Practise Exam
1. There will be one page of short answer questions.
2. Derive a second order Taylors method to solve the following initial value problem.
y = tey ,
y (0) = 1 .
3. Consider the boundary value problem,
y = sin(t)y ,
y (0) = 1 ,
y (1) = 2 .
(a
Math 3210 - Numerical Analysis
Homework #1 Solutions
1. Prove that the function f (t, y ) in
y = f (t, y ) = +
|y |
does not satisfy a Lipschitz condition on the rectangle |t| 1, |y | 1.
To satisfy a Lipschitz condition in a domain D, a function f (x) mus
Practise Midterm
1. There will be one page of short answer questions.
2. Find a second order Taylor method for the problem
y = y sin(t2 ) ,
y (0) = 1 .
3. (a) Find the local truncation error for the midpoint method:
w0 = ,
wi+1 = wi + hf (ti +
h
h
, wi +
Practise Midterm
1. There will be one page of short answer questions.
2. Find a second order Taylor method for the problem
y = y sin(t2 ) ,
y (0) = 1 .
For a second order method, we need the local truncation error to be O(h3 ). So we set
y (tn + h) = y (t
Math 3210 - Numerical Analysis
Homework #4 Due Dec 5
1. Consider the boundary value problem,
y 9y 10y = 0 ,
y (0) = 1.0001 , y (1) = 2.57052
(a) Solve the above boundary value problem using linear shooting. For the time integration
use rk2 method with tim
Math 3210 - Numerical Analysis
Homework #2 Due October 17th
1. Consider the initial value problem,
2y 3y 2
3,
t
t
y (1) = 1 .
y =
The exact solution to (1) is y (t) =
1 t 2,
(1a)
(1b)
t2
.
1+3 ln(t)
(a) Approximate the solution to (1) using Eulers method
Math 3210 - Numerical Analysis
Homework #3 Solutions
1. (Exercise 2.12) For simplicity and convenience, application codes often use a constant step
size when integrating with the backward Euler formula. This formula has very good stability
properties, but
Practise Exam Solutions
1. There will be one page of short answer questions.
2. Derive a second order Taylors method to solve the following initial value problem.
y = tey ,
y (0) = 1 .
To construct a Taylors Method solution, we look at a Taylor series exp
A variable time step method
We construct and test a variable time step method. We will use Eulers method and the Midpoint method for our
two approximations. This code bounds the relative error. The derivation in the text is for the absolute error. The
cod
Constructing a Taylors method of order O(h2 )
Consider the dierential equation
y = y 2 t ,
y (0) = 1 .
The exact solution to this problem is
2
.
+2
We wish to construct a second order approximation to the solution. First we look at the results of a rst or
Stability of Numerical Methods
We will consider two types of stability for numerical methods:
zero-stability Growth of round-o errors.
a-stability Growth of rapidly decaying modes of the solution
To study zero-stability, we consider the test problem
y = 0
Construction and Implementation of a Second Order Adams-Bashford Methd
We will now go through the process of constructing implementing and testing a second order Addams Bashford
method. We consider the dierential equation
dy
= f (t, y ) ,
dt
y (0) = y0 .
Implementation of Backward Euler Method Solving the Nonlinear System using Newtons Method.
As I showed in class the Backward Euler method has better stability properties than the normal Euler method.
Specically errors wont grow when approximating the solu