The explicit solution of linear, dissipative, second-order initial-boundary value problems with variable coefficients

TL;DR

This paper develops explicit solution formulas for linear, dissipative, second-order IBVPs with spatially varying coefficients, extending previous methods to variable-coefficient problems and enabling direct analysis of solution properties.

Contribution

It introduces a method to derive explicit solutions for variable-coefficient IBVPs by approaching them as limits of constant-coefficient interface problems, expanding the applicability of the Unified Transform Method.

Findings

Explicit solution representations for variable-coefficient IBVPs.

Ability to determine eigenvalues and eigenfunctions explicitly.

Application to heat and complex Ginzburg-Landau equations.

Abstract

We derive explicit solution representations for linear, dissipative, second-order Initial-Boundary Value Problems (IBVPs) with coefficients that are spatially varying, with linear, constant-coefficient, two-point boundary conditions. We 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. As explicit examples, we demonstrate the solution procedure for different IBVPs of variations of the heat equation, and the linearized complex Ginzburg-Landau (CGL) equation (periodic boundary conditions). We can use this to find the eigenvalues of dissipative second-order linear operators (including non-self-adjoint ones) as roots of a transcendental function, and we can write their eigenfunctions explicitly in terms of the eigenvalues.