Sathyabama University 2008 B.E Computer Science and Engineering Applied Numerical Methods - Question Paper

SATHYABAMA UNIVERSITY

(Established under section 3 of UGC Act, 1956)

 

Course & Branch: B.Tech CSE/ECE/EEE/MECH/CHEM/CIVIL (Part Time)

Title of the paper: Applied Numerical Methods

Semester: III Max. Marks: 80

Sub.Code: 6CPT0011 (2006/2007 JAN) Time: 3 Hours

 

PART A (10 x 2 = 20)

Answer All the Questions

 

1. State the principle of Least squares.

2. Define and express each of D and in terms of E.

 

3. Write down the Newtons divided difference formula for interpolation.

 

4. Explain the term numerical differentiation.

 

5. Prove that Newton-Raphsons iterative formula for is

x_{n + 1} = x_n(2 N x_n).

 

6. Solve the equations 3x + 4y = 8, 4x + 3y = 7 by Gauss-Jordan method.

 

7. Use modifieds Eulers method to find y(0.1) form the equation + xy² = 0; y(0) = 2.

 

8. Write down the Milnes predictor-corrector formulae.

 

9. Classify the Partial differential equation

(x + 1) f_xx + 2(x + 2) f_xy + (x + 3) f_yy = 0.

 

10. Derive the finite difference equation corresponding to ²u = 0.

 

PART B (5 x 12 = 60)

Answer All the Questions

 

11. (a) Fit a straight line to the following data by the method of moments:

x	0	1	2	3	4	5	6
y	0.4	0.7	1.1	1.6	1.9	2.3	2.6

 

(b) Fit a curve of the form y = to following data by the method of least squares.

x	2	4	6	8	10
y	8.8	13.7	17.0	18.9	20.4

(or)

12. (a) Fit a curve of the form y = a x^b + c to the following data by the method of group averages:

x	250	500	900	1200	1600	2000
y	0.25	0.38	0.80	1.38	2.56	4.10

 

(b) Express the operator D in series of ascending powers of (i) D and (ii) .

 

13. (a) Find the value of y at x = 6 using Newtons backward interpolation formula:

 

(b) The velocity v of a particle at a distances from a point on its linear path is given in the following data. Estimate the time taken by the particle to traverse the distance of 20 meters, using Simpsons one-third rule.

s	0	2.5	5.0	7.5	10.0	12.5	15.0	17.5	20.0
v	16	19	21	22	20	17	13	11	9

(or)

14. (a) Apply Lagranges interpolation formula to find f(6), if

f(1) = 2, f(2) = 4, f(3) = 8, f(4) = 16 and f(7) = 128.

 

(b)Solve the equation y_n+2 = 5y_n+1 + 6y_n = n² + n + 1.

 

15. (a) Solve the equation x³ + 3x² 4 = 0, using Graeffes root squaring method, squaring 4 times

x	1.5	2.0	2.5	3.0	3.5	4.0
y	3.375	7.000	13.625	24.000	38.875	59.000

 

(b) Solve the following system of equations by Gauss-Seidel iteration method x + y + 54z = 110, 27x + 6y z = 85,

6x + 15y + 2z = 72.

(or)

16. Solve the following system of equations by gauss-seidal iteration method: x + y + 54z = 110, 27x + 6y z = 85,

6x + 15y + 2z = 72.

 

17. Using Taylor series method of the fourth order, find y at x = 1.1, 1.2, 1.3 by solving the equation = x² + y²; y(1) = 2. Also find y(1.4) using Adams-Bashforths predictor-corrector formulae.

(or)

18. (a) Find the values of y(1.2) and y(1.4), using improved Eulers method with h = 0.2, given that = x³ +; y(1) = 0.5.

 

(b) Solve the equation = ; y(0) = 1 for y(0.1) using Runge-Kutta method of the fourth order.

 

19. Given that u(x, y) satisfies the equation ²u = 0 and the boundary conditions u(x,0) = 0, u(x, 4) = 8 + 2x, u(0, y) = and u(4, y) = y², Find the values of u(i, j), i = 1, 2, 3; j = 1, 2, 3 correct to 2 decimal places, by Liebmanns iteration method.

(or)

20. Solve, by Crank-Nicolsons method, the equation 0 < x < 4, t > 0, satisfying conditions u(0, t) = 0, u(4, t) = 0 and . Compare for two time-steps with h = 1 and a convenient value of k.