The Linear KdV Equation with an Interface

TL;DR

This paper develops an explicit solution method for the linear KdV equation with an interface in a piecewise domain, extending the Fokas Unified Transform to higher derivatives and providing conditions for solution construction.

Contribution

It introduces a new explicit solution framework for the linear KdV equation with interfaces, generalizing previous methods to higher-order derivatives.

Findings

Explicit solutions constructed for the interface problem

Derived sufficient conditions for interface compatibility

Extended Fokas's method to third-order derivatives

Abstract

The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas's Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.