MODERN MATHEMATICAL METHODS FOR SCIENTISTS AND ENGINEERS

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 is also an 'Application' section 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. The book presents several modern important and computationally efficient topics, including feedforward neural networks, wavelets, generalized functions, stochastic optimization methods, and numerical methods. A unique and novel feature of the book is the introduction of a recently developed method for solving partial differential equations (PDEs), called the unified transform. PDEs are the mathematical cornerstone for describing an astonishingly wide range of phenomena, from quantum mechanics to 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. The unified transform greatly facilitates this task. For example, 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, the unified transform derives in a simple manner a generalization of the d'Alembert solution, 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, the unified transform constructs a new solution to this ubiquitous PDE, with 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. 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. Contents: Functions of Real Variables: Functions of a Single Variable Functions of Many Variables Series Expansions Complex Analysis and the Fourier Transform: Functions of Complex Variables Singularities, Residues, Contour Integration Mappings Produced by Complex Functions The Fourier Transform Applications to Partial Differential Equations: Partial Differential Equations: Introduction Unified Transform I: Evolution PDEs on the Half-line Unified Transform II: Evolution PDEs on a Finite Interval Unified Transform III: The Wave Equation Unified Transform IV: Laplace, Poisson, and Helmholtz Equations Probabilities, Numerical and Stochastic Methods: Probability Theory Numerical Methods Stochastic Methods Readership: Advanced undergraduate students and graduate students in physical science and engineering departments; researchers and practitioners (both in industry and academia) in the same fields. Can be used as textbook for courses in: Applied Mathematics, Mathematics for Physicists, Mathematics for Engineers, at both the advanced undergraduate and graduate level. Can also be adopted for fields that use mathematical tools for modeling, such as finance. Key Features: Students of physical and engineering sciences, and increasingly those in the life and social sciences, need a solid grounding in mathematical methods to understand and advance within their own discipline. Their typical mathematics training is either too formal (from a purely mathematical perspective with few real-life applications) or inadequate (fearing, or not having adequate time to take, mathematics courses). Thus, they never acquire the confidence and strength to use sophisticated mathematical methods, readily and easily, in their own discipline. This book aims to correct this problem The...