HOMEWORK # 1
Due by March 16th, 2023
MEEN 630 – Advanced Engineering Mathematics
Prof. Rashid K. Abu Al-Rub
For full credit, please show all the details of your solution. Use MATLAB when needed. When
you use MATLAB, then attach your MATLAB solution sheets.
Problem # 1
A) Factor these symmetric matrices into  K    L  D  L  with the pivots in  D  :
T
1 3 
K     ,
3 2 
1 b 
K     ,
b c 
2 1 0
 K   1 2 1 
 0 1 2 
B) Using the Gauss elimination method, solve by hand the equation  K u   f  where
2 8 0 
0 
 K   6 27 5  and  f   3
6 
 0 6 17 
 
and check your answers using the backslash command in MATLAB (i.e., u  K \ f ). Show all
the details of the hand calculations.
Problem # 2
A) In the following symmetric matrices, what are the smallest numbers p , c , and d that still
leave each of the following  K  positive semi-definite?
 2 1 0 
 K    1 2 2 ,
 0 2 p 
c 1 0
 K   1 c 1 
 0 1 c 
C) Using all the tests for positive-definiteness, determine if each of the following symmetric
matrices are positive-definite, positive semi-definite, or indefinite matrix:
 2 1 0 1
 1 2 1 0 
 ,
 K   
0 1 2 1


 1 0 1 2 
1
1
 B   
1

1
1
1
1
1
1
1
1
1
1
1
,
1

1
Page 1 of 3
1
1 
1.01 1
 1 1.01 1
1 

C   
1
1 1.01 1 


1
1 1.01
 1
HOMEWORK # 1
Due by March 16th, 2023
MEEN 630 – Advanced Engineering Mathematics
Prof. Rashid K. Abu Al-Rub
Problem # 3
This problem is about the following diffusion-convection differential equation:
D
d 2u du

 f
dx 2 dx
Consider a one-dimensional bar of a unit length, and with D =1/25 and f  1 . There will be a
boundary layer of rapid change at one endpoint when the essential boundary conditions are
u (0)  0 , u (1)  0
such that the du dx convection term dominates.
1) Find the exact solution of the above boundary-value problem (BVP) and graph that exact
solution;
2) Solve the BVP using FDM using a uniform mesh size. Use three values h  1/ 4, 1/ 7, 1/10 .
Plot the results on the same graph of the exact solution. Approximate du dx using the central
difference.
3) Repeat the application of FDM by approximating du dx using the forward and backward
differences assuming h  1/ 7 and compare with the solution in (2).
4) Write a SHORT conclusion from your results. For example, which method(s) to use?
Problem # 4
This is a problem describing the steady state temperature distributing in a plate. This can be
described by the following PDE:
  2u  2u 
 D  2  2   f ( x, y )
 x y 
Consider 0  x  1 , 0  y  1 , and with D =1 and f  x 2  y with the boundary conditions
shown in the figure below. Using the finite difference mesh shown in the figure, calculate the
temperature at each node.
Page 2 of 3
HOMEWORK # 1
Due by March 16th, 2023
MEEN 630 – Advanced Engineering Mathematics
Prof. Rashid K. Abu Al-Rub
43
44
45
u0
46 47
36
37
38
39
40
41
42
29
u
0
x
22
30
31
32
33
34
35
23
24
25
26
27
28 u  0
15
16
17
18
19
20
21
10
11
12
13
14
8
9
y
1 x 2
3
5
4
u
0
y
Page 3 of 3
48
49
6
7