07a1bs09 Numerical Methods

Set No. 1

Code No: 07A1BS09

I B.Tech Regular Examinations, May/Jun 2008 NUMERICAL METHODS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Find the real root of the equation x3 -x-1=0 using bisection method. (b) Find the root of the equation xex -cosx=0 correct to three decimal places by regula falsi method . [6+10] 2. (a) If y=(3x+1)(3x+4)....(3x+22), prove that ∆4 y =136080(3x+13)(3x+16)(3x+19) (3x +22). (b) Prove that: i. ∇ = 1 − (1 − ∇)−1 . ii. (1 + ∆)(1 − ∇) = 1.

[8+4+4]

3. (a) Fit a straight line y=a+bx to the following data. Also estimate the value of y at x=70. x 71 68 73 69 67 65 66 64 y 69 72 70 70 68 67 68 64 (b) Obtain the least square polynomial approximation of degree one and two for 1 f(x) =x 2 on [0,1] [8+8] 4. (a) Is a spline of the form S(x)=

∞ P

ci Bik (x) uniquely determined by a finite set

i=−∞

of interpolation conditions S(ti ) = yi (0 ≤ i ≤ n)?Why or why not? R∞ xt (b) Find the Fourier transform of e−|x| and hence evaluate cos dt. 1+t2

[8+8]

0

5. (a) Fit the Cubic spline for x 0 1 2 y 1 2 5 Hence find f(0.75) and f(1.75). (b) Dividing the range into 10 equal parts, find the approximate value of

Rπ

sin x dx

0

by i. Trapezoidal rule ii. Simpson’s 1/3 rule.

[8+8]

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Set No. 1

Code No: 07A1BS09

6. (a) If a, b, c are the distinct non-zero numbers, Show rhat the homogeneous system      0 x a b c  a2 b2 c2   y  =  0  has no non trival solution. 0 z a3 b 3 c 3 (b) Using Gauss-Jordan method solve. 2x-3y+z=-1 x+4y+5z=25 3x-4y+z=2.

[8+8]

dy 7. (a) Find y(0.4) using Adam’s method if y(x) is the solution of dx = xy + y 2 given y(0)=1. Use RK fourth order to find y(0.1), y(0.2), y(0.3).

(b) By using Euler’s method compute y′ = 2xy with y(0) = 1.

[8+8]

8. (a) Solve the Laplace’s equation uxx + uyy = 0 for square mesh given u=0 on the 4 boundaries dividing the square into 16 sub-squares of length 1 unit. (b) Solve ∇2 u = 0 (the two dimensional heat conduction equation in steady-state) at the interior lattice points, given boundary values as follows. As shown in Figure 8b. [8+8]

Figure 8b ⋆⋆⋆⋆⋆

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Set No. 2

Code No: 07A1BS09

I B.Tech Regular Examinations, May/Jun 2008 NUMERICAL METHODS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Find the root of the equation ex =cotx correct to three decimal places by (a) Bisection method. (b) Method of false position.

[8+8]

2. Given the following values in the table, x √ y= x

150 152 154 156 12.247 12.329 12.410 12.490

Evaluate √ (a) 155 and √ (b) 151, using Lagrange’s interpolation formula.

[16]

3. (a) Fit a parabola to the data: x 0.5 1 2 4 8 12 y 160 120 94 75 62 56 (b) Fit a straight line to the data below: x 19 25 30 36 40 45 50 y 76 77 79 80 82 83 85

[8+8]

4. Show that the class of all spline functions of degree m have knots x0 , x1 , ..., xn includes the class of polynomials of degree m. [16] 5. (a) Evaluate

R5

dx 4x+5

by Simpson’s one- third rule and hence find the value of

0

.loge 5 (n = 10). R1 dx (b) Compute 1+x 2 by using Trapezoidal rule, taking h=0.5 and h=0.25. Com0

pare with exact integration.

[8+8]

6. (a) If a, b, c are the distinct non-zero numbers, Show rhat the homogeneous system      0 x a b c  a2 b2 c2   y  =  0  has no non trival solution. z a3 b 3 c 3 0 1 of 2

Set No. 2

Code No: 07A1BS09 (b) Using Gauss-Jordan method solve. 2x-3y+z=-1 x+4y+5z=25 3x-4y+z=2.

[8+8]

dy 7. (a) Solve dx = y − 2x , y(0) = 1, y(0.1) = 1.0954, y(0.2) = 1.1832, y (0.3) = 1.2649, y find y (0.4) by Milne’s method.

(b) Given 10y′ = x2 +y2 , given y(0)=1 for x = 0.1(0.1) 0.3 using R.K. method of fourth order. [8+8] 8. (a) Solve: ∇2 u = 0 for the square mesh. As shown in Figure 8a.

Figure 8a (b) Derive standard five point formula to solve Laplace equation, stating the assumptions you make. [8+8] ⋆⋆⋆⋆⋆

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Set No. 3

Code No: 07A1BS09

I B.Tech Regular Examinations, May/Jun 2008 NUMERICAL METHODS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Solve the equation x2 − logex = 1.2, correct to three decimal places by (a) Bisection method. (b) Method of false position.

[8+8]

2. (a) Find the cubic polynomial which takes the following values y(0)=1, y(1)=0, y(2)=1 and y(3)=10. Also obtain y(4). (b) If the interval of differencing is unity prove that y3 = y2 + ∆y1 + ∆2 y0 + ∆3 y0 . [8+8] 3. (a) Find the equation of a parabola of form y=ax2 +b for the data using method of least squares: x -1 0 1 y 3.1 0.9 2.9 (b) The distance from a point (x0 , y0 ) to a line ax+by=c is (ax0 +by0 −c)((a2 +b2 )−1/2 . Determine a straight line that fits a table of data points(xi , yi ), for 0 ≤ i ≤ m, in such a way that the sum of the squares of the distances from the points to the line is minimized. [6+10] 4. (a) Show that the functions fn(x)=cosnx are generated by this recursive definition: f0 (x) = 1, f1 (x) = cos x fn+1 (x) = 2f1 (x)fn (x) − fn−1 (x)(n ≥ 1). (b) Find Fourier transform of f (x) = 1 − x2 , |x| < 1 =0 elsewhere. 5. (a) Evaluate

R6 0

x2 dx 1+x4

[8+8]

by

i. Weddle’s rule ii. Boole’srule. (b) The velocity v of a particle at distance s from a point on its path is given by the following table: S(ft) 0 10 20 30 40 50 60 V[ft/s] 55 68 77 65 69 42 34 Estimate the time taken to travel 60ft using Simpson’s 1/3 rule, compare the result with Simpson’s 3/8 rule. [8+8] 1 of 2

Set No. 3

Code No: 07A1BS09

    8 x 1 2 1 6. (a) Solve the system  2 3 1   y  =  13  by LU decomposition method. 5 z 1 1 0 

(b) Using Gauss-Jordan method solve. 2x+y+z=10 3x+2y+3z=18 x+4y+9z=16. 7. (a) Using Milne’s method find y(2) if y(x) is the solution of y(0) = 2, y(0.5) = 2.636 ,y(1) = 3.595, y(1.5) = 4.968. (b) Solve y′ = 1+xy, y(0) by modified Euler’s method.

[8+8] dy dx

=

1 2

(x + y) given [8+8]

8. (a) Solve the elliptic equation uxx +uyy = 0 for the following square mesh with boundary values as shown in Figure 8a. Iterate until the maximum difference between successive values at any point is less than .005.

Figure 8a (b) Solve: ∇ u = 0 in the square region bounded by x = 0, x = 4, y = 0, y = 4 and with boundary conditions u (0, y) = 0, u (4, y) = 12+ y, u (x, 0) = 5 , u (x, 4) = 6 by taking h = k = 1. solve by Jacobi’s method. [8+8] 2

⋆⋆⋆⋆⋆

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Set No. 4

Code No: 07A1BS09

I B.Tech Regular Examinations, May/Jun 2008 NUMERICAL METHODS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Find the root of the equation ex =cotx correct to three decimal places by (a) Bisection method. (b) Method of false position.

[8+8]

2. (a) Find the seventh term of the sequence 2,9,28,65,126,217 and also find the general term. (b) √ If

√ √ √ 12500 = 111.803399, 12510 = 111.848111, 12520 = 111.892805, 12530 = √ 111.937483 find 12516 by Gauss’s backward formula. [6+10]

3. (a) Fit a straight line to the following data: x 0.0 0.2 0.4 0.6 0.8 1.0 y -1.85 -1.20 -0.55 0.15 0.80 1.35 (b) Fit the least square approximation of second degree for the discrete data below: x -2 -1 f(x) 15 1

0 1 2 1 3 19

[8+8]

4. Prove that every monomial xr ,0 ≤ r ≤ k, can be written as a sum of multiple of the Bik , with k fixed. [16] 5. (a) Fit the Cubic spline for x 0 y -5

1 -4

2 3

Hence find y(0.5) and y ′ (1). R3 (b) Find the value of (ln x + sin 3x + e−x )dx using Trapezoidal rule and Wed0

dle’s rule. Compare your result by integration.

[8+8]

6. (a) Using Gauss-Jordan method solve 10x − 2y + 3z = 23, 2x + 10y − 5z = −33, 3x − 4y + 10z = 41 (b) Solve the system of equations by Gauss elimination method 4x + 2y + z = 14, x + 5y − z = 10, x + y + 8z = 20 1 of 2

[8+8]

Set No. 4

Code No: 07A1BS09

7. (a) Using Euler’s method find y (0.2) given dy/dx = log(x + y) and y (0) = 1, h = 0.2. (b) Solve by Taylor series method dy/dx = y + x3 for x = 1.1, 1.2 given y (1) = 1. [8+8] 8. (a) Solve: ∇2 u = 0 in the square region bounded by x = 0, x = 4, y = 0, y = 4 and with boundary conditions u (0, y) = 0, u (4, y) = 4 + y, u (x, 0) = (x/2), u (x, 4) = 2x by taking h = k = 1. solve by Jacobi’s method. (b) Solve ∇2 u = 0 in the region shown below: As shown in Figure 8b.

Figure 8b ⋆⋆⋆⋆⋆

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[8+8]