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

Instructions

1. There are six questions in total, grouped into two sections. The two sections must be answered in separate books.
2. Section 1 comprises of questions 1, 2 and 3.
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$^{th}$ June. There will be 24 hours to answer question 6.
4. 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$\frac{1}{2}$ hours	(exam conditions)
			Total Marks (1-5) = 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]

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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]

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Four pages - Page three

PHYS400
PHYSICS HONOURS
June 2003 Advanced Techniques in Physics

4.

With respect to Gaussian Quadratures,

a)

Derive the result

$$\int_a^b f(x)\,dx = \sum_{k=0}^{m} a_k \overline f(x_k)$$

where $\overline f(x)$ is a polynomial interpolating the integrand $f(x)$. (10)

b)

Discuss the order of Gaussian quadratures w.r.t. the Newton-Cotes 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]

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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 $\phi_1$ and $\phi_2$ are approximate solutions corresponding to mesh sizes $h$ and $h/j$ respectively, then show that $(j^2\phi_2-\phi_1)/(j^2 - 1)$ is an improved solution. (5)

Total for Question 5 [10]

6.

To be a take home component on Friday 20$^{th}$ June.

Total for Question 6 [70]

Total Question 1-5 : [130 marks]