A Transform Pair for Doubly Connected Domains
Abstract
A new transform-based technique that generalizes the unified transform method is developed for bounded doubly connected domains as a novel way to numerically solve boundary value problems for holomorphic functions and solutions to the Laplacian. This work builds on the transform methods for multiply connected circular domains developed by Crowdy (2015, IMA J., 80, 1902-1931) and the methods for simply connected bounded domains developed by Hulse et al. (2025, Proc. A, 481 (2319)). The Szegő kernel of the annulus and a corresponding transformation law is pivotal in the derivation of this new technique. The modified Schwarz problem and a boundary value problem for a point vortex inside of a doubly connected domain are implemented in detail to demonstrate the effectiveness of this new method.
Related
- Strict Dirichlet Pólya Inequality for Simply Connected Planar Domains
- On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative
- Algebraic properties of Toeplitz operators on annulus and multiply connected domains
- A priori Dirichlet energy bounds for positive Trudinger--Moser critical points on simply connected planar domains
- Hot Spots: From Smooth to Nonsmooth Simply Connected Planar Domains
BibTeX
@misc{hulse2026transform,
title = {{A Transform Pair for Doubly Connected Domains}},
author = {Jesse J. Hulse},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.05220},
eprint = {2607.05220},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.05220},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Transform Pair for Doubly Connected Domains
AU - Jesse J. Hulse
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2607.05220
UR - https://arxiv.org/abs/2607.05220
AB - A new transform-based technique that generalizes the unified transform method is developed for bounded doubly connected domains as a novel way to numerically solve boundary value problems for holomorphic functions and solutions to the Laplacian. This work builds on the transform methods for multiply connected circular domains developed by Crowdy (2015, IMA J., 80, 1902-1931) and the methods for simply connected bounded domains developed by Hulse et al. (2025, Proc. A, 481 (2319)). The Szegő kernel of the annulus and a corresponding transformation law is pivotal in the derivation of this new technique. The modified Schwarz problem and a boundary value problem for a point vortex inside of a doubly connected domain are implemented in detail to demonstrate the effectiveness of this new method.
ER -