Final Examination 2009 1

Sulaimani University College of Sciences Physics Department

Final Examination 2008-2009 First trial

Numerical Analysis Third Stage Time: 3 hours

Q1/ (a) Write the advantage and disadvantage of Bisection method. (b) Explain the modification of Newton-Raphson method in the case of repeated roots, and in the case of two nearly roots. (7 mark)

Q2/ (a) Find the approximate solution of the function 2 Use Aitken’s method where 0.001. (b) Define the shift operator , Averaging operator show that: 3 3 / / /

2 cos

0

, and centering difference operator

. Then

/

(7 mark) / Q3/ Let Apply Lagrang’s formula to find an approximated polynomial based on: 1, 0, 1, 3

(6 mark)

Q4/ The relative density

of air was measured at various altitudes ; the results were: 0 1.5 3 4.5 6 7.5 9 1 0.86 0.74 0.63 0.53 0.43 0.36 Use a quadratic least-squares fit to determine the relative air density at 10 . (6 mark)

Q5/ Compare the error associated with Simpson 1/3, and Simpson 3/8, and use the best one to find: 4

,

6 (6 mark)

Q6/ Let

,

ln 3

4

Using five point formula with

2

15 estimate values of 0.1

,

,

,

(6 mark)

Q7/ Use the Runge-Kutta method to solve the initial value problem in the region 0.1 0.1 0.3, with initial condition 0 2. (6 mark)

Q8/ Solve the Laplace equation

0 for the following configuration, using finite deference method:

0V (6 mark) V2

V1

100 V

0V V3

V4

With best wishes 0V

Omed Ghareb