| Abstract: |
| The Unified Transform Method (UTM), introduced by A.S. Fokas in the late 1990s, is a method for analyzing boundary value problems for linear and integrable nonlinear PDEs. Since its inception, the UTM has attracted significant attention within the mathematics community and has been extended to address a wide range of problems.
For Laplace's equation, Fokas and Kapaev (2003) developed a transform method for boundary value problems in simply-connected convex polygons. Their approach initially utilized various techniques, including spectral analysis of parameter-dependent ODEs and Riemann--Hilbert methods. Later, Crowdy (2015) showed that this method can be reformulated within a complex function-theoretic framework, leading to the construction of new transform pairs tailored to circular domains (domains bounded by circular arcs, with convex polygons as a special case).
We present a new transform-based approach for boundary value problems for Laplace's equation in more general planar domains, specifically in simply connected Lipschitz domains (including non-convex domains). The key ingredient is the exploitation of the properties of the Szeg\H o kernel and its connection with the Cauchy kernel, which enables the construction of transform pairs for analytic functions in such domains. Several examples are presented, together with numerical implementations illustrating the effectiveness of the transform pairs. |
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