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.
El-Refaie, A., Rawy, E., & Z. Hassan, H. (2012). Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle. Journal of the Egyptian Mathematical Society, 20(2), 78-91. doi: org/10.1016/j.joems.2012.08.014
MLA
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, 20, 2, 2012, 78-91. doi: org/10.1016/j.joems.2012.08.014
HARVARD
El-Refaie, A., Rawy, E., Z. Hassan, H. (2012). 'Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle', Journal of the Egyptian Mathematical Society, 20(2), pp. 78-91. doi: org/10.1016/j.joems.2012.08.014
VANCOUVER
El-Refaie, A., Rawy, E., Z. Hassan, H. Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle. Journal of the Egyptian Mathematical Society, 2012; 20(2): 78-91. doi: org/10.1016/j.joems.2012.08.014
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