INSTITUTE OF AERONAUTICAL ENGINEERING
(AUTONOMOUS) Code No: BST003
M.Tech- I Semester Regular Examinations, February 2017
COMPUTER ORIENTED NUMERICAL METHODS
(Structural Engineering)
Time: 3 hours Max. Marks: 70
Answer ONE Question from each Unit All Questions Carry Equal Marks
All parts of the question must be answered in one place only
UNIT-I
1. (a) Solve the following system of linear equations by Gauss-Seidel method
1 2 3 4 1 2 3 4
1 2 3 4 1 2 3 4
2 3 4 3, 2 5 2,
5 3 4 1,10 2 2 4
x x x x x x x x
x x x x x x x x
[7M]
(b) Compute the largest Eigen value and its corresponding Eigen vector for the matrix
1 2
A 3 4 using power method [7M]
2. (a) Find the largest Eigen value and largest Eigen vector for the matrix A using inverse power method when one Eigen value of A is 4, if
0 11 5
2 17 7 4 26 10 A
[7M]
(b) Determine X from AX=B by Cholesky’s factorization method
1 2 3 2 5 8 3 8 22
A
[7M]
UNIT-II
3. (a) The following table contains the values of yf x( ).For what value of x does y equal 1
2
X 0.45 0.46 0.47 0.48 0.49 0.50
y 0.4754818 0.4846555 0.4937452 0.5027498 0.5116683 0.5204999
[7M]
MODEL QUESTION PAPER - I
(b) Find the cubic spline that passes through the data points (0,1),(1,-2),(2,1) and (3,16) with first derivative boundary conditionsy(0) 4 & (3) 23y
[7M]
4. (a) Construct the natural cubic spline for the data (0,-4),(1,-3),(1.5,-0.25) and (2,4). [7 M]
(b) Calculate y(1.01), y(1.12) and y(1.28) from the following data
X 1.00 1.05 1.10 1.15 1.20 1.25 1.30
y 1.0000 1.02470 1.04881 1.07238 1.09544 1.11803 1.14017
[7 M]
UNIT-III
5. (a) State three-point formula for numerical differentiation. How is it different from the two point formula? Illustrate the differences using geometric interpretations.
[7M]
(b) Calculate y(1), (1.03)y for the function y=f(x) given in the table
x 0.96 0.98 1.00 1.02 1.04
y(x) 1.8025 1.7939 1.7851 1.7763 1.7673
[7M]
6. (a) Find the values of 1.1 and 1.2 for the initial value problem x2y2on [0,3]using
Taylor series with y(1)=2.3.
[7M]
(b) Find the derivative of exsin( )x at the point x=1.0 using Richardson extrapolation starting with h=0.5 and continuing using step = 0.0001
[7M]
UNIT-IV
7. (a) Compute the first and second order derivatives of the function y=f(x) at x=1 and x=3 by Newton’s forward differences formula using the following table
x 1 2 3 4 5 6
y(x) 3.9183 4.5210 5.2552 6.1530 7.2496 8.5891
[7M]
(b)
Evaluate
1 1 0 0
I ex ydxdy using the trapezoidal and Simpson’s rule and compare with the exact value. Consider h=0.5, k=0.5
[7M]
8. (a) A rocket is launched from the ground vertically upwards. Its acceleration a is
registered during the first 80 seconds and is given in the table as follows
x 0 10 20 30 40 50 60 70 80
y(x) 30 31 33.4 35.4 37.7 40.3 43.2 46.6 50.6
[7M]
(b) Compute the following multiple integration by Simpson’s method
1 1 0 0
1 1 dxdy
xy
[7M]
UNIT-V
9. (a) Find an approximate solution of the differential equationy x y y, (0) 1 by using Backward Euler method.
[7M]
(b) Solve the ordinary differential equation y x2 y y, (0) 1 using Euler’s method.
Estimate y (0.2) for h=0.1, h=0.2
[7M]
10. (a)
Solve the two point boundary value problem
2
2 4
(1 ) y y
x subject to the conditions
(0) 0
y andy(1) 2 (1) 0 y . [7M]
(b) Solve the boundary value problem y64y10 0, (0) y y(1) 0 compute y(0.5) .
[7M]
