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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2010/
Module Title: Computing & Numerical Methods 1 (CA)
Module Code: MATH
School: School of Mechanical & Process Engineering
Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering
Bachelor of Engineering (Honours) in Chemical & Biopharmaceutical
Engineering
Bachelor of Engineering (Honours) in Structural Engineering
Programme Code: EMECH_8_Y
CSTRU_8_Y
ECPEN_8_Y
External Examiner(s): Dr. P. Robinson
Internal Examiner(s): Dr. R. Sheehy, Dr. P. Robinson
Instructions: Answer ALL questions in Section A.
Section B – Answer any two parts of TWO questions.
All questions carry equal marks.
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.
SECTION A
ANSWER ALL PARTS.
Q1. Describe any two of the following methods for obtaining roots:
Bisection, Newton Raphson, False – Position
Explain the terms convergence and stability as applied to numerical
Methods for obtaining roots and show that;
'
2
f x f x
f x ^ 1 is a necessary condition for convergence of Newton-Raphson Method (x near the
root)
Use root finding techniques to estimate
3 3 .Two iterations suffices
Q2. Describe the Gauss Seidel Method for solving a system of Linear Equations.
Briefly describe the main pitfalls in using Gauss Elimination Method and list techniques for
improving the solution.
Illustrate using a suitable example an ill – conditioned system.
Q3. (a.) Briefly describe the rationale behind:-
(i) Newton Cotes Integration formulae and
(ii) Gauss Quadrature.
.
(b) Use two point Gauss Quadrature to evaluate the
Integral of f ( x ) =
2 x between the limits x = 0 and x = 1
Compare result with the exact solution.
(c) Use central difference formulae of 0(h
2 ) to estimate
the first and second derivative of f ( x ) =
3 x at x= 0.5 step size h =0.
Use Richardson’s extrapolation to obtain 0(h
4 ) estimate of the first derivative at x = 0.
Q4. (a) Briefly describe the terms:
(i) Interpolation
(ii) Extrapolation.
(b) The points (1, 0), (4, 1.386), (6, 1.792) lie on the curve f(x) = ln( ) x
Fit a 2nd order interpolating polynomial to the data and use it to estimate
ln(2)
Q3. ( a) Briefly explain explicit and implicit finite difference methods in the solution of partial
differential equations
Use either an explicit or implicit method to obtain a solution to heat conduction equation
2
2 2
x
T
c t
T
in a thin rod of length 10cm.At time t=0 the temperature T = 0 and boundary
conditions are fixed at all times at T (0,t)=100°C and T(10,t) = 50°C
Note: the rod is aluminum with
2 c =.835cm ²/s h=2 cm
(b) Given that heat flow in a uniform rod is governed by the equation
2
2 2
x
T
c t
T
Where T(x, t) =Temperature
Show that the solution for a rod of length L whose ends are kept at 0°C is given by:-
T(x, t) =
2
1
sin n^
t n n
n x B L
L
cn n
2
If the rod has boundary conditions 0
x
T
at x=0 and x= L show the solution is:-
T(x, t) =
2
1
cos n^
t n n
n x A L
Given that the initial temperature T = f(x) at t=0 show how Bn and An can be computed
(c) Given that the solution for a rod of length L whose ends are kept at 0°C is given by:-
T(x, t) =
2
1
sin n^
t n n
n x B L
L
cn n
2
Extend the solution to solve heat flow in a rod with fixed boundary temperature’s T (0, t) = T 1 and
T(L, t) = 2 T .Show the solution is given by:
T(x, t) = T 1 (x) +
2
1
sin n^
t n n
n x B L
L
cn n
2 where T 1 (x) = 1
(T 2 - T 1 )
x T L
and
Bn = dx L
n x f x T x L
L
( ( ) ( )) sin
0
^1
