Differential Equations and Boundary Value Problems: Computing and Modeling (4e)

C. Henry Edwards

David E. Penney, University of Georgia, Athens

Title Differential Equations and Boundary Value Problems: Computing and Modeling

Edition 4th

ISBN 9780131561076

ISBN 10 0131561073

Published 26/07/2007

Published by Pearson Higher Ed USA

Pages 816

Format Cloth

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Description

For introductory courses in Differential Equations.

This best-selling text by these well-known authors blends the traditional algebra problem solving skills with the conceptual development and geometric visualization of a modern differential equations course that is essential to science and engineering students. It reflects the new qualitative approach that is altering the learning of elementary differential equations, including the wide availability of scientific computing environments like Maple, Mathematica, and MATLAB. Its focus balances the traditional manual methods with the new computer-based methods that illuminate qualitative phenomena and make accessible a wider range of more realistic applications. Seldom-used topics have been trimmed and new topics added: it starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the text.

1. First Order Differential Equations.

Differential Equations and Mathematical Models. Integrals as General and Particular Solutions. Slope Fields and Solution Curves. Separable Equations and Applications. Linear First Order Equations. Substitution Methods and Exact Equations.

2. Mathematical Models and Numerical Methods.

Population Models. Equilibrium Solutions and Stability. Acceleration-Velocity Models. Numerical Approximation: Euler's Method. A Closer Look at the Euler Method, and Improvements. The Runge-Kutta Method.

3. Linear Equations of Higher Order.

Introduction: Second-Order Linear Equations. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Nonhomogeneous Equations and Undetermined Coefficients. Forced Oscillations and Resonance. Electrical Circuits. Endpoint Problems and Eigenvalues.

4. Introduction to Systems of Differential Equations.

First-Order Systems and Applications. The Method of Elimination. Numerical Methods for Systems.

5. Linear Systems of Differential Equations.

Linear Systems and Matrices. The Eigenvalue Method for Homogeneous Systems. Second Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Matrix Exponentials and Linear Systems. Nonhomogenous Linear Systems.

6. Nonlinear Systems and Phenomena.

Stability and the Phase Plane. Linear and Almost Linear Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical Systems. Chaos in Dynamical Systems.

7. Laplace Transform Methods.

Laplace Transforms and Inverse Transforms. Transformation of Initial Value Problems. Translation and Partial Fractions. Derivatives, Integrals, and Products of Transforms. Periodic and Piecewise Continuous Forcing Functions. Impulses and Delta Functions.

8. Power Series Methods.

Introduction and Review of Power Series. Series Solutions Near Ordinary Points. Regular Singular Points. Method of Frobenius: The Exceptional Cases. Bessel's Equation. Applications of Bessel Functions.

9. Fourier Series Methods.

Periodic Functions and Trigonometric Series. General Fourier Series and Convergence. Even-Odd Functions and Termwise Differentiation. Applications of Fourier Series. Heat Conduction and Separation of Variables. Vibrating Strings and the One-Dimensional Wave Equation. Steady-State Temperature and Laplace's Equation.

10. Eigenvalues and Boundary Value Problems.

Sturm-Liouville Problems and Eigenfunction Expansions. Applications of Eigenfunction Series. Steady Periodic Solutions and Natural Frequencies. Applications of Bessel Functions. Higher-Dimensional Phenomena.

References.

Appendix: Existence and Uniqueness of Solutions.

Answers to Selected Problems.

Index.

New to this edition

Chapter 3 now includes new explanations of signs and directions of internal forces in mass-spring systems; new introduction and illustration of polar exponential forms of complex numbers; and fuller explanation of method of undetermined coefficients.
Chapter 5 now includes new exposition to explain the connection between matrix variation of parameters and (scalar) variation of parameters for second-order equations previously discussed in Chapter 3.
Chapter 7 now includes new discussion clarifying functions of exponential order and existence of Laplace transforms and new much-expanded discussion of the proof of the Laplace-transform existence theorem and its extension to include the jump discontinuities that play an important role in many practical applications.
New problems have been added to most chapters to provide students with further practice of what they have learned.
New figures and images help increase student understanding of concepts.
Additional discussions appear at the beginning of select chapters to provide students with more clarity on difficult topics.

Features & benefits

Approximately 2000 problems–These problems span the range from computational problems to applied and conceptual problems. There are over 300 new qualitative problems in this edition.
- Provides students with problem sets that are carefully graded so that the opening problems can be easily solved by most students, giving them encouragement to continue through the set.

Emphasis on the intersection of technology and ODEs–Recognizes the need to instruct students in the new methods of computing differential equations.
- Shows students the software systems tailored specifically to differential equations as well as the widely used Maple, Mathematica, and MATLAB.
300 new computer-generated graphics–Show vivid pictures of slope fields, solution curves, and phase plane portraits.
- Brings to life the symbolic solutions of differential equations through the visualizing of qualitative features.
Extensive expansion of qualitative solutions to the problem sets. Scan the Answer section to see the new geometric flavor.
Fresh numerical methods emphasis–Made possible by the early introduction of numerical solution techniques, mathematical modeling, stability and qualitative properties of differential equations. The text includes generic numerical algorithms that can be implemented in various technologies.
- Gives students and instructors a choice when implementing which type of technology they can utilize, as well as provides an innovative combination of topics usually dispersed later in other texts.
Application Modules–Follow key sections throughout the text; while many involve computational investigations, they are written in a technology-neutral manner. Technology-specific systems modules are available in the accompanying Applications Manual.
- Actively engages students, providing facility and experience in the geometric visualization and qualitative interpretation that play a prominent role in contemporary differential equations.
Leaner and more streamlined coverage–Shaped by the availability of computational aids.
- Allows students to learn traditional manual topics (like exact equations and variation of parameters) more easily.
Unusually flexible treatment of linear systems–Covers in Chapters 4 and 5 the necessary linear algebra followed by a substantial treatment of nonlinear systems and phenomena in Chapter 6. The use of matrix exponential methods plays an enhanced role in this edition.
- Reflects the current trends in science and engineering education and practice.
Accompanying 350-page Applications Manual–Provides detailed coverage of Maple, Mathematica, and MATLAB and is free when shrink-wrapped with text.
- Expands and enhances for students the text's applications material.
Very Robust DE Website–Presents text projects in the form of interactive Maple, Mathematica, and MATLAB notebooks and worksheets. http://www.prenhall.com/edwards

Provides students with the opportunity to download technology-specific versions of individual projects.
Also, Polkings DField and PPlane, are available here in Java applets.

Author biography

C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia's honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution's highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.

David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran's Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee's research team's primary focus was on the active transport of sodium ions by biological membranes. Penney's primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He is the author of research papers in number theory and topology and is the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.