A Unified Transform Method for Solving Linear and
Certain Nonlinear PDE's
Professor A.S. Fokas (Imperial and Clarkson)
The inverse spectral method is a nonlinear Fourier transform
method for solving initial value problems for certain
nonlinear PDE's in 2 and 3 dimensions. Is is based
on the solution of the so called Riemann-Hilbert and
\bar{\boundary} problems. After reviewing this method
we will present a new transform method for solving initial
boundary value problems for both linear and for
integrable nonlinear PDE's in 2 dimensions. This
method provides a unified approach to solving linear
equations with simple boundary conditions, linear
equations with complicated boundary conditions
(Wiener-Hopf type problems) and nonlinear
integrable equations.