Differential Equations: Theory,Technique and Practice with Boundary Value Problems, 1st Edition (Hardback) book cover

Differential Equations

Theory,Technique and Practice with Boundary Value Problems, 1st Edition

By Steven G. Krantz

Chapman and Hall/CRC

464 pages | 96 B/W Illus.

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Hardback: 9781498735018
pub: 2015-10-16
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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.

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

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

About the Author

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.

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations