A transform-based technique for solving boundary value problems on convex planar domains

A new technique is presented that can be used to solve mixed boundary value problems for Laplace’s equation and the complex Helmholtz equation in bounded convex planar domains. This work is an extension of Crowdy (2015, CMFT, 15, 655–687) where new transform-based 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.