Riemann Mapping Theorem as a Conformal mapping Methods in solving Dirichlet Boundary Value Problems and its’ Application

I O Ismaila¹ , N H Manjak² , A D Hina³

Section:Research Paper, Product Type: Journal-Paper
Vol.11 , Issue.3 , pp.92-96, Jun-2024

Online published on Jun 30, 2024

Copyright © I O Ismaila, N H Manjak, A D Hina . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
This study started by illustating 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 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.

Key-Words / Index Term :
conformal map, self map, upper half plane, unit disc, Riemann map, dirichlet problem, boundary value problem

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