Applied Mathematics / Edition 4

by J. David Logan

Overview

This examination of mathematical methods covers scaling and dimensional analysis, regular and singular perturbation methods, nonlinear wave propagation, and stability and bifurcation.

The outstanding Second Edition — skillfully revised and thoroughly updated

Since its initial publication over ten years ago, J. David Logan's Applied Mathematics has established itself as an ideal modern introduction to the principles of this important subject. Now, energized with fresh material and refined to achieve optimum clarity, this Second Edition brings Logan's highly acclaimed text fully up to date for the '90s and beyond.

Like the original, this revised volume covers not only such standard topics as fluid mechanics and calculus of variations, but also more contemporary methods, including dimensional analysis and scaling, nonlinear wave propagation, bifurcation, and singular perturbation. New sections have been added on the WKB approximation, the asymptotic expansion of integrals, and numerical methods. The treatment of partial differential equations, integral equations, and Green's functions has also been expanded, and there are many new exercises throughout to help readers develop and improve problem-solving skills with the support of the Maple programs provided.

Classical in approach and broad in scope, Applied Mathematics, Second Edition is firmly rooted in the interdependence of mathematics and the physical sciences, making it an accessible and truly relevant resource for students and practitioners across a wide range of scientific and engineering fields.

Product Details

ISBN-13:	2901118475804
Publication date:	05/28/2013
Pages:	688
Product dimensions:	6.30(w) x 9.20(h) x 1.60(d)

About the Author

J. DAVID LOGAN, PHD, is Willa Cather Professor of Mathematics at the University of Nebraska, Lincoln. He is also the author of An Introduction to Nonlinear Partial Differential Equations, Second Edition and Mathematical Methods in Biology, both published by Wiley. Dr. Logan has served on the faculties at the University of Arizona, Kansas State University, and Rensselaer Polytechnic Institute, and he has been affiliated with Los Alamos Scientific Laboratory, Lawrence Livermore National Laboratory, and the Aerospace Research Laboratory.

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Table of Contents

	Preface	xiii
1	Dimensional Analysis, Scaling, and Differential Equations	1
1.1	Dimensional Analysis	2
1.1.1	The Program of Applied Mathematics	2
1.1.2	Dimensional Methods	5
1.1.3	The Pi Theorem	8
1.1.4	Proof of the Pi Theorem	13
1.2	Scaling	19
1.2.1	Characteristic Scales	19
1.2.2	A Chemical Reactor Problem	22
1.2.3	The Projectile Problem	25
1.3	Differential Equations	35
1.3.1	Review of Elementary Methods	36
1.3.2	Stability and Bifurcation	44
1.4	Two-Dimensional Problems	54
1.4.1	Phase Plane Phenomena	54
1.4.2	Linear Systems	63
1.4.3	Nonlinear Systems	68
1.4.4	Bifurcation	76
2	Perturbation Methods	85
2.1	Regular Perturbation	87
2.1.1	Motion in a Resistive Medium	88
2.1.2	Nonlinear Oscillations	90
2.1.3	The Poincare-Lindstedt Method	93
2.1.4	Asymptotic Analysis	95
2.2	Singular Perturbation	104
2.2.1	Algebraic Equations	104
2.2.2	Differential Equations	107
2.2.3	Boundary Layers	108
2.3	Boundary Layer Analysis	112
2.3.1	Inner and Outer Approximations	112
2.3.2	Matching	114
2.3.3	Uniform Approximations	116
2.3.4	General Procedures	119
2.4	Initial Layers	123
2.4.1	Damped Spring-Mass System	123
2.4.2	Chemical Reaction Kinetics	127
2.5	The WKB Approximation	135
2.5.1	The Non-oscillatory Case	137
2.5.2	The Oscillatory Case	138
2.6	Asymptotic Expansion of Integrals	142
2.6.1	Laplace Integrals	142
2.6.2	Integration by Parts	146
2.6.3	Other Integrals	147
3	Calculus of Variations	153
3.1	Variational Problems	153
3.1.1	Functionals	153
3.1.2	Examples	155
3.2	Necessary Conditions for Extrema	159
3.2.1	Normed Linear Spaces	159
3.2.2	Derivatives of Functionals	163
3.2.3	Necessary Conditions	165
3.3	The Simplest Problem	168
3.3.1	The Euler Equation	168
3.3.2	Solved Examples	171
3.3.3	First Integrals	172
3.4	Generalizations	177
3.4.1	Higher Derivatives	177
3.4.2	Several Functions	179
3.4.3	Natural Boundary Conditions	181
3.5	The Canonical Formalism	185
3.5.1	Hamilton's Principle	185
3.5.2	Hamilton's Equations	191
3.5.3	The Inverse Problem	194
3.6	Isoperimetric Problems	199
4	Eigenvalue Problems, Integral Equations, and Green's Functions	207
4.1	Orthogonal Expansions	207
4.1.1	Orthogonality	207
4.1.2	Classical Fourier Series	216
4.2	Sturm-Liouville Problems	220
4.3	Integral Equations	226
4.3.1	Introduction	226
4.3.2	Volterra Equations	230
4.3.3	Fredholm Equations with Degenerate Kernels	236
4.3.4	Symmetric Kernels	239
4.4	Green's Functions	247
4.4.1	Inverses of Differential Operators	247
4.4.2	Physical Interpretation	250
4.4.3	Green's Function via Eigenfunctions	255
4.5	Distributions	258
4.5.1	Test Functions	258
4.5.2	Distributions	261
4.5.3	Distribution Solutions to Differential Equations	265
5	Discrete Models	271
5.1	One-Dimensional Models	272
5.1.1	Linear and Nonlinear Models	272
5.1.2	Equilibria, Stability, and Chaos	277
5.2	Systems of Difference Equations	285
5.2.1	Linear Models	285
5.2.2	Nonlinear Interactions	296
5.3	Stochastic Models	303
5.3.1	Elementary Probability	303
5.3.2	Stochastic Processes	310
5.3.3	Environmental and Demographic Models	314
5.4	Probability-Based Models	321
5.4.1	Markov Processes	321
5.4.2	Random Walks	327
5.4.3	The Poisson Process	331
6	Partial Differential Equations	337
6.1	Basic Concepts	337
6.1.1	Linearity and Superposition	341
6.2	Conservation Laws	346
6.2.1	One Dimension	347
6.2.2	Several Dimensions	349
6.2.3	Constitutive Relations	354
6.2.4	Probability and Diffusion	358
6.2.5	Boundary Conditions	361
6.3	Equilibrium Equations	367
6.3.1	Laplace's Equation	367
6.3.2	Basic Properties	370
6.4	Eigenfunction Expansions	374
6.4.1	Spectrum of the Laplacian	374
6.4.2	Evolution Problems	377
6.5	Integral Transforms	383
6.5.1	Laplace Transforms	383
6.5.2	Fourier Transforms	387
6.6	Stability of Solutions	398
6.6.1	Reaction-Diffusion Equations	398
6.6.2	Pattern Formation	400
6.7	Distributions	406
6.7.1	Elliptic Problems	406
6.7.2	Tempered Distributions	411
6.7.3	Diffusion Problems	412
7	Wave Phenomena	419
7.1	Wave Propagation	419
7.1.1	Waves	419
7.1.2	The Advection Equation	425
7.2	Nonlinear Waves	430
7.2.1	Nonlinear Advection	430
7.2.2	Traveling Wave Solutions	435
7.2.3	Conservation Laws	440
7.3	Quasi-linear Equations	445
7.3.1	Age-Structured Populations	449
7.4	The Wave Equation	454
7.4.1	The Acoustic Approximation	454
7.4.2	Solutions to the Wave Equation	458
7.4.3	Scattering and Inverse Problems	463
8	Mathematical Models of Continua	471
8.1	Kinematics	472
8.1.1	Mass Conservation	477
8.1.2	Momentum Conservation	478
8.1.3	Thermodynamics and Energy Conservation	482
8.1.4	Stress Waves in Solids	487
8.2	Gas Dynamics	493
8.2.1	Riemann's Method	493
8.2.2	The Rankine-Hugoniot Conditions	499
8.3	Fluid Motions in R[superscript 3]	502
8.3.1	Kinematics	502
8.3.2	Dynamics	508
8.3.3	Energy	515
	Index	525

What People are Saying About This

From the Publisher

“…can be thoroughly recommended to all who want an up-to-date approach to their subject.” (Zentralblatt MATH, 2007)

"Future mathematicians, scientists and engineers should find the book to be an excellent introductory text for coursework or self-study as well as worth its shelf space for reference." (MAA Reviews, October 12, 2006)

Editorial Reviews

This second edition, published some ten years after the first, is considerably updated, but retains coverage of such standard topics as fluid mechanics and calculus of variations, as well as more contemporary methods such as dimensional analysis and scaling, nonlinear wave propagation, bifurcation, and singular perturbation. New sections are included on the WKB approximation, the asymptotic expansion of integrals, and numerical methods, while treatment of partial differential equations, integral equations, and Green's functions is expanded. Annotation c. by Book News, Inc., Portland, Or.

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