The Linearized Classical Boussinesq System on the Half-Line

TL;DR

This paper explicitly solves the linearized classical Boussinesq system on the half-line using the unified transform method, providing a new solution representation that satisfies initial and boundary conditions.

Contribution

It introduces a novel explicit solution formula for the linearized Boussinesq system on the half-line using two different frameworks of the unified transform method.

Findings

Derived a new explicit solution formula for the system.

Proved the solution satisfies initial and boundary conditions.

Showed uniform convergence at the boundary.

Abstract

The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half-line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokas's method to systems of equations; (ii) by expressing the linearized classical Boussinesq system as a single, higher-order equation which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classical Boussinesq system on the half-line. Moreover, thanks to the uniform convergence at the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data via a direct calculation.