On the potential in a strip with a pair of U notches. (English) Zbl 0595.31004
In \({\mathbb{R}}^ 2\), consider the strip \(y=0\), \(y=2a\) with a pair of U notches. The upper U notch is of the same size as the lower one and located symmetrically with respect to the line \(y=a\). The paper deals with the problem of finding a function in \({\mathbb{R}}^ 2\) which gives potential on the lower boundary as 1 and that on the upper boundary as - 1.
In an earlier paper [Q. Appl. Math. 41, 369-377 (1983; Zbl 0557.31002)] the author showed that the solution could be written as a linear combination of a set of harmonic functions up to additive linear terms. In this paper, he gives an alternate proof, deriving the set of harmonic functions from a single function (and not from an integral as earlier) and adjusting the boundary conditions on the curves without the use of conformal transformation. Some numerical examples are also given.
In an earlier paper [Q. Appl. Math. 41, 369-377 (1983; Zbl 0557.31002)] the author showed that the solution could be written as a linear combination of a set of harmonic functions up to additive linear terms. In this paper, he gives an alternate proof, deriving the set of harmonic functions from a single function (and not from an integral as earlier) and adjusting the boundary conditions on the curves without the use of conformal transformation. Some numerical examples are also given.
Reviewer: V.Anandam
MSC:
31A25 | Boundary value and inverse problems for harmonic functions in two dimensions |
31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |
31C20 | Discrete potential theory |