A transform pair for bounded convex planar domains
Complex Variables
2023-12-04 v1
Abstract
A new transform pair which can be used to solve mixed boundary value problems for Laplace's equation and the complex Helmholtz equation in bounded convex planar domains is presented. This work is an extension of Crowdy (2015, CMFT, 15, 655--687) where new transform techniques were developed for boundary value problems for Laplace's equation in circular domains. The key ingredient of the method is the analysis of the so called global relation which provides a coupling of integral transforms of the given boundary data and of the unknown boundary values. Three problems which involve mixed boundary conditions are solved in detail, as well as numerically implemented, to illustrate how to apply the new approach.
Cite
@article{arxiv.2312.00668,
title = {A transform pair for bounded convex planar domains},
author = {Jesse Hulse and Loredana Lanzani and Stefan Llewellyn Smith and Elena Luca},
journal= {arXiv preprint arXiv:2312.00668},
year = {2023}
}