Sathyabama University 2008 B.Tech Chemical 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
Date: 23-04-2008 Session: FN
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
xn + 1 = xn(2 N xn).
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) fxx + 2(x + 2) fxy + (x + 3) fyy = 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 xb + 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:
x |
1 |
2 |
3 |
4 |
5 |
y |
41.66 |
34.46 |
28.28 |
22.94 |
18.32 |
(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 yn+2 = 5yn+1 + 6yn = 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.
Earning: Approval pending. |