Fokas's Uniform Transform Method for linear systems

TL;DR

This paper applies Fokas's Unified Transform Method to boundary value problems for linear systems, demonstrating its effectiveness on equations like Klein-Gordon and Fitzhugh-Nagumo, and addressing singularity issues.

Contribution

It extends the Fokas method to systems of linear PDEs, showing how to handle branch singularities and applying it to specific equations.

Findings

Successfully applied to Klein-Gordon and Fitzhugh-Nagumo systems

Showed branch singularities are removable in the global relation

Provided detailed treatment for wave equations in an appendix

Abstract

We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial differential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, allowing the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the Klein-Gordon equation and the linearized Fitzhugh-Nagumo system. The case of wave equations is treated separately in an appendix.