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.