About the Author
R. Kent Nagle (deceased) taught at the University of South Florida. He was a research mathematician and an accomplished author. His legacy is honored in part by the Nagle Lecture Series which promotes mathematics education and the impact of mathematics on society. He was a member of the American Mathematical Society for 21 years. Throughout his life, he imparted his love for mathematics to everyone, from students to colleagues.
Edward B. Saff received his B.S. in applied mathematics from Georgia Institute of Technology and his Ph.D. in Mathematics from the University of Maryland. After his tenure as Distinguished Research Professor at the University of South Florida, he joined the Vanderbilt University Mathematics Department faculty in 2001 as Professor and Director of the Center for Constructive Approximation. His research areas include approximation theory, numerical analysis, and potential theory. He has published more than 240 mathematical research articles, co-authored 9 books, and co-edited 11 volumes. Other recognitions of his research include his election as a Foreign Member of the Bulgarian Academy of Sciences (2013); and as a Fellow of the American Mathematical Society (2013). He is particularly active on the international scene, serving as an advisor and NATO collaborator to a French research team at INRIA Sophia-Antipolis; a co-director of an Australian Research Council Discovery Award; an annual visiting research collaborator at the University of Cyprus in Nicosia; and as an organizer of a sequence of international research conferences that helps foster the careers of mathematicians from developing countries.
Arthur David Snider has 50+ years of experience in modeling physical systems in the areas of heat transfer, electromagnetics, microwave circuits, and orbital mechanics, as well as the mathematical areas of numerical analysis, signal processing, differential equations, and optimization. He holds degrees in mathematics (BS, MIT; PhD, NYU) and physics (MA, Boston U), and is a registered professional engineer. He served 45 years on the faculties of mathematics, physics, and electrical engineering at the University of South Florida. He worked 5 years as a systems analyst at MIT's Draper Instrumentation Lab, and has consulted for General Electric, Honeywell, Raytheon, Texas, Instruments, Kollsman, E-Systems, Harris, and Intersil. He has authored nine textbooks and roughly 100 journal articles. Hobbies include bluegrass fiddle, acting, and handball.
Table of Contents
(Most chapters end with a Chapter Summary, Review Problems and Group Projects.) 1. Introduction.
Solutions and Initial Value Problems.
The Phase Line.
The Approximation Method of Euler.
2. First Order Differential Equations.
Special Integrating Factors.
Substitutions and Transformations.
3. Mathematical Models and Numerical Methods Involving First Order Equations.
Heating and Cooling of Buildings.
Improved Euler's Method.
Higher-Order Numerical Methods: Taylor and Runge-Kutta.
Professional Codes for Solving Initial Value Problems.
4. Linear Second Order Equations.
Linear Differential Operators.
Fundamental Solutions of Homogeneous Equations.
Reduction of Order.
Homogeneous Linear Equations with Constant Coefficients.
Auxiliary Equations with Complex Roots.
Superposition and Nonhomogeneous Equations.
Method of Undetermined Coefficients.
Variation of Parameters.
Qualitative Considerations for Variable-Coefficient and Nonlinear Equations.
A Closer Look at Free Mechanical Vibrations.
A Closer Look at Forced Mechanical Vibrations.
5.Introduction to Systems and Phase Plane Analysis.
Introduction to the Phase Plane.
Elimination Method for Systems.
Coupled Mass-Spring Systems.
Numerical Methods for Higher-Order Equations and Systems.
Dynamical Systems, Poincaré Maps, and Chaos.
6. Theory of Higher-Order Linear Differential Equations.
Homogeneous Linear Equations with Constant Coefficients.
Undetermined Coefficients and the Annihilator Method.
Method of Variation of Parameters.
7. Laplace Transforms.
Definition of the Laplace Transform.
Properties of the Laplace Transform.
Inverse Laplace Transform.
Solving Initial Value Problems.
Transforms of Discontinuous and Periodic Functions.
Impulses and the Dirac Delta Function.
Solving Linear Systems with Laplace Transforms.
8. Series Solutions of Differential Equations.
Power Series and Analytic Functions.
Power Series Solutions to Linear Differential Equations.
Equations with Analytic Coefficients.
Cauchy-Euler (Equidimensional) Equations Revisited.
Method of Frobenius.
Finding a Second Linearly Independent Solution.
9. Matrix Methods for Linear Systems.
Review 1: Linear Algebraic Equations.Review 2: Matrices and Vectors.
Linear Systems in Normal Form.
Homogeneous Linear Systems with Constant Coefficients.
Nonhomogeneous Linear Systems.
The Matrix Exponential Function.
10. Partial Differential Equations.
Method of Separation of Variables.
Fourier Cosine and Sine Series.
The Heat Equation.
The Wave Equation.
11. Eigenvalue Problems and Sturm-Liouville Equations.
Eigenvalues and Eigenfunctions.
Regular Sturm-Liouville Boundary Value Problems.
Nonhomogeneous Boundary Value Problems and the Fredholm Alternative.
Solution by Eigenfunction Expansion.
Singular Sturm-Liouville Boundary Value Problems.
Oscillation and Comparison Theory.
12. Stability of Autonomous Systems.
Linear Systems in the Plane.
Almost Linear Systems.
Lyapunov's Direct Method.
Limit Cycles and Periodic Solutions.
Stability of Higher-Dimensional Systems.
13. Existence and Uniqueness Theory.
Picard's Existence and Uniqueness Theorem.
Existence of Solutions of Linear Equations.
Continuous Dependence of Solutions.
Method of Least Squares.
Runge-Kutta Precedure for n Equations.
Answers to Odd-Numbered Problems.
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The first math class that I actually liked.