Fokas diagonalization of piecewise constant coefficient linear differential operators on finite intervals and networks

TL;DR

This paper introduces a new diagonalization approach for linear differential operators with piecewise constant coefficients on finite intervals and networks, enabling the solution of boundary value problems with complex boundary and interface conditions.

Contribution

It presents a novel, weaker form of diagonalization that extends the Fokas transform method to piecewise constant coefficient operators on networks, including multipoint and interface problems.

Findings

Applicable to operators with varying coefficients across subintervals

Solves both homogeneous and inhomogeneous boundary value problems

Extends to networks with arbitrary interface and boundary conditions

Abstract

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included. Both homogeneous and inhomogeneous problems are solved.