Explicit solutions to linear, second-order, initial and boundary value problems with variable coefficients

Abstract

I derive explicit solution representations for linear, second-order Initial-Boundary Value Problems (IBVPs) with coefficients that are spatially varying, with linear, constant-coefficient, two-point boundary conditions. I accomplish this by considering the variable-coefficient problem as the limit of a constant-coefficient interface problem, previously solved using the Unified Transform Method of Fokas. Our method produces an explicit representation of the solution, allowing us to determine properties of the solution directly. I prove that these representations are solutions to fully and partially dissipative problems under general conditions. As explicit examples, I demonstrate the solution procedure for different IBVPs of variations of the heat equation, and the linearized complex Ginzburg-Landau (CGL) equation (with periodic boundary conditions). The solution can be used to find the eigenvalues of second-order linear operators (including non-self-adjoint ones) as roots of a transcendental function, and their eigenfunctions may be written explicitly in terms of the eigenvalues.