# Uthm Final Exam

Topics: Trigraph, Partial differential equation, Finite element method Pages: 35 (4361 words) Published: April 26, 2013
CONFIDENTIAL

UNIVERSITI TUN HUSSEIN ONN MALAYSIA FINAL EXAMINATION SEMESTER I SESSION 2011/2012 COURSE NAME COURSE CODE PROGRAMME : : : ENGINEERING MATHEMATICS IV BWM 30603/BSM 3913 1 BEE 2 BDD/BEE/BFF 3 BDD/BEE/BFF 4 BDD/ BFF JANUARY 2012 3 HOURS ANSWER ALL QUESTIONS IN PART A AND TWO (2) QUESTIONS IN PART B. ALL CALCULATIONS AND ANSWERS MUST BE IN THREE (3) DECIMAL PLACES.

EXAMINATION DATE : DURATION INSTRUCTION : :

THIS EXAMINATION PAPER CONSISTS OF SEVEN (7) PAGES

CONFIDENTIAL

BWM 30603 /BSM 3913

PART A Q1 (a) Consider the heat conduction equation

 2 T ( x, t )   2 T ( x, t ), t x

0  x  10, t  0 ,

where  is thermal diffusity  10, since   c 2 . Given the boundary conditions, T (0, t )  0, T (10, t )  100

and initial condition,
T ( x,0)  x 2 .

By using explicit finite-difference method, find T ( x,0.055) and T ( x,0.11) with 5 grid intervals on the x coordinate. (10 marks) (b) Let y( x, t ) denotes displacement of a vibrating string. If T is the tension in the string,  is the weight per unit length, and g is acceleration due to gravity, then y satisfies the equation

 2 y Tg  2 y  , 0 x  2, t 0. t 2  x 2 Suppose a particular string is 2 m long and is fixed at both ends. By taking T  1.5 N,   0.01 kg/m and g  10 m/s2, use the finite-difference method to solve for y up to level 2 only. The initial conditions are

 x 0  x 1  2,  y ( x , 0)   2  x , 1 x  2  2 

and

y ( x , 0)  x( x  2). t

Performed all calculations with x  0.5 m and t  0.01 s. (15 marks)

Q2

The steady state temperature distribution of heated rod follows the one-dimensional form of Poisson’s equation d 2T  Q( x)  0 . dx 2 Solve the above equation for a 6 cm rod with boundary conditions of T (0, t )  10 and T (6, t )  50 and a uniform heat source Q( x)  40 with 3 equal-size elements of length by using finite-element method with linear approximation. (25 marks) 2

BWM 30603 /BSM 3913

PART B Q3 (a) Given f  x   7e x sin  x   1 for 3  x  3 . (i) Find the least positive root of f  x  by using Secant method (iterate until f  xi    ).

(ii)

If the exact solution of f  x  is 0.118, find the absolute error for the method. (8 marks)

(b)

A mixture company has three sizes of packs of nuts. The Large size contains 2 kg of walnuts, 2 kg of peanuts and 1 kg of cashews. The Mammoth size contains 3 kg of walnuts, 6 kg of peanuts and 2 kg of cashews. The Giant size contains 1 kg of walnuts, 4 kg of peanuts and 2 kg of cashews. Suppose that the company receives an order for 14 kg of walnuts, 26 kg of peanuts and 12 kg of cashews. (i) (ii) (iii) By taking a, b and c represent Large, Mammoth and Giant size, obtain the system of linear equations for this company. By using Gauss elimination method, determine how can this company fill this order with the given sizes of packs. Suppose that the mixture company above is planning to expand their mixing productions. Their planning can be summarized as Table Q3(b) below: Walnuts Large Mammoth Giant Small 10 1 4 3 Peanuts 4 5 14 3 Table Q3(b) Cashews 2 11 2 2 Almond 2 1 1 9

If the company receives a new order for 13 kg of walnuts, 34 kg of peanuts, 15 kg of cashews and 28 kg of Almond, determine the possible solutions of the system by using Gauss-Seidel iteration method. (17 marks)

3

BWM 30603 /BSM 3913

Q4

(a)

A certain lab experiment produced the following data (Table Q4(a)): x 0 20 40 60 80 Table Q4(a) Predict y, when x  70 by using (i) Lagrange polynomial and (ii) Newton divided-difference. (16 marks) y -100 280 1460 3440 6220

(b)

Evaluate

 e
1 0

x2

sin x dx by using 3-point Gauss Quadrature.
(9 marks)

Q5

(a)

Given the matrix

 3 4 1   A   4 3 0  1 4 3   (i) (ii) Using Power method, compute the dominant eigenvalue, Largest of A and its associated eigenvector v1 . Then, find smallest eigenvalue, Smallest of A by using Shifted...