Are calculus and ""post"" calculus (such as differential equations) playing an important role in research and development done in industry? Are these mathematical tools indispensable for improving industrial products such as automobiles, airplanes, televisions, and cameras? Do they play a role in understanding air pollution, predicting weather and stock market trends, and building better computers and communication systems? This book was written to convince the reader, by examples, that the answer to all the above questions is YES! Industrial mathematics is a fast growing field within the mathematical sciences. It is characterized by the origin of the problems that it engages; they all come from industry: research and development, finances, and communications. The common feature running through this enterprise is the goal of gaining a better understanding of industrial models and processes through mathematical ideas and computations. The authors of this book have undertaken the approach of presenting real industrial problems and their mathematical modeling as a motivation for developing mathematical methods that are needed for solving the problems. Each chapter presents and studies, by mathematical analysis and computations, one important problem that arises in today's industry. This book introduces the reader to many new ideas and methods from ordinary and partial differential equations, integral equations, and control theory. It brings the excitement of real industrial problems into the undergraduate mathematical curriculum. The problems selected are accessible to students who have taken the first two-year basic calculus sequence. A working knowledge of Fortran, Pascal, or C language is required.
Introduction Preface to the Student Chapter 1: Crystal Precipitation. The Road Ahead-Some Helpful Hints to the Student Background The Model Some Facts About Differential Equations (BASIC) Picard's Method of Successive Approximations (BASIC) Non-rectangular Regions (BASIC) The Euler Method (or Method of Polygons) (BASIC) Crystals of Single Size Remark on Theorems 1.8.3 - 1.8.5 Reminder on Newton's Method The Runge-Kutta Method (BASIC) Discussion and Motivation (BASIC) Crystals of Several Sizes Summary References Chapter 2: Air Quality Modeling. Background The Model The Advection Equation Numerical Methods The General Advection Equation Enters Diffusion The von Neumann Stability Criterion Stability, Consistency and Convergence Summary References Chapter 3: Electron Beam Lithography. Background The Mathematical Model The Heat Equation The Proximity Effect The Use of Fourier Series The Inclusion of Backward Scattering Computational Experiments Summability of Fourier Series Summary References Appendix: Proof of Fejer's Theorem Chapter 4: Development of Color Film Negative. The Process The Mathematical Model The Bulk Reaction Problem Analysis of the Solution Late Development of Film Implicit Methods for Solving the (4.12)-(4.16) Numerically Summary References Appendix: Proof of the Strong Maximum Principle Chapter 5: How Does a Catalytic Converter Function?. Background The Model The Control Problem A Simplified Model The Calculus of Variations The Euler-Lagrange Equation The Simplified Control Problem Determining the Optimal Control Summary References Chapter 6: The Photocopy Machine. Background The Photocopy Machine The Electric Image Modeling the Electric Image Solving Poisson's Equations Numerically Transmission Conditions Computing the Electric Image A Simple Method for Solving the Equations (6.18) (6.17) Summary References Chapter 7: The Photocopy Machine (Continued). The Visible Image Impossibility of a Precise Image Summary References Index.
'This is REAL. It has a different spirit. It gives students the distinct feeling that they could go into industry and actually work on problems like those in this book. The standard teaching of 'here is the mathematics, use it to solve this problem' has been replaced with 'here is a problem, use mathematics to solve it.' This book refreshes the interest of students in mathematics and motivates them to learn more of it. It helps them understand the nature and the importance of mathematics in real world applications.' Oscar Bruno, Assistant Professor of Mathematics, Georgia Institute of Technology
