MEC706 ADVANCED NUMERICAL ANALYSIS
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MEC706
ADVANCED NUMERICAL ANALYSIS
JAN 2023
TEST 3
ANSWER ALL QUESTIONS.
QUESTION 1
a) Describe three most commonly used numerical methods used to solve partial differential equations. (10 marks)
b) Compute the temperature distribution for the L-shaped plate as shown in the figure below (Figure Q1). Use Liebman’s method. All temperatures given are in oC. The nodes for unknown temperatures are shown in the figure. You should use Excel or MATLAB to solve this question. Attach the final answer in the answer sheet and also pass up the Excel/Matlab solution file together with your answer sheet. Label your Excel/MATLAB sheet as Q1_Your name. Solve the question where error is less than 0.01%.
Figure Q1
(20 marks)
QUESTION 2
a) Explain what you know about Fibonacci number.
b) Consider the following function:
(10 marks)
f(x) = 2x +(3/x)
i) Use Golden-section search. (xl = 0, xu=2, es <1%). Perform iterations until you reach es <1%. You may use Excel or MATLAB for this. Compare your answer using analytical graphical solution. (10 marks)
ii) Parabolic interpolation (xo = 0.1, x1= 0.5, x2= 5, es <1%). Perform iterations until you reach es <1%. You may use Excel or MATLAB for this. (10 marks)
iii) Newton’s method (x0 = 0.1, es<1%). Perform iterations until you reach es <1%. You may use Excel or MATLAB for this.
Comment on the convergence of your results for (i), (ii) and (iii) (10 marks)
QUESTION 3
a) Briefly explain why optimization is important in engineering application? (10 marks)
b) Solve the following problem numerically:
dy/dt = -2y+5e-t
dz/dt = (-yz2)/2
over the range t = 0 to 0.4 using a step size of 0.1 with y (0) = 2 and z (0) =4
i) Euler’s method
ii) Third order RK method.
iii) Fourth order RK method (20 marks)
2023-01-29