## Overview

In 1747 d'Alembert derived the first PDE in the history of mathematics,
namely the wave equation. Soon after, d'Alembert and Euler
discovered a general method for constructing large classes of
solutions of PDEs, namely the method of *separation of variables*.
Bernoulli, in his attempt to solve the wave equation introduced the
infinite sine series, whereas Euler discovered the standard formula for
the coefficients of a Fourier series. Fourier, in his attempts to
understand heat diffusion, inaugurated in 1807 the era of
linearization which dominated mathematical physics for the first
half of the nineteenth century. In 1814 Cauchy wrote an essay on
using complex variables for the evaluation of certain integrals. In
1828 Green introduced the powerful approach of integral
representations that can be obtained via Green's functions (or more
precisely via the fundamental solutions). Separation of variables led
to the *spectral analysis* of ordinary differential operators
and to the solution of PDEs via a *transform pair*. The
prototypical such pair is the Fourier transform; variations include
the sine, the cosine, the Laplace, the Mellin transforms, as well as their
discrete analogues.

In the second half of the 20th century it was realised that certain
*nonlinear evolution* PDEs, called *integrable*, can be formulated
as the compatibility condition of two linear eigenvalue equations, called a
*Lax pair*, and that this formulation gives rise to a method for
solving the initial value problem of these equations, called the
*inverse scattering transform* method. It has been
emphasised by Fokas and Gelfand
[Fok2009b],
[FG1994a]
that this method is based on a deeper form of separation
of variables. Indeed, the spectral analysis of the *t*-independent
part of the Lax pair yields an appropriate *nonlinear Fourier
transform pair*, whereas the *t*-dependent part of the Lax pair
yields the time evolution of the nonlinear Fourier data. In this
sense, inspite of the fact that the inverse scattering transform is
applicable to nonlinear PDEs, this method still follows the logic of
separation of variables by deriving a nonlinear Fourier transform pair.

After the emergence of a method for solving the *initial value
problem* for nonlinear integrable evolution equations in one and two
spatial variables, the most
outstanding open problem in the analysis of these equations became
the solution of *initial-boundary value problems*. A general
approach for solving such problems for evolution equations in one
spatial variable was announced in
[Fok1997a]
and developed in the work of more than sixty researchers
[Fok2008a].
It is remarkable that these results have motivated the discovery of a
new transform method for solving *linear* PDEs in two variables
[Fok2000a].

The new method, which is usually refered to as the “Fokas
method” or the “Fokas Transform method”, is
based on two novel ideas (steps): (1) *Perform the simultaneous
spectral analysis of both equations defining the Lax pair of the
given integrable PDE* (this is to be contrasted with the case of
initial value problems, where the spectral analysis of only the
*t*-independent part of the Lax pair is performed). (2)
*Analyze a certain global relation which couples the given initial and
boundary data with the unknown boundary values*. The new method goes
*beyond* separation of variables. Indeed, since it is based on
the *simultaneous* spectral analysis of both parts of the Lax
pair, it corresponds to the *synthesis* as opposed to separation
of variables. As a consequence of this fundamental difference, the
form of the solution obtained by the new method for linear PDEs differs
drastically from the classical representations. It should be noted
that the integral representations obtained classically via Green's
functions, retain global features. Actually, it is shown in
[FZ2002a] and
[FS2010a] that in
the case of linear PDEs the easiest way to construct the novel integral
representations obtained by the new method is *not* to perform
the simultaneous spectral analysis of the associate Lax pair, but to
use appropriate contour deformations and Cauchy's theorem
starting from the integral representations obtained via Green's
functions. In this sense, the Fokas method reveals a deep relationship
between the seminal contributions of Fourier, Cauchy and Green and extends these
contributions to integrable *nonlinear* PDEs. Indeed, it is shown in
[FS2010a]
that for linear PDEs this method provides a unification as
well as a significant extension of the classical transforms, of the method of
images, of the Green's functions representations, and of the
Wiener-Hopf technique (the latter technique through a series of
ingenious steps gives rise to a Wiener-Hopf factorization problem,
which is actually equivalent to a Riemann-Hilbert problem; in the new method such
Riemann-Hilbert problems can be *immediately* obtained using the global
relation). Furthermore, the new approach provides an appropriate
“nonlinearisation” of some of the above concepts.

## Further details

Fokas' method has been implemented for many classes of problems. Please see the applications page for further details.