Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle | ||||
| Journal of the Egyptian Mathematical Society | ||||
| Volume 20, Issue 2, 2012, Page 78-91 PDF (506.67 K) | ||||
| DOI: org/10.1016/j.joems.2012.08.014 | ||||
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| Authors | ||||
A. O. El-Refaie; E. K. Rawy  ; H. A. Z. Hassan
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| Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt | ||||
| Abstract | ||||
| 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. | ||||
| Keywords | ||||
| Laplace’s equation; Mixed boundary-value problem; Trefftz’s method; Boundary Collocation Method; Corner singularity | ||||
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