B. Pelloni is a Professor of Mathematics at the University of Reading. She has published over 30 papers on the unified transform and lectured on it as the Olga Taussky-Todd prize lecturer at ICIAM 2011. She is a fellow of the Institute for Mathematics and Its Applications and a member of the London Mathematical Society. A. S. Fokas is the Chair of Nonlinear Mathematical Science in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. Among his many awards are the Naylor Prize, the Excellence Prize of the Bodossaki Foundation, and a Guggenheim Fellowship. He is a member of the Academy of Athens and is included in the list of most highly cited researchers in mathematics. He is the originator of the unified transform.

• Chapter 1: IntroductionPart I: Theoretical ResultsChapter 2: Evolution Problems: LinearChapter 3: Evolution Problems: NonlinearChapter 4: Elliptic Problems: LinearChapter 5: Elliptic Problems: NonlinearPart II: Between Theory and ApplicationsChapter 6: Overview of Variational Formulations for Linear Elliptic PDEsPart III: ApplicationsChapter 7: Numerical Methods for Elliptic ProblemsChapter 8: Numerical Methods for Evolution Problems.