MAT 3240 Numerical Methods For Dierential Equations
(2nd Term, 20112012)
Tutorial 11
1. Finite dierence method and nite element method
Example 1
Consider the forwardtime backwardspace (FTBS) scheme
n
n
n
n
vm vm1
vm+1 vm
+a
= 0,
k
h
i.e.,
n
n
n
vm+1 =
MAT 3240 Numerical Methods For Dierential Equations
(2nd Term, 20112012)
Tutorial 10
1. Shooting method and nite dierence method
Example 1
Solve the following boundary value problem by the shooting method:
x (t) = x(t), 0 < t <
x(0) = 1, x( ) = 3
2
2
Sol
MAT 3240 Numerical Methods For Dierential Equations
(2nd Term, 20112012)
Tutorial 9
1. Stiness , Astability , absolute stability of ODEs
Example 1
When testing a linear ODE system to see if it is sti, it is convenient to write it as
X (t) = AX (t) + W (
MAT 3240 Numerical Methods For Dierential Equations
(2nd Term, 20112012)
Tutorial 8
1. Higherorder ODEs and numerical solutions of systems of rstorder ODEs
Example 1
(a)Convert the following initialvalue problem
x sin(x ) = 25
x(0) = 5, x (0) = 3
into
MAT 3240 Numerical Methods For Dierential Equations
(2nd Term, 20112012)
Tutorial 7
1. Consistency, stability and convergence of linear multistep methods
Example 1 Check the consistency and stability of the following schemes
(a)
xn+1 = 4xn 3xn1 2hfn1 ,
2
MAT 3240 Numerical Methods For Dierential Equations
(2nd Term, 20112012)
Tutorial 6
1. Predictorcorrector methods:
(0)
Denition We rst use some explicit multistep method to predict a value xn+1 , then use the
(k +1)
(k )
(0)
xedpoint iteration: xn+1 =
MAT 3240 Numerical Methods For Dierential Equations
(2nd Term, 20112012)
Tutorial 2
1. Important Theorems:
1. Existence of solutions to an ODE
Theorem 1 If f (t, x) is continuous in a rectangle R centered at (t0 , x0 ), i.e.,
R = cfw_(t, x) : t t0  ,

MAT 3240 Numerical Methods For Dierential Equations
(2nd Term, 20112012)
Tutorial 2
1. Denitions:
1. ODE (Ordinary Dierential Equation)
An ODE is a equation which involves a unknown function of one independent variable
and some derivatives of unknown fun
Midterm Examination for
Numerical Methods For Dierential Equations
(MAT 3240; Second Term, 20112012)
1. (a) It is because nearly all higher order ODEs, no matter whether they are linear or
nonlinear, can be transformed into a system of ODE in the follow
clc,clear % clean the screen and erase the memory of matlab
tic % begin to measure the running time of the following codes
x0=2; % initial value of x
t=1; % initial value of t
M=100; % number of iterations
h=0.01; % stepsize
x=x0; % initialize x
fprintf('
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Solution to Assignment 6
Note: Please contact Mr. KjLiu or Ytchow should there be any problems.
1. (a) The nite dierence scheme is as follows:
cfw_
1
2
1
2 +1 + ( 2 + () 2 1 = (
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Assignment 6
Note: You do not need to hand in this assignment.
1. Consider the following dierential equation:
cfw_
() + ()() = ()
(0) = (1) = 0
for (0, 1)
(1)
where () > 0 for s
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Solution to Assignment 5
Note: Please contact Mr. KjLiu or Ytchow should there be any problems.
1. (a) The nite dierence scheme is as follows:
cfw_
+1 2 +1
2
=
0 = = 0
or
cfw_
+
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Assignment 5
Note: Please hand in your assignment to the assignment box.
Due day: 5:00 p.m., 12th April (Thursday). Overdue assignments will not be counted.
1. Consider the foll
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Solution to Assignment 4
Note: Please contact Mr. KjLiu or Ytchow should there be any problems.
cfw_
(1) = 14 1 = 0
1. (a) The scheme is consistent
8
(1) = 2 (1)4 + 3 (1)4 + 2
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Assignment 4
Note: Please hand in your assignment to the assignment box.
Due day: 5:00 p.m., 29th March (Thursday). Overdue assignments will not be counted.
1. Consider the mult
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Assignment 3
Note: Please hand in your assignment to the assignment box.
Due day: 5:00 p.m., 8th March (Thursday). Overdue assignments will not be counted.
1. Consider the follo
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Solution to Assignment 2
Note: Please contact Kjliu or Ytchow at LSB 222C if there is any problems.
Please note that ( ) stands for the exact value of () at the point , while re
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Assignment 2
Note: Please hand in your assignment to the assignment box.
Due day: 5:00 p.m., 23rd February (Thursday).
counted.
1. Consider the following integral
() =
0
()
Ove
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Solution to Assignment 1
Note: Please contact Kjliu or Ytchow at LSB 222C if there is any problems.
1. (a)
i.
+ = 2
The system is nonlinear.
Let 1 = , 2 = then
(
1
2
)
ii.
(
=
2
MAT 3240 Numerical Methods For Dierential Equations
(20112012, Second Term)
Assignment 1
Note: Please hand in your assignment to the assignment box.
Due day: 5:00 p.m., 2nd February (Thursday). Overdue assignments will not be counted.
1. Consider the fo
MAT 3240 Numerical Methods For Differential Equations
(2nd Term, 20112012)
Tutorial 4
Made by Liu Keji
RungeKutta methods
Some remarks:
1.Also Taylor series
2.Eliminate all partial derivatives.
3.Only need to compute values of function
Secondorder Rung
MAT 3240 Numerical Methods For Differential Equations
(2nd Term, 20112012)
Tutorial 5
Made by Liu Keji
Multistep methods
A numerical method which uses more than one previous value
from x0 , x1 , xn1 , xn ,to evaluate xn+1 x(tn+1) is called a
multistep m