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Posted By: rama Member Level: Silver Posted Date: 16 Jun 2008
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2008 Anna University B.E Electrical and Electronics MA1251-NUMERICAL METHODS Question paper
B.E/B.Tech DEGREE EXAMINATION, APRIL/MAY 2008
Fourth Semester
(Regulation 2004)
Civil Engineering
MA1251-NUMERICAL METHODS
(Common to Aeronautical Engineering, Electrical and Electronics Engineering, Mechatronics Engineering /Metallurgical Engineering and Petroleum Engineering)
(Common to B.E (Part-time) Third Semester Regulation 2005)
Time:Three hours Maximum:100 marks
Answer ALL questions
PART A –(10*2=20marks)
1. Find an iterative formula for finding vN where N is a real number, using
Newton-Raphson formula.
2. State the conditions for the convergence of Gauss-Seidel iterative method
for solving system of equations.
3. Obtain a divided difference table for the following data:
x: 5 7 11 13 17
y: 150 392 1452 2366 5202
4. Write the Newton`s forward difference interpolation formula.
5. State Newton’s backward difference formula to find ( dy/dx)x=xn and
(d²y/dx²)x=xn.
6. State two point Gaussian quadrature formula.
7. Find y(1.1) , using Euler’s method, from dy/dx = x²+ y², y(1)=1.
8. State Adam’s predictor and corrector formula for solving initial value
problem.
9. State finite difference approximation for d²y/dx² and state the order of
truncation error.
10. State explicit finite difference scheme for one dimensional wave equation
d² u/d t² = a² d²u/dx².
PART B(5*16=80 marks)
11. (a)(i) Find a real root of the equation x³-2x-5=0 by the method of false position
correct to three decimal places.
(ii) Apply Gauss-Seidel iterative method, solve the system of equations:
20x+y-2z = 17
3x+20y-z = -18
2x-3y+20z = 25
Or 8 -4 0
(b)(i) Using Gauss-Jordan method, find the inverse of the matrix -4 8 -4
0 -4 8
(ii) Find numerically largest eigen value and the corresponding eigen vector
1 6 1
of A= 1 2 0 .
0 0 3
12. (a)(i) The following values of x and y are given:
x: 1 2 3 4
y: 1 2 5 11
Find the cubic splines and evaluate y(1.5).
Or
(b)(i) Use Lagrange’s formula to fit a polynomial to the data:
x: - 1 0 2 3
y: -8 3 1 12
and hence find y at x=1.
(ii) The following data are taken from the steam table:
Temp °C : 140 150 160 170 180
Pressure Kgf/cm² : 3.685 4.854 6.302 8.076 10.225
Find the pressure at temperature t=142° and t=175°.
13. (a)(i) Find f’(4) and f’’(4) from the table:
x: 0 2 3 5
y: 8 6 20 108 1
(ii) Use Romberg’s method to compute ? dx / 1+ x² by taking h=0.5 and
0
0.25.
Or
(b)(i) Given that
x: 1.1 1.2 1.3 1.4 1.5
y: 8.403 8.781 9.129 9.451 9.75
Find dy/dx and d²y/dx² at x=1.1.
1
(ii) Evaluate ? d t / 1 + t by Gaussian formula with three points.
0
14. (a) Apply Runge-Kutta method to find approximate value of y for x=0.2 in
steps of 0.1 if d y/ d x = x+y² given that y=1 when x=0.
Or
(b)(i) Find by Taylor’s series method, the values of y at x=0.1 and x=0.2, to
four decimal places from dy/dx = x²y-1, y(0)=1.
(ii) Given dy/dx=xy+y²,y(0)=1,y(0.1)=1.1169,y(0.2)=1.2773,y(0.3)=1.5049,
evaluate y(0.4) by using Milne’s method.
15. (a)(i) Solve the equation ?² u=-10 (x²+y²+10) over the square with sides
x=0,y=0,x=3 and y=3 with u=0 on the boundary and mesh length=1.
(ii) State implicit finite difference scheme for one dimensional heat equation.
Or
(b)(i) Find the values of u(x,t) satisfying the parabolic equation du/dt=4d²u/dx²
and the boundary conditions u(0,t)=0,u(8,t)=0 and u(x,0)4x-x²/2 at the
points x=i , i =0,1,2,…7 and t=1/8 j , j=0,1,2,3.
(ii) Solve the equation y’’=x+y with conditions y(0)=y(1)=0, by finite
difference method , taking h=0.25.
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