Chapter 9
Collocation Methods
9.1 Introduction
Let's continue the discussion of collocation methods with the second-order linear BVP
Ly = y + p(x)y + q(x)y = r(x)
00
a<x<b
0
y(a) = A
(9.1.1a)
y(b) = B:
(9.1.1b)
To pick up where we left o in Section 6.4, c
Chapter 8
Finite Di erence Methods
8.1 Introduction
Let's continue the discussion of nite di erence procedures for two-point boundary value
problems that we began in Section 6.3 with a brief review of the basic theoretical concepts.
Recall that we had exa
Chapter 7
Initial Value Methods
7.1 Introduction
We'd like to extend the initial-value or shooting procedures to vector systems of m rstorder ODEs. However, before studying nonlinear problems, let's focus on the linear
system
y = Ay + b
(7.1.1a)
Ly(a) = l
Part 3: Two-Point Boundary Value
Problems
1
2
Chapter 6
Fundamental Problems and Methods
6.1 Problems to be Solved
Several problems arising in science and engineering are modeled by di erential equations
that involve conditions that are speci ed at more t
Chapter 5
Multivalue or Multistep Methods
5.1 Introduction
One-step methods only require information about the solution at one time, say t = tn;1
to compute the solution at an advanced time t = tn. After integrating away from the
initial point, we have se
Chapter 4
Extrapolation Methods
4.1 Polynomial and Rational Extrapolation
In Chapter 3, we used extrapolation as a tool to estimate the discretization errors of highorder Runge-Kutta methods. We learned that these error estimates are \asymptotically
corre
Chapter 3
One-Step Methods
3.1 Introduction
The explicit and implicit Euler methods for solving the scalar IVP
y = f (t y)
y(0) = y0:
0
(3.1.1)
have an O(h) global error which is too low to make them of much practical value. With
such a low order of accur
Part 2: Initial Value Problems
1
2
Chapter 2
Introduction
2.1 The Explicit Euler Method: Convergence and
Stability
As an introductory example, we examine, perhaps, the simplest method for solving the
rst-order scalar IVP
y
0
=( )
fty
t>
0
y
(0) =
(2.1.1)
Part 1: Overview of Ordinary Di erential Equations
1
2
Chapter 1
Basic Concepts and Problems
1.1 Problems Leading to Ordinary Di erential Equations
Many scienti c and engineering problems are modeled by systems of ordinary di erential
equations (ODEs). So