An iterative analytic series method for Laplacian problems with free and mixed boundary conditions

Mixed boundary value problems occur in a wide variety of applications in applied mathematics. These problems are characterised by a combination of Dirichlet and Neumann conditions along at least one boundary. For example, problems in both saturated and unsaturated flow usually contain mixed boundary conditions. Historically, only a small subset of these problems could be solved using analytic series methods, by using an appropriate coordinate transformation or choice of axes. However, there are some striking similarities between the mixed boundary problem and the free boundary problem, where the location of one boundary is initially unknown. This unknown boundary is subject to two boundary conditions, and so the problem can be fully defined. In this paper, I will point out the similarities between mixed boundary and free boundary problems. I will consider mixed boundary conditions of the form ?? where ? satisfies Laplace's equation. Finally, I will present an iterative method to find analytic series solutions for problems of this type.