An introduction to the unified transform method and applications

Abstract: The unified transform method (UTM) was born out of developments in the theory of integrable partial differential equations. Well known integrable PDEs, such as Korteweg-deVries equation, can be expressed as the equivalence of two ordinary differential equations (known as a Lax pair). The standard theory considered only initial value problems. However, the extension of these ideas to boundary-value problems was found equally fruitful. In particular, many of the ideas had exact analogues for linear PDEs. In this talk, I will motivate the need for an extension to Fourier theory for linear boundary-value problems and then introduce the unified transform method. Following this, I will provide examples of where UTM is efficient in providing solution expressions and how these may be used for a variety of problems in partial differential equations.