Use of Riemann Mapping Theorem in Solving Dirichlet Problems and Its Application

Ismaila OI, Manjak NH, Kwami AM, Okai J and Hina AD

Published on: 2024-02-29

Abstract

This study started by illustrate the Riemann Mapping Theorem and find also finding a function that is harmonic on the unit disc to the upper half plane and vice-versa using an inverse map with the help of a self-conformal map and we discovered that those self-map and its inverse are both harmonic. We were able to map the sector onto the upper half plane with the help of conformal self map which was transform back the unit disc by taking its inverse. Further, we showed the application of Riemann Mapping Theorem on steady state temperature in a thin infinite plate onto the upper half plane since there are three points of discontinuity on that steady state temperature by using a best conformal map (Mobius transformation) and then removing all points of discontinuity before obtaining our result then transforming back to the original plane since it has the same conformal mapping property.