Linear and nonlinear generalized Fourier transforms BY BEATRICE PELLONI * Department of Mathematics, University of Reading, Reading RG6 6AX, UK This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method. Keywords: Fourier transform methods; integral transforms; spectral representations; boundary value problems; nonlinear evolution PDEs 1. Introduction The Fourier transform has been, over the last two centuries, one of the most important mathematical tools in every branch of science, and particularly in physics, engineering, computing and image processing. The one-dimensional Fourier transform formulae are the following. Let f (x) be a smooth function defined on R such that lim x/GN f (x)Z0 sufficiently fast (I will not give the precise requirement on differentiability and decay). Then the direct and inverse Fourier transforms are defined by direct : ^ f ðk Þ Z ð N K N e Kiky f ðyÞdy; inverse : f ðx Þ Z 1 2p ð N K N e ikx ^ f ðk Þdk : ð1:1Þ The function ^ f ðk Þ is known as the Fourier transform of f (x). An important application of the Fourier transform is the construction of the solution of initial value problem for the basic evolutionary partial differential equations (PDEs) of mathematical physics. These equations model processes evolving in time from a given initial state. One fundamental example is the initial value problem for the heat equation, which models how the heat diffuses starting from a certain initial temperature. Mathematically, this is the problem of finding the function T(x, t) such that T t ðx ; t Þ Z T xx ðx ; t Þ; t O 0; K N! x !N ; T ðx ; 0Þ Z T 0 ðx Þ; K N! x !N ; ð1:2Þ where x is a space variable; t denotes time; and the solution T(x, t) represents the temperature, at time t and position x, of an infinite one-dimensional object, which is initially at temperature T 0 (x). Phil. Trans. R. Soc. A (2006) 364, 3231–3249 doi:10.1098/rsta.2006.1893 Published online 18 October 2006 One contribution of 23 to a Triennial Issue ‘Mathematics and physics’. *b.pelloni@reading.ac.uk 3231 q 2006 The Royal Society