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Four pages  Page one
University of the Witwatersrand
Advanced Techniques in Physics : 2003
Examination : June 2003

Instructions
 There are six questions in total, grouped into two sections.
The two sections must be answered in separate books.
 Section 1 comprises of questions 1, 2 and 3.
 Section 2 comprises of questions 4 and 5.
Section two also contains a take  home question. This will be question 6
which does not appear in this paper and will not be answered in the answer book.
You will receive question 6 on Friday 8:30 20 June. There will be
24 hours to answer question 6.
 Answer all questions, 1 to 5. Questions can be attempted in any order.
Start each new question on a new page.
Time: 
Questions 1 to 5 
2 hours 
(exam conditions) 



Total Marks (15) = 130 

Question 6 
One day 
(take home conditions) 



Total Marks (6) = 70 
 1.
 Section 1 Question 1
 a)
 Section 1 Question 1a
(10)
 b)
 Section 1 Question 1b
(10)
 c)
 Section 1 Question 1c
(10)
Total for Question 1 [30]
2/Page two...
Four pages  Page two
PHYS400
PHYSICS HONOURS
June 2003 Advanced Techniques in Physics
 2.
 Section 1 Question 2
Total for Question 2 [30]
 3.
 Section 1 Question 3
Total for Question 3 [40]
3/Page three...
Four pages  Page three
PHYS400
PHYSICS HONOURS
June 2003 Advanced Techniques in Physics
 4.
 With respect to Gaussian Quadratures,
 a)
 Derive the result
where
is a polynomial interpolating the integrand .
(10)
 b)
 Discuss the order of Gaussian quadratures w.r.t.
the NewtonCotes formulae.
(5)
 c)
 Discuss the relationship between higher order and higher
accuracy. How may these considerations be accomodated
in the scheme of Gaussian quadratures ?
(5)
Total for Question 4 [20]
4/Page four...
Four pages  Page four
PHYS400
PHYSICS HONOURS
June 2003 Advanced Techniques in Physics
 5.
 Finite difference representations discretising
P.D.E. operators are the starting point for numerical
methods of solution.
 (a)
 Write down the finite difference formula for
the Laplacian for a scalar function of two variables.
Use the Taylor expansion to demonstrate explicitly the
largest error term and its order.
(5)
 (b)
 If and are approximate solutions
corresponding
to mesh sizes and respectively, then show
that
is an
improved solution.
(5)
Total for Question 5 [10]
 6.
 To be a take home component on Friday 20 June.
Total for Question 6 [70]
Total Question 15 : [130 marks]
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