A Unified Approach To Boundary Value Problems

  title={A Unified Approach To Boundary Value Problems},
  author={Athanassios S. Fokas},
This book presents a new approach to analyzing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, a method that the author first introduced in 1997 and which is based on ideas of the inverse scattering transform. This method is unique in also yielding novel integral representations for the explicit solution of linear boundary value problems, which include such classical problems as the heat equation on a finite interval and the Helmholtz… 


A new method for investigating boundary value problems in two dimensions has recently been introduced by one of the authors. The main achievement of this method is that it yields explicit integral

Initial and boundary value problems in two and three dimensions

This thesis: (a) presents the solution of several boundary value problems (BVPs) for the Laplace and the modified Helmholtz equations in the interior of an equilateral triangle; (b) presents the

A Hybrid Method for Solving Inhomogeneous Elliptic PDEs Based on Fokas Method

The proposed approach provides a framework for solving inhomogeneous elliptic PDEs using the unified transform that relies on the derivation of the global relation, containing certain integral transforms of the given boundary data as well as of the unknown boundary values.

A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line

On the numerical solution of the generalized dirichlet-neumann map for the 2D laplace equation using modified generic factored approximate sparse inverse preconditioning

The proposed method for the 2D Laplace equation in several regular convex polygons with an arbitrary number of edges is considered, along with discussions on the implementation details of the method.

The numerical solution of semidiscrete linear evolution problems on the finite interval using the Unified Transform Method

A semidiscrete analogue of the Unified Transform Method introduced by A. S. Fokas is studied, showing how it treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and the notion of “natural” discretizations is proposed.