Faculty of Science, Technology and Human Development

UNIVERSITI TUN HUSSEIN ONN MALAYSIA

FACULTY OF SCIENCE, TECHNOLOGY AND HUMAN DEVELOPMENT

SEMESTER I SESSION 2012/2013

BWM 30603 / BSM 3913 / TEST 2 (20%) /

DURATION: 75 MIN

ANSWER ALL QUESTIONS. ALL THE CALCULATION MUST BE IN 3 DECIMAL PLACES. USE .

Q1 (a) Table Q1 gives the water vapor capacity of air () in gram per cubic meter for selected

temperatures () in degrees Celsius. Determine the slope of the water vapor capacity curve at 12 using the 2-point forward, 2-point backward and 3-point central difference formula.

Table Q1

/ 10 / 12 / 14 / 16 / 18
/ 9.401 / 10.812 / 12.403 / 14.195 / 16.206

(b) The moment of mass about -axis of a thin plate with constant density, bounded by two curves and on the interval is given by

.

Determine for the plate bounded by and on the interval by using 3 point Gauss quadrature. Assume the constant density 2.

(30 marks)

Q2 Consider ordinary differential equation

with initial condition , and .

Solve the initial value problem by using:

(a) Euler method.

(b) Classic fourth-order Runge-Kutta method.

For both method, calculate absolute error for each approximation if the true solution for the differential equation is given by .

(45 marks)

Q3 Given the following matrix

, with .

Approximate the smallest eigenvalue (in absolute value) for matrix by inverse power method.

(25 marks)

30th November 2012 /

STRIVE FOR SUCCESS

/ [Total marks : 100]
LIST OF FORMULA
Numerical differentiation
2-point forward difference:
2-point backward difference:
3-point central difference:
Numerical integration
Gauss quadrature:
(a) For ,
3-points:
(b) Natural coordinates in isoparametric function:
Gaussian Quadrature coefficients:
/ /
1 / 0.0 / 2.0
2 / 0.5773502692 / 1.0
3 / 0.7745966692
0.0 / 0.555555556
0.888888889
Eigen value
Power Method :
Ordinary differential equations (Initial value problems):
Euler’s method:
Classic 4th order Runge-Kutta method:
where

Marking Scheme [ M – method ; A – answer] / Marks / Total
Q1 (a) / Let ,
2-point forward difference

2-point backward difference

3-point central difference
/ A1
M1
A1
A1
A1
M1
A1
A1
A1
M1
A1
A1
A1 / 30
Q1 (b) /

Therefore


By using 3-point Gauss quadrature
/ M1
A1
M1
A1-x
A1-dx
M1A1
M1-subs
A1-x
A1-dx
A1
A1
M1
A3
A1
Q2 (a) /
------
Euler method with ,

/ / / exact / abs error
0 / 2 / 5 / 5 / -
1 / 2.25 / 5.5 / 5.506 / 0.006
2 / 2.5 / 6.010 / 6.021 / 0.011
3 / 2.75 / 6.528 / 6.543 / 0.015
4 / 3 / 7.018 / 7.071 / 0.053
/ M1
M1
M1
A1-all x
A5-each y
A1-all exact
A5-each error / 45
Q2
(b) / Classic 4th order Runge-Kutta method with

where

/ / / / / / / exact / abs error
0 / 2 / 5 / 0.5 / 0.506 / 0.506 / 0.511 / 5 / -
1 / 2.25 / 5.506 / 0.511 / 0.515 / 0.515 / 0.519 / 5.506 / 0
2 / 2.5 / 6.021 / 0.519 / 0.522 / 0.522 / 0.525 / 6.021 / 0
3 / 2.75 / 6.543 / 0.525 / 0.528 / 0.528 / 0.530 / 6.543 / 0
4 / 3 / 7.071 / 7.071 / 0
/ M1
M1
A1-all x
A5-each y
A16-each k
A1-all exact
A5-each error
Q3 /
/ / /
0 / 0 / 1 / 0.5 / -0.477 / 0.303 / 0.884 / 0.884
1 / -0.539 / 0.342 / 1 / -0.826 / 0.255 / 1.978 / 1.978
2 / -0.418 / 0.129 / 1 / -0.732 / 0.170 / 1.904 / 1.904
3 / -0.384 / 0.089 / 1 / -0.710 / 0.152 / 1.884 / 1.884
4 / -0.377 / 0.081 / 1 / -0.705 / 0.148 / 1.880 / 1.880
5 / -0.375 / 0.079 / 1
Since , .
Therefore,
/ M1
A3-each row
M1
A6-
A5-
A5-
M1A1
M1
A1 / 25