The analytic extension of solutions to initial-boundary value problems outside their domain of definition

TL;DR

This paper explores how solutions to linear initial-boundary value problems can be analytically extended beyond their original spatial domains using the Fokas method, revealing the nature of extended initial conditions and their regularity.

Contribution

It introduces a method to extend solution representations outside their domains via Taylor series and characterizes the extended initial conditions, including their regularity properties.

Findings

Extended solutions are obtained through Taylor series outside the original domain.

The extended initial condition may lack differentiability or continuity without compatibility conditions.

The approach applies to dissipative, dispersive, and both continuous and discrete spatial problems.

Abstract

We examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation for solutions to half-line and finite-interval initial-boundary value problems as integrals of kernels with explicit spatial and temporal dependence. These solution representations are defined within the spatial domain of the problem. We obtain the extension of these representation formulae via Taylor series outside these spatial domains and find the extension of the initial condition that gives rise to a whole-line initial-value problem solved by the extended solution. In general, the extended initial condition is not differentiable or continuous unless the boundary and initial conditions satisfy compatibility conditions. We analyze dissipative and dispersive problems, and problems with continuous and discrete spatial variables.