Boundary behaviour of the solution of the heat equation on the half line via the Fokas unified transform method

TL;DR

This paper analyzes the boundary behavior of solutions to the heat equation on a half line using the Fokas method, providing conditions for smooth extension to boundaries and examining derivatives.

Contribution

It offers a detailed analysis of boundary behavior and derivative properties of heat equation solutions via the Fokas unified transform method, with new boundary extension conditions.

Findings

Boundary behavior characterized near semi-axes, infinity, and origin.

Conditions established for smooth extension of solutions to boundaries.

Analysis of derivatives' boundary behavior.

Abstract

We consider the Fokas method expression for the solution of the heat equation on the half line with Dirichlet data and we study in detail its boundary behaviour near the spatiotemporal domain boundaries, i.e., the semi-axes, infinity and the origin, by analyzing the integrals involved. We also study the boundary behaviour of the derivatives of the solution. In particular we give conditions on the data which guarantee the extension of the solution to a smooth function up to the semi-infinite boundaries.