To browse Academia. Skip to main content. Log In Sign Up. Download Free PDF. Applied Mathematics 4th Edition, - J. David Logan Joe Bar. Download PDF. A short summary of this paper. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section or of the United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc.
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David Logan. The emphasis is on how mathematics interrelates with the applied and natural sciences. Prerequisites include a good command of concepts and techniques of calculus, and sophomore-level courses in differential equations and matrices; a genuine interest in applications in some area of science or engineering is a must. Readers should understand the limited scope of this text. Being a broad introduction to the methods of applied mathematics, it cannot cover every topic in depth.
Indeed, each chapter could be expanded into a one, or even several, full-length books. In fact, readers can find elementary books on all of the topics; some of these are cited in the references at the end of the chapters.
Secondly, readers should understand the mathematical level of the text. Some books on applied mathematics take a highly practical approach and ignore technical mathematical issues completely, while others take a purely theoretical approach; both of these approaches are valuable and part of the overall body of applied mathematics.
Here, we seek a middle ground by providing the physical basis and motivation for the ideas and methods, and we also give a glimpse of deeper mathematical ideas. There are major changes in the fourth edition. The material has been rearranged and basically divided into two parts.
Chapters 1 through 5 involve models leading to ordinary differential equations and integral equations, while Chapters 6 through 8 focus on partial differential equations and their applications.
Motivated by problems in the biological sciences where quantitative methods are becoming central, Chapter 9 deals with discrete-time models, which include some material on random processes. Sections reviewing elementary methods for solving systems of ordinary differential equations have been added in Chapters 1 and 2. Many additional examples and figures are included in this edition, and several new exercises appear throughout.
Some exercises from the last edition have been revised for better clarity, and many new exercises are included. The length of the text has expanded over pages. The Table of Contents details the specific topics covered. Note that equations are numbered within sections. Thus, equation label 3. My colleagues in Lincoln, who have often used the text in our core sequence in applied mathematics, deserve special thanks.
Former students Bill Wolesensky and Kevin TeBeest read parts of the earlier manuscripts and both were often a sounding board for suggestions. I am extremely humbled and grateful to those who used earlier editions of the book and helped establish it as one of the basic textbooks in the area; many have generously given me corrections and suggestions, and many of the typographical errors from the third edition have been resolved.
Because of the extensive revision, some new ones, but hopefully not many, have no doubt appeared. Solutions to some of the exercises and an errata will appear when they become available. My editor at Wiley, Susanne Steitz-Filler, along with Jackie Palmieri, deserves praise for her continued enthusiasm about this new revision and her skill in making it an efficient, painless process.
Finally, my wife, Tess, has been a constant source of support for my research, teaching, and writing, and I again take this opportunity to publicly express my appreciation for her encouragement and affection. Suggestions for use of the text.
The full text cannot be covered in a two-semester, 3-credit course, but there is a lot of flexibility built into the text. There is significant independence among chapters, enabling instructors to design special one- or two- semester courses in applied mathematics that meet their specific needs.
Portions of Chapters 1 through 5 can form the basis of a one-semester course involving differential and integral equations and the basic core of applied mathematics. Chapter 4 on the calculus of variations is essentially independent from the others, so it need not be covered. If students have a strong background in differential equations, then only small portions of Chapters 1 and 2 need to be covered.
A second semester, focused around partial differential equations, could cover Chapters 6,7, and 8. Students have the flexibility to take the second semester, as is often done at the University of Nebraska, without having the first, provided small portions of Chapter 5 on Fourier-type expansions is covered. Chapter 9, like Chapter 3, is independent from the rest of the book and can be covered at any time.
The text, and its translations, have been used in several types of courses: applied mathematics, mathematical modeling, differential equations, mathematical biology, mathematical physics, and mathematical methods in chemical or mechanical engineering. In every physical setting a good grasp of the possible relationships and comparative magnitudes among the various dimensioned parameters nearly always leads to a better understanding of the problem and sometimes points the way toward approximations and solutions.
In this chapter we introduce some of the basic concepts from these two topics. A statement and proof of the fundamental result in dimensional analysis, the Pi theorem, is presented, and scaling is discussed in the context of reducing problems to dimensionless form. The notion of scaling points the way toward a proper treatment of perturbation methods, especially boundary layer phenomena in singular perturbation theory as well as algebraic equations with small parameters.
The first part of Section 1. This material may be perused or used as a reference by readers familiar with the basic concepts and elementary solution methods. The last part includes a discussion of stability and bifurcation; it may be less familiar. In a very limited sense it is a set of methods that are used to solve the equations that come out of science, engineering, and other areas. Traditionally, these methods were techniques used to examine and solve ordinary and partial differential equations, and integral equations.
At the other end of the spectrum, applied mathematics is applied analysis, or the theory that underlies the methods. By a mathematical model we mean an equation, or set of equations, that describes some physical problem or phenomenon having its origin in science, engineering, economics, or some other area. By mathematical modeling we mean 10th Ncert Textbook 8th Edition Pdf the process by which we formulate and analyze the model. This process includes introducing the important and relevant quantities or variables involved in the model, making model- specific assumptions about those quantities, solving the model equations by some analytic or numerical method, and then comparing the solutions to real data and interpreting the results.
This latter process, confronting the model with data, is often the most difficult part of the modeling process. It involves determining parameter values from the experimental data. This book does not address these important issues, and we refer to texts on data-fitting techniques. This confrontation may lead to revision and 10th Ncert Textbook 4th Edition refinement until we are satisfied that the model accurately describes the physical situation and is predictive of other similar observations. This process is depicted schematically in Fig.
Thus, the subject of mathematical modeling involves physical intuition, formulation of equations, solution methods, analysis, and data fitting. A good mathematical model is simple, applies to many situations, and is predictive. Figure 1. In summary, in mathematical modeling the overarching objective is to make quantitative sense of the world as we observe it, often by inventing caricatures of reality.
Scientific exactness is sometimes sacrificed for mathematical tractability. Model predictions depend strongly on the assumptions, and changing the assumptions changes the model. If some assumptions are less critical than others, we say the model is robust to those assumptions. They help us clarify verbal descriptions of nature and the mechanisms that make up natural law, and they help us determine which parameters and processes are important, and which are unimportant.
Another issue is the level of complexity of a model. With modern computer technology it is tempting to build complicated models that include every possible effect we can think of, with large numbers of parameters and variables.
Simulation models like these have their place, but computer runs do not always allow us to discern which are the important processes and which are not. Finally, authors have tried to classify models in several ways�stochastic vs. In this book we are interested in modeling the underlying reasons for the phenomena we observe explanatory rather than fitting the data with formulas descriptive as is often done in statistics.
For example, fitting measurements of the size of an animal over its lifetime by a regression curve is descriptive, and it gives some information. But describing the dynamics of growth by a differential equation relating growth rates, food assimilation rates, and energy maintenance requirements tells more about the underlying processes involved.
Models are a blend of physical laws, such as conservation of mass, energy, etc. The reader is already familiar with many models. The first step in modeling is to select the relevant variables independent and dependent and parameters that we need to describe the problem. Usually these are based on available experimental data and natural laws.
Physical quantities have dimensions like time, distance, degrees, and so on, or corresponding units like seconds, meters, and degrees Celsius. The equations we write down as models must be dimensionally correct. Apples cannot equal oranges. Verifying that each term in our model has the same dimensions is the first task in obtaining a correct equation.
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