Differential Equations and Boundary Value Problems Computing and Modeling

by Edwards, C. Henry; Penney, David E.

• ISBN13: 9780131561076
• ISBN10: 0131561073
• Edition: 4th
• Format: Hardcover
• Copyright: 2007-07-26
• Publisher: Pearson

Summary

This practical book reflects the new technological emphasis that permeates differential equations, including the wide availability of scientific computing environments likeMaple, Mathematica,and MATLAB; it does not concentrate on traditional manual methods but rather on new computer-based methods that lead to a wider range of more realistic applications.The book starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the book.For mathematicians and those in the field of computer science and engineering.

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.

Table of Contents

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

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

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

Introduction to Systems of Differential Equations

First-Order Systems and Applications

The Method of Elimination

Numerical Methods for Systems

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

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

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

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

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

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

Table of Contents provided by Publisher. All Rights Reserved.