Boundary tracing and boundary value problems: I. Theory

Given an exact solution of a partial differential equation in two dimensions, which satisfies suitable conditions on the boundary of the domain of interest, it is possible to deform the boundary curve so that the conditions remain fulfilled. The curves obtained in this manner can be patched together in various ways to generate a remarkably broad range of domains for which the boundary constraints remain satisfied by the initial solution. This process is referred to as boundary tracing and works for both linear and nonlinear problems. This article presents a general theoretical framework for implementing the technique for two-dimensional, second-order, partial differential equations with a general flux condition imposed around the boundary. A couple of simple examples are presented that serve to demonstrate the analytical tools in action. Applications of more intrinsic interest are discussed in the following paper.