Overview
Research addressed interface problems involving combinations of constant coefficient linear evolution partial differential equations (PDEs) spanning different spatial orders. The methodology employed extended Fokas's unified transform method, with solution formulae evaluated using Filon quadrature.
Research Context
The study focused on a class of interface problems arising from the conjunction of distinct linear evolution PDEs. Specifically, these problems are configured on a one-dimensional spatial domain (a line), where one type of equation applies to one spatial half-line, and a different type applies to the other. Solutions were sought under specified interface conditions linking the two domains.
Approach
The approach involved solving interface problems for constant coefficient linear evolution PDEs. The PDEs considered as prototypical examples included: The heat equation, the linear Schrödinger equation, the Airy equation, the linearized Korteweg de Vries (KdV) equation, and the biharmonic Schrödinger equation. The core of the solution method involved novel extensions of Fokas's unified transform method. After deriving explicit solution formulae, these were subsequently evaluated using Filon quadrature.
Findings
- Interface problems on the line between various constant coefficient linear evolution partial differential equations can be solved.
- The specific PDEs for which solutions were found include the heat, linear Schrödinger, Airy, linearized Korteweg de Vries, and biharmonic Schrödinger equations.
- These problems involve situating one of the listed equations on one spatial half-line and another on the other spatial half-line.
- Appropriate interface conditions linking the two half-lines were incorporated into the problem structure.
- Solutions were achieved through novel extensions of Fokas's unified transform method.
- Explicit solution formulae derived from this method were evaluated through the application of Filon quadrature.