[Book Cover]

Differential Equations and Boundary Value Problems: Computing and Modeling, 2/e

C. Henry Edwards
David E. Penney, both of the University of Georgia

Published April, 1999 by Prentice Hall Engineering/Science/Mathematics

Copyright 2000, 787 pp.
Cloth
ISBN 0-13-079770-7


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Summary

For introductory courses in Differential Equations. This text provides the conceptual development and geometric visualization of a modern differential equations course while maintaining the solid foundation of algebraic techniques that are still essential to science and engineering students. It reflects the new excitement in differential equations as the availability of technical computing environments like Maple, Mathematica, and MATLAB reshape the role and applications of the discipline. New technology has motivated a shift in emphasis from traditional, manual methods to both qualitative and computer-based methods that render accessible a wider range of realistic applications. With this in mind, the text augments core skills with conceptual perspectives that students will need for the effective use of differential equations in their subsequent work and study.

Features


NEW—Coverage of seldom-used topics trimmed—There is more streamlined coverage of certain traditional, manual topics like exact equations and variation of parameters in Chapters 1, 3, and 5. Symbolic, graphic, and numerical solution methods are combined wherever it seems advantageous.

  • Allows for greater emphasis on core techniques as well as qualitative aspects of the subject associated with direction fields, solutions curves, phase plane portraits, and dynamical systems.
NEW—Contemporary topics added—Elementary introduction to period-doubling in biological and mechanical systems, the pitchfork diagram, and the Lorenz strange attractor (all illustrated with vivid computer graphics).
NEW—Increased emphasis on and flexible treatment of linear systems of differential equations—With coverage in Chapters 4 and 5 (along with the necessary linear algebra) followed by a substantial treatment of nonlinear systems and phenomena in Chapter 6 (including chaos in dynamical systems). Chapter 4 offers an early, intuitive introduction to first-order systems, models, and numerical approximation techniques. Chapter 5 begins with a self-contained treatment of the linear algebra required, then presents the eigenvalue approach to linear systems. Section 5.5 now includes a more extensive treatment of matrix exponentials. A new section (5.6) on nonhomogeneous linear systems was added to this edition.
  • Reflects current trends in science and engineering education and practice.
NEW—About half of the over 300 computer-generated graphics are new—Most were constructed using MATLAB and show vivid pictures of direction fields, solutions curves, and phase plane portraits. For instance, the cover graphic shows an eigenfunction of the three-dimensional wave equation that illustrates surface waves on a spherical planet and was constructed using associated Legendre functions (see section 10.5).
  • Brings symbolic solutions of differential equations to life.
This text is unique in its blend of traditional algebraic material with the modern geometric approach—This computer friendly text still stresses the development of strong algebraic skills by offering many challenging problem sets. However, the first three chapters introduce a carefully prepared introduction to qualitative issues, especially the geometric side.
Coverage begins and ends with discussions and examples of the mathematical modeling of real-world phenomena.
  • Students learn through mathematical modeling and empirical investigation to balance the questions of what equation to formulate, how to solve it, and whether a solution will yield useful information.
Includes about 45 Computing Projects following key sections throughout the text. Half of these are NEW or substantially revised from the previous edition—These “technology neutral” project sections contain much additional and extended problem material and illustrate the use of computer algebra systems like Maple, Mathematica, and MATLAB.
The projects are expanded considerably in the Computing Projects Manual that accompanies the text. Provided are parallel subsections entitled Using Maple, Using Mathematica, and Using MATLAB that detail the applicable methods and techniques of each system. This manual is free when wrapped with text. Project notebooks and worksheets can be downloaded from the supporting web site: www.prenhall.com/edwards
  • Actively engages students in the exploration and application of computational technology. Affords them the opportunity to compare the merits and styles of different computational systems.
Approximately 2000 problems—over 200 of which are NEW to this edition—Problems in each section span the range from computational problems to applied and conceptual problems. Answers to most odd-numbered problems can be found in the answer section at the back of the book.
Supported by a fully dedicated DE Website.
—A superb direction field and phase portrait plotter is provided. Written as Java applets, all you need is a computer with a browser—no other software.
—Includes expanded versions of the in-text projects. Each project is supported by a Maple V worksheet, a Mathematica (version 3) notebook, and a MATLAB (version 5) script that illustrates the computing techniques used in the project. Students can download these technology-specific versions of the individual projects directly from the site: www.prenhall.com/edwards for FREE.
—An Internet Tutor—available on Sunday nights to answer student questions concerning the problem sets. Free to adopters.
—Selected text examples are animated to provide better visual understanding of DEs.
—Quizzes, verbal in nature, are provided for each section to help ensure that the students will actually read the core text.


Table of Contents
    1. First-Order Differential Equations.

      Differential Equations and Mathematical Models. Integrals as General and Particular Solutions. Direction Fields and Solutions 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. 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 the Method of 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 Eignvalue Solutions. Matrix Exponentials and Linear Systems. Nonhomogeneous 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. Fourier Sine and Cosine Series. 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.

      Strum-Liouville Problems and Eigenfunction Expansions. Applications of Eigenfunction Series. Steady Periodic Solutions and Natural Frequencies. Cylindrical Coordinate Problems. Higher-Dimensional Phenomena.


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