Modern Mathematical Methods for Computational Sciences and Engineering: A Street-smart Introduction

Partial differential equations (PDEs) are the mathematical cornerstone for describing an astonishingly wide range of phenomena, from quantum mechanics and ocean waves, to the diffusion of heat in matter and the behavior of financial markets. Despite the efforts of many famous mathematicians, physicists and engineers, the solution of partial differential equations remains a challenge. This book's authors introduce a novel method, the unified transform, which greatly facilitates this challenge.

Two and a half centuries after Jean d'Alembert formulated the wave equation and presented a solution for solving a simple problem for this equation, this book introduces a generalization of the d'Alembert solution, which is valid for general boundary value problems. Moreover, two centuries after Joseph Fourier introduced the classical tool of the Fourier series for solving the heat equation, it offers a new solution of this problem, which has important analytical and numerical advantages in comparison to the classical solutions. The authors present the unified transform pedagogically, building all the necessary background, including functions of real and of complex variables and the Fourier transform, illustrating the method with numerous examples.

Modern Mathematical Methods for Scientists and Engineers is a modern introduction to basic topics in mathematics at the undergraduate level, with emphasis on explanations and applications to real-life problems. There are also "Application" sections at the end of each chapter, with topics drawn from a variety of areas, including neural networks, fluid dynamics, and the behavior of put and call options in financial markets. In addition to the unified transform, the book presents several modern important and computationally efficient topics, including feed-forward neural networks, wavelets, generalized functions, stochastic optimization methods, and numerical methods.

Broad in scope, but pedagogical in style and content, the book is an introduction to powerful mathematical concepts and modern tools for students in science and engineering.

Since 2002, Professor Athanassios Fokas has held the newly inaugurated Chair of Nonlinear Mathematical Sciences at the University of Cambridge, UK (prior to this appointment he was Chair of Applied Mathematics at Imperial College London, UK). In addition, since 2015 he has been Adjunct Professor at the departments of Civil and Environmental Engineering, and of Biomedical Engineering at the University of Southern California, USA. Fokas holds a BSc in Aeronautics from Imperial College (1975), a PhD in Applied Mathematics from Caltech (1979), and an MD from the University of Miami (1986).

Fokas has made broad scientific contributions with seminal results in several areas in mathematics, including the 'Fokas method', which features in the textbook as a method of greatly simplifying calculations for undergraduates.

In 2000, Fokas was awarded the Naylor Prize, the most prestigious prize in Applied Mathematics and Mathematical Physics in the UK (in 1999 this prize was awarded to Stephen Hawking). He is also a recipient of a Guggenheim Fellowship, and has been elected in several academies, including the prestigious Academy of Athens, and is the only applied mathematician to be elected at this academy. His contributions to the areas of medicine and biology were recently recognized by his election to the American Institute of Medical and of Biological Engineering (one of only a handful of mathematicians to be members of this Institute). His overall contributions were recognized in 2015 with a Senior Fellowship by EPSRC, UK, which covers his salary at the University of Cambridge, so that he can concentrate on research, released from all teaching and administrative responsibilities.

Professor Fokas has published close to 400 papers. He has appeared in the list of the most highly cited researchers in mathematics for Web of Science. He has co-edited eight books and has authored or co-authored five monographs. His book, Complex Variables and Applications, with M J Ablowitz published by Cambridge University Press, has sold nearly 20,000 copies.

Professor Efthimios Kaxiras was educated at the Massachusetts Institute of Technology where he received a PhD in theoretical condensed matter physics. He joined the faculty of Harvard University in 1991, where currently holds the title of John Hasbrouck Van Vleck Professor of Pure and Applied Physics in the Department of Physics and Professor of Applied Mathematics in the School of Engineering and Applied Sciences (SEAS). He served as Chair of Applied Mathematics, SEAS, from 2017 to 2020 and was appointed Chair of the Department of Physics in 2020. He is the Founding Director of the Institute for Applied Computational Science and served as its Director for two terms (2010–2013 and 2017–2020). He also served as Director of the Initiative on Innovative Computing (2007–2009). He has held faculty appointments and administrative positions in Switzerland (Ecole Polytechnique Federal de Lausanne) and Greece (University of Crete, University of Ioannina, Foundation for Research and Technology Hellas). He is a Fellow of the American Physical Society, and Chartered Physicist and Fellow of the Institute of Physics (UK).