Boundary behavior of the solution to the linear Korteweg-De Vries equation on the half line

In this paper, we consider the solution to the linear Korteweg-De Vries (KdV) equation, both homogeneous and forced, on the quadrant $\{x\in {\mathbb{R}}^{+},t\in {\mathbb{R}}^{+}\}$ via the unified transform method of Fokas and we provide a complete rigorous study of the integrals of the formula provided by the method, especially focusing on the explicit verification of the considered initial-boundary-value problems (IBVPs), with generic data, as well as on the uniform convergence of all its derivatives, as $(x,t)$ approaches the boundary of the quadrant, and their rapid decay as $x\to \infty$.