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Four pages - Page one

University of the Witwatersrand
Advanced Techniques in Physics : 2002
Examination : June 2002

  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 Wednesday 8:30 27$^{th}$ 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

Questions 1-3 not included.

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Four pages - Page three

June 2002 Advanced Techniques in Physics

Consider the ordinary differential equation

\frac{d^2y}{dx^2} + k^2(x)y = S(x)

where initial values are specified, and $k^2$ may have either sign.
Discuss the condition (ill- or otherwise) of an attempted numerical integration of this O.D.E. by, for example, a Runge-Kutta method.
Relate the above discussion to the condition of numerical generation of special functions by recurrence relations. As a specific example consider recurrence relation for the cylindrical Bessel functions

C_{n-1}(x) + C_{n+1}(x) = \frac{2n}{x} C_n(x)

$C_0$ and $C_1$ are replaced by $J_0, J_1$ or $Y_0, Y_1$ for the regular or irregular solution respectively. Demonstrate (using the recurrence relation) why forward recurrence is stable for generation of the $Y_n$ but not for generation of the $J_n$ unless $n<x$.
Describe an alternative method for generation of the $J_n$ when $n>x$.

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Four pages - Page four

June 2002 Advanced Techniques in Physics

In discussing computer representation of numbers and arithmetic we can write :-
true stored
$a$ $x = a(1+\delta_1)$
$b$ $y = b(1+\delta_2)$
$a \circ b$ $x \hat\circ y = x \circ y(1+\delta_3)$

The symbol $\circ$ represents any primary operation.

Find the relative error in addition to first order and show that if $a$ and $b$ differ in sign with $a + b \approx 0$ then there is a disasterous loss of accuracy.
Show in addition that error propogation in summing truncated series is minimised by adding terms in ascending order of magnitude.

To be a take home component on Friday 21$^{st}$ June.

Total Question 1-5 : [130 marks]

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Connell 2004-04-15