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Faculty of Mechanical and Aerospace Engineering
Institut Teknologi Bandung
WF2202 Partial Differential Equation and Numerical Method
Take Home Exam
Given out: THUR 05 JUN 2025 , Due Date WED 20 JUN 2025 ( 17 .00 WIB)
Please write this etical statement on the top of your answer sheet and sign:
“I hereby declare that all answers in the exam are from my independent work. I did not commit or
facilitate any improper conduct during exam. If i am proven to be in violation, I am ready to accept the
consequences in accordance with the applicable regulations”
sign
(Your Name)
The objective of the learning on the numerical method is to educate the students to be able to
solve many mathematical problems computationally such as system of linear equations, partial
differential equations, finite difference, and integral. In this take home exam, the students are
expected to provide theoretical basis, flow chart, and programming source code for the
following problems.
1.Create a flowchard and programming code to solve the system simultaneous linear algebraic
equations, which is defined as the following:
a
11
x
1
+ a
12
x
2
+... + a
1j
x
j
+... + a
1(n-1)
x
(n-1)
+ a
1n
x
n
= b
1
a
21
x
1
+ a
22
x
2
+... + a
2j
x
j
+... + a
2(n-1)
x
(n-1)
+ a
2n
x
n
= b
2
a
i
x
1
+ a
i
x
2
+... + a
ij
x
j
+... + a
i(n-1)
x
(n-1)
+ a
in
x
n
= b
i
(E.1.1)
a
(n-1)
x
1
+ a
(n-1)
x
2
+... + a
(n-1)j
x
j
+... + a
(n-1)(n-1)
x
(n-1)
+ a
(n-1)n
x
n
= b
(n-1)
a
n
x
1
+ a
n
x
2
+... + a
nj
x
j
+... + a
n(n-1)
x
(n-1)
+ a
nn
x
n
= b
n
where the a’s are constant coefficients and the b’s are constants, and x are unknown variables.
Create a matlab programming code that can be used to solve the system simultaneous linear
algebraic equations. Please include in your programming solution the following:
a) Create the solve the system simultaneous linear algebraic equations that will have inputs
from the keyboards. Inputs are :
a.1. the number of equations/number of unknown variables =n
a.2. the a’s constant coefficients and
a.3. the b’s constants.
b) There are two choice for the method:
Choice A = Using Gauss Elimination
Choice B = Using Gauss Seidel
c) Print the system simultaneous linear algebraic equations, and validate the code for the
solution given below
Number of equations = 4
10 X1 - 2 X2 - X3 - X4 = 3
- 2 X1 + 10 X2 - X3 - X4 = 15
- X1 - X2 + 10 X3 - 2 X4 = 27
- X1 - X2 - 2 X3 + 10 X4 = - 9
The solutions are printed in the format as the following
The answers after 7 iterations are the following (4 digits of accuracy):
X1= 0.
X2= 1.
X3= 2. 9999
X4= - 0.
(for cases of n= 7 )
d) Use the validated code to solve the following system of equations using both Gauss
Elimination and Gauss-Seidel Methods.
Figure P. 3 .1. Heated Plate
- Deflection of aluminum beam that is clamped at one end and given the load as shown in
figure P.4.1 can be calculated using the Bernoulli-Euler theory as follows:
𝑀
𝐸𝐼
𝑦′′
[ 1 +(𝑦
′
)
2
]
3 ⁄ 2
(E.4.1)
Figure P.4.1.
Equation (E.4.1) can be solved numerically by writing it into:
′′
𝑀
𝐸𝐼
[ 1 +
′
2
]
3 ⁄ 2
(E.4.2)
and then, equation (E.4.2) can be written into two equations:
′
= 𝑧 (E.4.3)
′
𝑀
𝐸𝐼
[ 1 + 𝑧
2
]
3 ⁄ 2
(E.4.4)
23
o
C
0
o
C
02
o
C
136
o
C
M
EI
with the boundary condition y (0) = z (0) = 0
Data of beams are : E= Five last numbers of your NIM kg/mm
2
, I=1500 mm
4
, L=200 mm,
M= Four last numbers of your NIM mm kg. For Example: NIM 13621021, E = 21021
kg/mm
2
and M=1021 mm kg.
a) Write the formula of Runge-Kutta with c/p coefficient using Modified Euler for second
order R-K method and sketch the graphic to get y m+
from y m
and z m+
from z m
b) Using the boundary condition y 0
= y(0) = 0 dan z 0
= z(0) = 0 , determine y 1
and z 1
if
h=20 mm. (use the 2
nd
order R-K method).
c) Using the same boundary conditions in b), determine the y and z with the 4
th
order
Runge-Kutta R-K Method
d) Plot your results in b) and c) in the same graph
Good Luck for the exam.
