Fokas, Athanassios (Univ Of Cambridge, Uk & Univ Southern California, Usa)

Modern Mathematical Methods For Scientists And Engineers: A Street-smart Introduction Fokas, Athanassios (Univ Of Cambridge, Uk & Univ Southern California, Usa)

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.