CHAPTER 7 Introduction to Difference Equations
T
his Chapter concerns the dynamical behavior of systems in which time can be treated as a discrete quantity as opposed to a continuous one. For
example, some mathematical models of the onset of cardiac arrhy
Guide to Commonly Used Notation
Symbol R Rn t u, v x, y x0 , y0 x , y f A, M, P D N tr( A) det( A) , Es , Eu , Ec Ws, Wu, Wc AB spancfw_v1 , v2 , . . . vn t (x0 ) t f : Rn Rm f Jf uv v 2 u-v 2 B(x, ) V (x) (t)
Usual Meaning the set of real numbers n-dime
CHAPTER 13 The Laplace and Poisson Equations
U
p to now, we have dealt almost exclusively with pde s for which one independent variable corresponds to time. Now, we will analyze a pde
for which this is not the case: Laplace's equation. To motivate where L
CHAPTER 12 Introduction to Fourier Series
In each example, we were able to construct series representations of the solutions provided that the initial conditions themselves had special series representations (i.e., Fourier sine and cosine series). In this
CHAPTER 11 Initial-Boundary Value Problems
T
he infinite spatial domains considered in the previous chapter give insight regarding the behavior of waves and diffusions. However, since such
domains are not physically realistic, we need to develop new techn
CHAPTER 10 The Heat and Wave Equations on an Unbounded Domain
A
t first glance, the heat equation ut - u xx = 0 and the wave equation
to be positive constants, the only apparent distinction is between the ut in the
utt - c2 u xx = 0 appear very similar. S
CHAPTER 9 Linear, First-Order Partial Differential Equations
I
n this chapter, we will discuss the first of several special classes of pde s that can be solve via analytical techniques. In particular, we will investigate linear,
first-order pde s a( x, t)
CHAPTER 8 Introduction to Partial Differential Equations
perature T depends not only upon time, but also upon spatial location. If x and y denote latitude and longitude and t denotes time, then the function T ( x, y, t) describes how temperature varies in
CHAPTER 6 Introduction to Delay Differential Equations
I
n this Chapter, we turn our attention to delay differential equations (dde s), a major departure from the ordinary differential equations that were considered
up to now. A basic reference for this m
CHAPTER 5 Bifurcations
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n practice, we often deal with ode s which contain parameters (unspecified constants) whose values can profoundly influence the dynamical behavior of
the system. For example, suppose we model population changes for a species. The
CHAPTER 4 Periodic, Heteroclinic, and Homoclinic Orbits
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n this chapter, we shift our attention away from equilibria, instead seeking more "interesting" solutions of nonlinear systems x = f (x). Much of our
discussion involves planar systems (i.e., f : R
CHAPTER 3 Nonlinear Systems: Local Theory
systems is usually impossible, so we must settle for qualitative descriptions of the dynamics. On the other hand, nonlinear systems can exhibit a wide variety of behaviors that linear systems cannot. Moreover, mos
CHAPTER 2 Linear, Constant-Coefficient Systems
T
here are few classes of ode s for which exact, analytical solutions can be obtained by hand. However, for many systems which cannot be solved
explicitly, we may approximate the dynamics by using simpler sys
CHAPTER 1 Introduction
T
he mathematical sub-discipline of differential equations and dynamical systems is foundational in the study of applied mathematics. Differential equations
arise in a variety of contexts, some purely theoretical and some of practic
Ordinary and Partial Differential Equations
An Introduction to Dynamical Systems
John W. Cain, Ph.D. and Angela M. Reynolds, Ph.D.
Mathematics Textbook Series. Editor: Lon Mitchell 1. Book of Proof by Richard Hammack 2. Linear Algebra by Jim Hefferon 3. A