Differential Equations Computing and Modeling

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  • Edition: 4th
  • Format: Hardcover
  • Copyright: 7/31/2007
  • Publisher: Pearson
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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
Existence and Uniqueness of Solutions
Answers to Selected Problems
Table of Contents provided by Publisher. All Rights Reserved.

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