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Differential Equations
Theory,Technique and Practice with Boundary Value Problems




ISBN 9781498735018
Published October 16, 2015 by Chapman and Hall/CRC
480 Pages 96 B/W Illustrations

 
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Book Description

Differential Equations: Theory, Technique, and Practice with Boundary Value Problems presents classical ideas and cutting-edge techniques for a contemporary, undergraduate-level, one- or two-semester course on ordinary differential equations. Authored by a widely respected researcher and teacher, the text covers standard topics such as partial differential equations (PDEs), boundary value problems, numerical methods, and dynamical systems. Lively historical notes and mathematical nuggets of information enrich the reading experience by offering perspective on the lives of significant contributors to the discipline. "Anatomy of an Application" sections highlight applications from engineering, physics, and applied science. Problems for review and discovery provide students with open-ended material for further exploration and learning.

Streamlined for the interests of engineers, this version:

  • Includes new coverage of Sturm-Liouville theory and problems
  • Discusses PDEs, boundary value problems, and dynamical systems
  • Features an appendix that provides a linear algebra review
  • Augments the substantial and valuable exercise sets
  • Enhances numerous examples to ensure clarity

A solutions manual is available with qualifying course adoption.

Differential Equations: Theory, Technique, and Practice with Boundary Value Problems delivers a stimulating exposition of modeling and computing, preparing students for higher-level mathematical and analytical thinking.

Table of Contents

What Is a Differential Equation?
Introductory Remarks
A Taste of Ordinary Differential Equations
The Nature of Solutions
Separable Equations
First-Order Linear Equations
Exact Equations
Orthogonal Trajectories and Families of Curves
Homogeneous Equations
Integrating Factors
Reduction of Order
The Hanging Chain and Pursuit Curves
Electrical Circuits
Anatomy of an Application
Problems for Review and Discovery

Second-Order Linear Equations
Second-Order Linear Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
The Use of a Known Solution to Find Another
Vibrations and Oscillations
Newton’s Law of Gravitation and Kepler’s Laws
Higher-Order Equations
Historical Note: Euler
Anatomy of an Application
Problems for Review and Discovery

Power Series Solutions and Special Functions
Introduction and Review of Power Series
Series Solutions of First-Order Equations
Second-Order Linear Equations: Ordinary Points
Regular Singular Points
More on Regular Singular Points
Gauss’s Hypergeometric Equation
Historical Note: Gauss
Historical Note: Abel
Anatomy of an Application
Problems for Review and Discovery

Numerical Methods
Introductory Remarks
The Method of Euler
The Error Term
An Improved Euler Method
The Runge–Kutta Method
Anatomy of an Application
Problems for Review and Discovery

Fourier Series: Basic Concepts
Fourier Coefficients
Some Remarks about Convergence
Even and Odd Functions: Cosine and Sine Series
Fourier Series on Arbitrary Intervals
Orthogonal Functions
Historical Note: Riemann
Anatomy of an Application
Problems for Review and Discovery

Sturm–Liouville Problems and Boundary Value Problems
What Is a Sturm–Liouville Problem?
Analyzing a Sturm–Liouville Problem
Applications of the Sturm–Liouville Theory
Singular Sturm–Liouville
Anatomy of an Application
Problems for Review and Discovery

Partial Differential Equations and Boundary Value Problems
Introduction and Historical Remarks
Eigenvalues, Eigenfunctions, and the Vibrating String
The Heat Equation
The Dirichlet Problem for a Disc
Historical Note: Fourier
Historical Note: Dirichlet
Problems for Review and Discovery
Anatomy of an Application

Laplace Transforms
Introduction
Applications to Differential Equations
Derivatives and Integrals of Laplace Transforms
Convolutions
The Unit Step and Impulse Functions
Historical Note: Laplace
Anatomy of an Application
Problems for Review and Discovery

Systems of First-Order Equations
Introductory Remarks
Linear Systems
Homogeneous Linear Systems with Constant Coefficients
Nonlinear Systems: Volterra’s Predator-Prey Equations
Anatomy of an Application
Problems for Review and Discovery

The Nonlinear Theory
Some Motivating Examples
Specializing Down
Types of Critical Points: Stability
Critical Points and Stability for Linear Systems
Stability by Liapunov’s Direct Method
Simple Critical Points of Nonlinear Systems
Nonlinear Mechanics: Conservative Systems
Periodic Solutions: The Poincaré–Bendixson Theorem
Historical Note: Poincaré
Anatomy of an Application
Problems for Review and Discovery

Appendix: Review of Linear Algebra

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Author(s)

Biography

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has written more than 175 scholarly papers and more than 65 books, including the following books published by CRC Press: Foundations of Analysis (2014), Convex Analysis (2014), Real Analysis and Foundations, Third Edition (2013), and Elements of Advanced Mathematics, Third Edition (2012). An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D from Princeton University.

Reviews

Praise for Differential Equations: Theory, Technique, and Practice, Second Edition

"Krantz is a very prolific writer. He … creates excellent examples and problem sets."
—Albert Boggess, Professor and Director of the School of Mathematics and Statistical Sciences, Arizona State University, Tempe, USA

 

A first course in differential equations lends itself to the introduction of many interesting applications of mathematics. In this well-written text, Krantz (mathematics, Washington Univ. in St. Louis) emphasizes the differential equations needed to succeed as an engineer. This work is similar to Krantz and Simmons’s Differential Equations: Theory, Technique, and Practice (2007), yet the current work adds the necessary exposure to Sturm-Liouville problems and boundary value problems for the intended engineering audience. This enables the reader access to the all-important introduction to the partial differential equations; namely, the heat and wave equations, as well as the Dirichlet problem. This text has two features that differentiate it from all others on the market at this level: the sections entitled, “Anatomy of an Application” and “Problems for Review and Discovery.” The former analyzes a particular application, while the latter introduces open-ended material for further student exploration. These features will serve students well in their pursuit of garnishing the applied fruits of the subject. This text sets a new standard for the modern undergraduate course in differential equations.
--J. T. Zerger, Catawba College