A New Green’s Function Method for Solving Linear PDE’s in Two Variables

A new spectral method for solving linear and integrable nonlinear PDE’s in two variables has recently been developed. This method is based on a Lax pair formulation, and generates an elegant integral representation of the solution to a given initial-boundary value problem that is particularly useful for determining long-time asymptotics. In this paper, it is shown that the integral representation obtained using Lax pairs can also be obtained using Green’s function techniques. This new Green’s function method is illustrated by solving a variety of initialboundary value problems for the diffusion equation, the linearized Schr¨odinger equation, and the linearized KdV equation.