Unified Transform Method

for boundary value problems



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.