What non-linear methods offered to linear problems? The Fokas transform method

Abstract

In 1750 D’ Alembert demonstrated how a linear partial differential equation can be solved via separation of variables, a method that decomposes a PDE into a set of ODEs. This method was the basis for the development of many branches of contemporary analysis, from function spaces to spectral analysis of operators and the theory of special functions. A condition for the method of separation of variables to work is the existence of a coordinate system that fits the boundary of the fundamental domain and at the same time it separates the PDE. It is remarkable that two and a half centuries later a generalization is introduced that has its origin in the analysis of non-linear integrable equations. In the present work, this promising new transform method is outlined and applied to particular boundary value problems. A crucial part of the method is the introduction of a global relation which, if properly used, can provide the missing boundary data in a very elegant and effective way. We show how this can be used to generate separable solutions of partial differential equations even when no system, that fits the geometry of the fundamental domain, is available. This is shown for the case of the Dirichlet problem for the modified Helmholtz equation in the interior of an equilateral triangle. Furthermore, the connection of the Fokas method to the classical moment problem is investigated. It is shown that, in this case, the global relation is decomposed into a sequence of global relations, directly associated with the Fourier coefficients of the Dirichlet and Neumann boundary values.