The generalised Dirichlet to Neumann map for moving initial-boundary value problems

TL;DR

This paper introduces an algorithm to characterize the Dirichlet to Neumann map for moving initial-boundary value problems, combining the global relation with a novel integral inversion method based on spectral analysis and d-bar formalism.

Contribution

It presents a new algorithm that integrates the global relation and spectral analysis techniques to invert integrals for moving boundary problems, advancing boundary value problem solutions.

Findings

Successfully determines Neumann boundary values from Dirichlet data for the linearised Schrödinger equation.

Develops a new integral inversion method based on spectral analysis and d-bar formalism.

Provides a general framework applicable to moving initial-boundary value problems.

Abstract

We present an algorithm for characterising the generalised Dirichlet to Neumann map for moving initial-boundary value problems. This algorithm is derived by combining the so-called global relation, which couples the initial and boundary values of the problem, with a new method for inverting certain one-dimensional integrals. This new method is based on the spectral analysis of an associated ODE and on the use of the d-bar formalism. As an illustration, the Neumann boundary value for the linearised Schroedinger equation is determined in terms of the Dirichlet boundary value and of the initial condition.