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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2010/
Module Title: Computing & Numerical Methods II
Module Code: MATH
School: School of Computing & Mathematics
Programme Title: Bachelor of Engineering in Mechanical Engineering
Programme Code: EMECH_7_Y
External Examiner(s): Dr. P. Robinson Internal Examiner(s): Dr. R. Sheehy
Instructions: Answer all questions.
Duration: 2 Hours Sitting: Autumn 2011
Requirements for this examination:
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination. If in doubt please contact an Invigilator.
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Q1. (a) Briefly explain explicit and implicit finite difference methods in the solution of partial differential equations
(b) Use both explicit and implicit finite difference methods to obtain a solution to the heat conduction equation 2 2
T T
k
t x
in a thin rod of length 10cm.
At t=0 the temperature of the rod is zero and the boundary temperatures are fixed for all time at T (0)
= 100 C and T (10) = 50 C
Note: the rod is aluminium with k= .835 cm 2 /s h = 2 cm
Q2.. (a) Illustrate using a suitable example both an Initial Value and Boundary Value Problem. Use the Shooting Method to solve the 1D steady state heat equation
For a rod of length 10 m with h’ = .01 m-2 Ta = 20 and boundary conditions T (0) = 40 T (10) = 200 (b) Use a Finite Difference Method with 4 interior nodes to solve the same problem as Part (a) Q3. (a) Obtain the temperature distribution at 1cm grid points of a (4 x 4) cm square plate bounded by the lines x = ± 2, y = ± 2. Laplaces Equation
2 2
T T
X Y
holds for all interior points and the boundary conditions are: T = -80 on the line y = 2 T = -40y on the lines x = ± 2 T = 240 + 80x for -2 ≤ x ≤ 0 on the line y = - T = 240 – 80x for 0 ≤ x ≤ 2 on the line y = - Find the temperatures at 9 interior grid points. One iteration of Gauss-Seidel Method suffices. Use a suitable relaxation factor. (b) Calculate the flux at any interior grid point. Plate Aluminium k= .49.w/m^2
Q4. (a) Explain the concepts of (i) Stability and (ii) Convergence as applied to the numerical solution of ordinary differential equations. (b) Briefly describe (i) Single Step (ii) Multistep Methods, give an example in each case
2
2 (^ a )^0
d T
h T T
dx
