solution to a singular, two-dimensional mixed boundary-value problem for Laplace’s equation in a rectangular sheet with one curve...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2012-07-01
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| Series: | Journal of the Egyptian Mathematical Society |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S1110256X12000296 |
| _version_ | 1818292686978809856 |
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| author | A.O. El-Refaie E.K. Rawy H.A.Z. Hassan |
| author_facet | A.O. El-Refaie E.K. Rawy H.A.Z. Hassan |
| author_sort | A.O. El-Refaie |
| collection | DOAJ |
| description | In this paper, we use a hybrid method based on a variant of Trefftz’s method (TM), in combination with the usual Boundary Collocation Method (BCM) to find the approximate solution to a singular, two-dimensional mixed boundary-value problem for Laplace’s equation in a rectangular sheet with one curved side.
After expressing the solution as a finite linear combination of harmonic trial functions, the usual BCM is used to enforce the boundary condition on the curved side, while a variant of TM is applied to the three remaining sides. The singularity at one corner of the rectangle is treated via the enrichment of the expansion with a specially built harmonic function which has a singularity at one corner.
The procedure ultimately produces a rectangular set of linear algebraic equations, which is solved by QR factorization method.
Numerical results are presented and discussed, in order to assess the efficiency of the proposed method. |
| first_indexed | 2024-12-13T03:03:55Z |
| format | Article |
| id | doaj.art-2785be7bf8754c3191187cf8ccfc47a3 |
| institution | Directory Open Access Journal |
| issn | 1110-256X |
| language | English |
| last_indexed | 2024-12-13T03:03:55Z |
| publishDate | 2012-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of the Egyptian Mathematical Society |
| spelling | doaj.art-2785be7bf8754c3191187cf8ccfc47a32022-12-22T00:01:45ZengSpringerOpenJournal of the Egyptian Mathematical Society1110-256X2012-07-01202879110.1016/j.joems.2012.08.014Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangleA.O. El-RefaieE.K. RawyH.A.Z. HassanIn this paper, we use a hybrid method based on a variant of Trefftz’s method (TM), in combination with the usual Boundary Collocation Method (BCM) to find the approximate solution to a singular, two-dimensional mixed boundary-value problem for Laplace’s equation in a rectangular sheet with one curved side. After expressing the solution as a finite linear combination of harmonic trial functions, the usual BCM is used to enforce the boundary condition on the curved side, while a variant of TM is applied to the three remaining sides. The singularity at one corner of the rectangle is treated via the enrichment of the expansion with a specially built harmonic function which has a singularity at one corner. The procedure ultimately produces a rectangular set of linear algebraic equations, which is solved by QR factorization method. Numerical results are presented and discussed, in order to assess the efficiency of the proposed method.http://www.sciencedirect.com/science/article/pii/S1110256X12000296Laplace’s equationMixed boundary-value problemTrefftz’s methodBoundary Collocation MethodCorner singularity |
| spellingShingle | A.O. El-Refaie E.K. Rawy H.A.Z. Hassan Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle Journal of the Egyptian Mathematical Society Laplace’s equation Mixed boundary-value problem Trefftz’s method Boundary Collocation Method Corner singularity |
| title | Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle |
| title_full | Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle |
| title_fullStr | Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle |
| title_full_unstemmed | Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle |
| title_short | Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle |
| title_sort | approximate solution to a singular plane mixed boundary value problem for laplace s equation in a curved rectangle |
| topic | Laplace’s equation Mixed boundary-value problem Trefftz’s method Boundary Collocation Method Corner singularity |
| url | http://www.sciencedirect.com/science/article/pii/S1110256X12000296 |
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