This thesis is concerned with new analytical and numerical methods for solving boundary
value problems for the 2nd order linear elliptic PDEs of Poisson, Helmholtz, and modified
Helmholtz in two dimensions.

In 1967 a new method called the Inverse Scattering Transform (IST) method was introduced to solve the initial value problem of certain non-linear PDEs (so-called "integrable"
PDEs) including the celebrated Korteweg-de Vries (KdV) and nonlinear Schrödinger
(NLS) equation. The extension of the IST method from initial value problems to boundary
value problems (BVPs) was achieved by Fokas in 1997 when a unified method for solving
BVPs for integrable nonlinear and linear PDEs was introduced. This thesis applies "the
Fokas method" to the basic elliptic PDEs in two dimensions.

It is perhaps suprising that ideas from the theory of integrable nonlinear PDEs can be
used to obtain new results in the classical theory of linear PDEs. In fact, the new method
has a beautiful connection with the classical integral representations of the solutions of
these PDEs due to Green. Indeed, this thesis shows that the Fokas method provides
the analogue of Green's integral representation (IR) in the transform, or spectral, space.
Both Irs contain boundary values which are not given as boundary conditions, and the
main difficulty with BVPs is determining these unknown boundary values. In addition to
the novel IR, the Fokas method provides a relation coupling the transforms of both the
known and unknown boundary values known as "the global relation", which is then used
to determine the contribution of the unknown boundary values to the solution.

One of the conclusions of this thesis is that the new method (applied to these 2nd order
linear elliptic PDEs) does three things: (a) solves certain BVPs which cannot be solved
by classical techniques, (b) yields novel expressions for the solutions of BVPs which have
both analytical and computational advantages over the classical ones, and (c) provides an alternative, simpler, method for obtaining the classical solutions.

Chapter 2 is about the novel integral representations. Chapter 3 is about the global
relation. In Chapter 4, a variety of boundary value problems in the separable domains
of the half plane, quarter plane and the exterior of the circle are solved. In Chapter
5, boundary value problems are solved in a non-separable domain, the interior of a right
isosceles triangle. Just as Green's integral representation gives rise to a numerical method
for solving these PDEs (the boundary integral method), the Fokas method can also be
used to design new numerical schemes; Chapter 6 presents these for the Laplace and
modfied Helmholtz equation in the interior of a convex polygon.