The unified transform for evolution equations on the half‐line with time‐periodic boundary conditions*

Abstract

This paper elaborates on a new approach for solving the generalized Dirichlet-to-Neumann map, in the large time limit, for linear evolution PDEs formulated on the half-line with time-periodic boundary conditions. First, by employing the unified transform (also knownThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Long‐time asymptotics and the radiation condition with time‐periodic boundary conditions for linear evolution equations on the half‐line and experiment

“…Here, we consider the Dirichlet problem on the half‐line for the general third‐order model (), which includes the linear KdV equation discussed in Ref. 18 but additionally involves a mixed spatiotemporal derivative like the one present in the linear BBM equation. Through our analysis, we demonstrate that the Fokas method for the characterization of the D‐N map in the case of asymptotically time‐periodic boundary conditions remains effective for this general third‐order model.…”

“…This analysis addresses a relevant uniqueness issue that was identified in Ref. 18. 4.In the case of real roots of the aforementioned cubic, the root $$k_0(n)$$ is the only one that corresponds to a nonnegative group velocity.…”

The Airy equation with nonlocal conditions

“…In Section 3, we study a problem related to problem (1) in which the data are time periodic with common period, but the initial condition is replaced with a time periodicity condition on 𝑢. We use the "𝑄 equation" method [23][24][25] to obtain a solution representation and criteria on its validity. The greater part of Section 4 is dedicated to an asymptotic analysis of the solution representation for 𝑢, whereby the long time decay result is derived.…”

“…We modify the arguments of Refs. [24,25] for the present setting, informed by the nonlocal global relation concepts that were introduced in Ref. [2] and adapted to the Airy equation above.…”

“…When evaluated at $$y=0$$, $$z=1$$, this equation is named “the Q equation” in Ref. [24], after a change of notation $$Q(\lambda ) = -\hat{q}(\lambda ;t)$$. To emphasize the simpler derivation of this equation presented herein (and to preserve the symbol Q for later use), we do not adopt this notation.…”