Interface Problems using the Fokas Method

Abstract

Interface problems for partial differential equations are initial boundary value problems for which the solution of an equation in one domain prescribes boundary conditions for the equations in adjacent domains. These types of problems occur widely in applications including heat transfer, quantum mechanics, and mathematical biology. These problems, though linear, are often not solvable analytically using classical approaches. In this dissertation I present an extension of the Fokas Method appropriate for solving these types of problems. I consider problems with both dissipative and dispersive behavior and consider general boundary and interface conditions. An analog for the Dirichlet to Neumann map for interface problems is also constructed.