Open AccessApproximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle
Author(s) -
A. O. El-Refaie,
E. K. Rawy,
Heba Hassan
Publication year - 2012
Publication title -
journal of the egyptian mathematical society
Language(s) - English
Resource type - Journals
eISSN - 2090-9128
pISSN - 1110-256X
DOI - 10.1016/j.joems.2012.08.014
Subject(s) - mathematics , singularity , mathematical analysis , rectangle , laplace's equation , boundary value problem , boundary (topology) , algebraic equation , laplace transform , singular boundary method , geometry , nonlinear system , finite element method , physics , quantum mechanics , boundary element method , thermodynamics
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