This page gives a list of works on the topic of Fokas' method, organised thematically. The list is growing and we hope to make it comprehensive soon. In the mean time, if you know of any omissions please send us an email with as much information as possible on the work.

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## Problems in two dimensions (either 1+1 or 2+0)

### Linear evolution PDEs formulated on the half-line

The novel *integral representations* obtained by the new method have
both analytical and numerical advantages in comparison with the classical
integral and series representations: (i) They are uniformly convergent at the
boundaries; analytically, this makes it easier to prove rigorously the validity
of such representations *without* the apriori assumption of existence,
whereas numerically, using appropriate contour deformations, it makes it
possible to obtain integrands which decay exponentially as |*k*|
→∞ and this leads to efficient numerical computations
[FF2008a],
[KP2009a],
[Vet2010a].
(ii) These representations retain their form even for more complicated boundary
conditions, whereas the classical representations involve infinite series over
a spectrum determined by a transcendental equation. For example, in the case
of the heat equation with Robin boundary conditions, the classical
representation involves an infinite *series* over
where
*k*_{n} satisfy a transcendental equation, whereas the new
method yields an explicit *integral* representation.

### Linear evolution PDEs formulated on a finite interval

The new transform method yields integral representations for evolution PDEs
containing *x*-derivatives of arbitrary order formulated on the finite
interval [Fok2008a],
whereas it can be shown that even for the linearised KdV on the finite interval
with generic boundary conditions, there does *not* exist an infinite
series representation
[Pap2011a],
[Smi2011a]! Note
that some of the works listed descibing linear evolution PDEs formulated on the
half-line also refer to finite interval formulations.

### Linear evolution PDEs formulated in a time-dependent domain

The new method can be used to study linear evolution equations in
domains of the form *l*(*t*)<*x*<∞,
*0*<*t*<*T*, with prescribed initial and boundary
conditions and with *l*(*t*) a given differentiable function.
In this case, the solution is characterised in general by a system of
linear integral equations with explicit, weakly singular kernel. For
example, for a second order problem such as the heat equation
*q*_{t}=*q*_{xx}, given the
Dirichlet boundary datum *q*(*l*(*t*),*t*), the
unknown Neumann boundary value
*q*_{x}(*l*(*t*),*t*) can be computed
as the solution of a single such linear Volterra integral equation.

### Linear elliptic PDEs on the interior of a polygon

The new method yields novel *integral representations* for the
solution of several classical PDEs including the Laplace, the Helmholtz, and
the modified Helmholtz equations formulated in the interior of a convex
polygon. These representations involve certain transforms of the associated
boundary values. There now exist *two* global relations. For a simple
domain, it is possible, using the global relations and their invariant
properties, to express all transforms in terms of the given boundary conditions
using only algebraic manipulations. For example, the
solution of the Dirichlet problem for the modified Helmholtz
equation in the interior of an equilateral triangle is expressed in
terms of an *integral* in the complex *k*-plane which involves an
integrand which decays exponentially as |*k*| →∞.
This solution has analytical and numerical advantages in comparison
with the classical *series* solution of Lamé; furthermore, problems
with more complicated boundary conditions (for which the classical
approach apparently fails) can also be solved in a similar way. For more
complicated domains, the global relation provides novel effective numerical
techniques for the determination of the unknown boundary values
[FFSS2009a],
[FSS2010a],
[FFSS2008a],
[FF2011a],
[FS2009a]
and [FFX2004a].
Several boundary value problems for the biharmonic equation are analysed in
[CF2004a]
and [DF2011a].

### Linear elliptic PDEs in polar coordinates

The new method has also been implemented in the case of boundary value problems formulated in polar coordinates, and in particular has produced: (a) novel integral representations for the Poissonand Helmholtz equations, and (b) the solutions of several boundary value problems for these equations.

### Spectral theory for non-self-adjoint boundary value problems

The new method permits investigation of initial-boundary value problems on rectangular domains without a priori assumptions on the eigenfunctions of the spatial differential operator. Moreover, the global relation may be used to determine the eigenfunctions directly and the methods of spetral theory applied to test their completeness and basic properties.

### Linear elliptic PDEs on the exterior of a polygon

### Linear and integrable nonlinear hyperbolic PDEs

Boundary value problems for these equations are simpler than boundary value problems for evolution and elliptic PDEs because all boundary values needed for the solution representation are prescribed as boundary conditions.

### Integrable nonlinear evolution PDEs formulated on the half-line

The new method yields novel integral representations formulated in the
complex *k*-plane (the Fourier plane). These integrals, in addition to
the exponentials which appear in the integrals of the linearized version of
these nonlinear PDEs, also contain the entries of a matrix-valued function
*M*(*x*,*t*,*k*), which is the solution of a matrix
Riemann-Hilbert (RH) problem. This RH problem is uniquely defined in terms
of appropriate nonlinear transforms of boundary values, called *spectral
functions*. These functions are defined in the Fourier space, i.e. they do
not involve the independent variables of the PDE. This crucial feature of the
new method makes it possible to obtain useful asymptotic information about the
solution even before characterising the spectral functions in terms of the
given boundary conditions. This can be achieved by using the Deift-Zhou (for
the long-time asymptotics) [DZ1992a], [DZ1993a] and the
Deift-Zhou-Venakides (for the zero-dispersion limit) [DVZ1994a], [DVZ1997a]
techniques for the asymptotic analysis of these RH problems
[Kam2003a],
[FI1992a],
[FI1996a],
[FI1992b] and
[FI1994a]. For
certain nonlinear boundary value problems, called *linearisable*, it is
possible, by analysing the global relation, to express the spectral functions
in terms of the given initial and boundary conditions using only algebraic
manipulations. Thus, for linearisable boundary conditions, the new method
yields a representation which is as effective as the classical representation
for the Cauchy problem obtained by the inverse scattering transform method. For
general, non-linearisable boundary conditions the global relation yields an
effective characterisation of the generalised Dirichlet to Neumann map. This
important development was first achieved in
[BFS2003a] by
employing the so-called Gelfand-Levitan-Marchenko (GLM) representations
[BK2000a],
and was implemented numerically in
[Zhe2006a] and
[Zhe2007a]. It was
later realised that this approach also provides a direct characterisation of
the spectral functions in terms of the given initial and boundary conditions,
bypassing the GLM representation,
[Fok2005a] and
[FT2008a]. Further
progress in this problem was achieved in
[FL2011b], where the
relevant formulae were obtained without the need to introduce the GLM
representations. However, it must be noted that for non-linearisable boundary
value problems, the above characterisation involves a *nonlinear*
equation.

### Integrable nonlinear evolution PDEs on the finite interval

The most interesting aspect of this class of problems is the realization
that the

### Integrable nonlinear elliptic PDEs formulated in the interior of a polygon

The new method again yields novel integral representations in the complex
*k*-plane. These integrals, in adddition to the exponentials which appear
in the integrals of the linearised versions of these nonlinear PDEs, also
contain the entries of a matrix-valued function
*M*(*x*,*y*,*k*) which is the solution of a matrix RH
problem. The spectral functions, i.e. the entries of the relevant jump matrix
which depend only on *k*, can be computed in terms of the associate
boundary values. For elliptic PDEs involving only second order derivatives, by
employing the *two* relevant global relations, it is possible to express
the unknown boundary values via a system of *nonlinear* Fredholm
integral equations. However, for linearisable boundary conditions, it is again
possible to avoid this nonlinear step and to express the spectral functions
directly in terms of the given boundary conditions. The new method has already
been implemented to the elliptic sine-Gordon formulated in the quarter plane
and in a semistrip, as well as to the celebrated Ernst equation formulated in
several complicated domains. Furthermore, particular linearisable boundary
value problems have been solved for these two physically significant PDEs.

## Problems in three dimensions

### Linear evolution PDEs in two space variables

The Fokas method has been applied to simple domains such as the quarter plane, as well as to more complicated domains such as the interior of the isosceles triangle.

### Linear elliptic PDEs in spherical coordinates

The new method has also been implemented in the case of boundary value
problems formulated in spherical coordinates and, in particular, has produced:
(a) the classical Poisson integral formula for the solutions of the Dirichlet
problem for the Poisson equation in the interior of a sphere, the analogous
formula for the Neumann problem, and the generalisations of these formulae to
*n* dimensions, (b) the solutions of various boundary value problems for
the inhomogeneous Helmholtz equation in the interior of a sphere, and (c) the
solution of the Dirichlet problem for the Laplace equation in the interior of a
spherical sector.

### Linear elliptic PDEs with nonlocal boundary conditions

In particular, the Fokas method has been applied to the problem of water waves, which involves the Laplace equation in three dimensions supplemented with certain nonlocal boundary conditions. This has led to a novel nonlocal formulation of this classical problem.

### Quaternion methods

### Elasticity and fluid loaded plate

The Fokas method yields novel results for the classical boundary value problems arising in elasticity and in fluid loaded plate in 2 and in 3 dimensions.

### Nonlinear evolution PDEs in two spatial dimensions

The new method uses either a d-bar formalism (as in the case of the Davey-Stewartson (DS) equation), or a combination of the a d-bar formalism with a nonlocal Riemann-Hilbert formalism (as in the case of the Kadomtsev-Petviashvili (KP) equation), to produce novel integral representations together with the associated global relations. These representations depend on certain spectral functions which themselves are characterized via certain linear integral equations of Fredholm type in terms of the initial conditions and all the boundary values. The characterization of the spectral functions in terms of the given initial and boundary conditions remains open. Moreover, the question of identifying linearisable boundary conditions for the DS or the KP also remains open.

## Differential-difference equations

### Linear and integrable nonlinear differential-difference equations

Following the development of the Fokas method for PDEs, it has been shown that the method can also be used for solving differential-difference equations (DDEs) on the natural numbers. These equations can be considered as discrete analogues of IBVPs for PDEs, since lattice spacing are taken from the natural numbers instead of the integers. The new method can deal with IBVPs for discrete linear and integrable discrete nonlinear Schrödinger equations (also known as the Ablowitz-Ladik system [AL1975a], [AL1976a]) on the natural numbers [BH2008a]. Linearizable BCs for the integrable discrete nonlinear Schrödinger equation were analysed, which are the discrete analogue of the homogenous Robin-type BCs. When such BCs are given, the unknown boundary values can be eliminated using only algebraic manipulation of the global relation as shown in continuum problems. It is worth emphasizing that the Fokas method can be generalized to solve IBVPs for any discrete linear evolution equations with more general boundary conditions just as effectively [BW2010a]; this is to be contrasted with the classical Fourier series approach which is very limited.