MA1251 -NUMERICAL METHODS Questions Bank

Anna University, Chenna

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SRIIVASAN ENGINEERING COLLEGE MA1251 -NUMERICAL METHODS UNIT - I

SOLUTIONS OF EQUATIONS AND EIGEN VALUE PROBLEMS

PART – B

1. (a) Solve 3x – cos x – 1 = 0 by Newton’s method

(b) By using Gauss Seidel method. Solve the following system of equation. x + y +54z = 110, 27x + 6y – z = 85, 6x +15y + 2z = 72.

2. (a) Solve

x3 - 4x +1 = 0

by using Regula falsi method.

(b) Solve x + 3y + 3z = 16 ; x + 4y + 3z = 18 ; x + 3y + 4z = 19 by Gauss elimination method.

3. (a) Using the Gauss Jacobi method to solve the following equations.

10x + y + z = 12, 2x + 10y + z = 13, x +y + 5z = 7.

clip_image001(b) Find a positive root of 3x -

1+ sin x

= 0 by fixed point method.

4. (a) Using the Gauss Jordan method to solve the following equations.

10x + y + z = 12, x + 10y - z = 10, x -2y + 10z = 9. (b) Solve x = cos x by Newton- Raphson method.

é 1 1 1 ù

5. (a) Find the inverse of A =

ê 0 1

-2ú

by Gauss Jordan method.

ê ú

êë-1 1 1 úû

(b) Solve x + y + 2z = 4 ; 3x + y - 3z = - 4 ; 2x – 3y – 5z = - 5 by Gauss elimination method.

6. (a) Find a positive root of 2x = 3 + cos x by fixed point method.

(b) By using Gauss seidel method. Solve the following system of equation.

20x – y -2z = 17, 3x + 20y – z = -18, 2x – 3y + 20z = 25.

7. (a) Using the Gauss Jordan method to solve the following equations

2x – 6y + 8z = 24, 5x + 4y - 3z = 2, 3x + y + 2z = 16.

(b) Solve

xex = 2 by using Regula falsi method.

8. (a) Solve 4x + 2y + z = 14; x + 5y - z = 10; x + y + 8z = 20 by Gauss Jacobi method.

é3 1 2 ù

(b) Find the inverse of A = ê2

-3 -1ú

by Gauss Jordan method.

ê ú

êë1 2 1 úû

é1 -3 2 ù

9. Find the all Eigen values and eigen vectors of A =

ê4 4

-1ú

by Power method of iteration.

10. Find the all Eigen values and eigen vectors of A =

ê ú

êë6 3 5 úû

é25 1 2 ù

ê ú

ê 1 3 0 ú

by Power method of iteration.

2 0 -4

PART – B

1. The population of a town is shown below. By using corresponding interpolation

Find the increase in the population b/w 1916 and 1948

Year

1911

1921

1931

1941

1951

1961

population

12

13

20

27

39

52

2. From the table find the pressure at t=142o and t=175o

Tempo C

140

150

160

170

180

Pressure

3.685

4.854

6.302

8.076

10.225

3. From the data find the number of students whose weight is between 60 to 70

Weight

0-40

40-60

60-80

80-100

100-120

No. of students

250

120

100

70

50

4. (a) Estimate

e-1.9

from the data.

x

1.00

1.25

1.50

1.75

2.00

e- x

0.3679

0.2865

0.2231

0.1738

0.1353

(b) Using Lagrange’s interpolation formula find y at x = 3

X

0

1

2

4

5

Y

0

16

48

88

0

5. (a) Using Lagrange’s interpolation formula

P.T y1 = y3 – 0.3(y5 – y-3) + 0.2(y-3 – y-5)

(b) Construct a polynomial for the data and hence find y(5) given below.

X

4

6

8

10

Y

1

3

8

16

6. (a) Construct a polynomial using Lagrange’s method , hence find f(2.5), f(3.8)

X

0

1

3

4

Y

-12

0

6

12

(b) Express

3x2 + x + 1

clip_image009( x - 1)( x - 2)( x - 3)

as a sum of partial fraction using Lagrange’s

Interpolation formula.

7. (a) Find a polynomial of degree two for the data

X

0

1

2

3

4

5

6

7

Y

1

2

4

7

11

16

22

29

(b)Using Newton’s divided difference formula find y at x = 5

X

0

1

3

6

Y

1

4

88

1309

¢¢ ¢¢

8. Fit a cubic spline for the giving data given y0

= y2

= 0 and hence find f(0.75), f(1.75).

X

0

1

2

Y

1

2

3

¢¢ ¢¢

9. Fit a cubic spline for the giving data given y0

= y2

= 0 and hence find y(1.5), y’(1).

X

1

2

3

Y

-8

-1

18

10. Using Newton’s divided difference formula, find the values of f(2), f(8) given below.

X

4

5

7

10

11

13

Y

48

100

294

900

1210

2028

PART –B

1. (a) Find the first, second, third derivatives of the function tabulated below at x = 1.5 and x = 4.

X

1.5

2

2.5

3

3.5

4

Y

3.375

7

13.625

24

38.875

59

(b) Find the value of cos (1.74 ) from the following table

x

1.7

1.74

1.78

1.82

1.86

sinx

0.9916

09857

09781

09691

0.9584

2. (a) From the following data find

f ¢(5),

f ¢¢(5)

& f ¢¢¢ (5)

X

0

2

3

4

7

9

F(x)

4

26

58

112

466

922

3 dt

clip_image020(b) Using Gaussian two point and three point formula evaluate ò

2 1 + t

3. The table below gives velocity of a moving particle at time t seconds. Find the distance covered by a particle in 12 seconds and also acceleration at t = 2 seconds.

t

0

2

4

6

8

10

12

v

4

6

16

34

60

94

136

4. (a) Find the first and second derivative of the function tabulated below at x = 0.6.

x

0.4

0.5

0.6

0.7

0.8

y

1.5836

1.7974

2.0442

2.3275

2.6511

(b) Using Gaussian two point formula evaluate

1 2

p 2

clip_image021ò log(1 + x)dx

0

5. (a) Find the value of log 2 1/3 from

x dx

ò 3

by using Simpson’s 1/3 rule, h = 0.25

(b) Evaluate

1.4 2.4

ò ò

1 2

clip_image0221 dxdy xy

0 1 + x

by using Simpson’s rule.

6 dx

clip_image0236. Evaluate ò 2

0 1 + x

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

iii) Simpson’s3/8 rule iv) Actual integration

2

clip_image024

2

7. Evaluate ò

dx ( n= 8), by i) Trapezoidal rule ii) Simpson’s 1/3 rule

0 1 + x + x

iii) Simpson’s3/8 rule iv) Actual integration

1 dx

clip_image0258. (a) Evaluate ò 2

by using Romerg’s method correct to 4 decimal places.

0 1 + x

Hence deduce an approximate value of Π.

clip_image026

1 3

2 4 1

(b) Evaluate

òò (x + y)2

dxdy

by using Trapezoidal, Simpsons rules with h=k=0.5.

9. (a)Evaluate

2 dx

clip_image027ò by i) Trapezoidal rule ii) Simpson’s 1/3 rule Take h=0.25.

1 x

1 1 1

clip_image028(b) Evaluate òò

dxdy

by using Trapezoidal, Simpsons rules.

0 0 1 + x + y

4

ex dx

10. (a) Using Simpson’s rule find

ò , with h=1

0

12 dx

clip_image029

x

(b) Using Gaussian two point and three point formula evaluate ò

5

PART – B

1. (a) Using Taylor series method , find , correct to four decimal places ,

the value of y(0.1) and y(0.2), given

dy = x2 + y 2 and y(0) = 1

dx

(b) Using Modified Euler and Euler method find y(0.2)

given

dy = y - x2 + 1 y(0) = 0.5

dx

2.(a) Using Taylor series method , find the value of y(0.2) and y(0.4), given

dy =1 - 2 xy and y(0) = 0

dx

(b) Using Modified Euler and Euler method find y(0.2), y(0.1)

given

dy = y 2 + x2 y(0) = 1

dx

clip_image033clip_image034dy y 2 - x2

3. Using R.K Method of fourth order, Solve

given y(0) = 1 find y at x = 0.2 x = 0.4.

=

dx y 2 + x2

4. Using R.K Method of fourth order Solve

given y(0) = 2 find y at x = 0.2 x = 0.4

dy = x3 + y dx

5.Using Milne’s method find y(2) given

y¢ = æ 1 ö ( x + y) given y(0) = 2

2

ç ÷

è ø

y(0.5) = 2.636 y(1) = 3.595 y(1.5) = 4.968.

6. Using Milne’s method find y(4.4) given 5xy¢ + y2 - 2 = 0 given y(4) = 1 y(4.1) = 1.0049 y(4.2) = 1.0097 y(4.3) = 1.0143.

7. Given

y¢= xy , y(0) = 1, y(0.1) = 1.01, y(0.2) = 1.022, y(0.3) = 1.023, find y(0.4) using

2

Adam’s method.

8. Given y‘ = 1 + xy , y(0) = 2, find y(0.4) by using Adam’s method.

9. Given

y¢ = x2 + y , y(0) = 1, find y(0.1) by Taylors method, y(0.2) by modified Euler’s

method, y(0.3) by Runge-kutta method and y(0.4) by Milne’s method.

10. Given

y¢ = x + y 2 , y(0) = 1, find y(0.1) by Taylors method, y(0.2) by modified Euler’s

method, y(0.3) by Runge-kutta method and y(0.4) by Adam’s method.

PART – B

1. Solve y’’ – y = x,

x Î ( 0, 1) , given y(0) = y(1) = 0 using finite differences dividing the

interval into four equal parts.

2. Solve uxx - 2ut =0, given u(0.t)=0, u(4,t)=0 and u(x, 0) = x(4 - x) , taking h=1 find the values of

u up to t=5.

2

3. Given

f = f , f (0, t ) = f (5, t) = 0,

f ( x, 0) = x2 (25 - x2 )

find f in the range taking h=1

x2 ¶t

and up to 5 seconds.

4. Using Crank-Nicholson’s scheme, solve uxx = 16ut , 0<x<1, t>0 given u(x,0)=0, u(0,t)=0,u(1,t)=100t compute u for one step in t direction taking h=1/4

5. Using Crank-Nicholson’s scheme, solve uxx = ut , 0<x<1, t>0 given u(x,0)=0, u(0,t)=0, u(1,t)=t compute u for one step in t direction taking h=1/4

6. Solve Ñ2u = -10( x2 + y2 + 10) over the square mesh with sides x = 0 , y = 0 ,

x = 3 , y = 3 with u = 0 on the boundary and mesh length 1 unit.

7. Solve Uxx + Uyy = 0 over the square mesh of side 4 units satisfying the following boundary

conditions i) u( 0 , y ) = 0 for 0 £ y £ 4 ii) u( 4 , y ) = 12 + y for 0 £ y £ 4

for 0 £ x £ 4 iv) u( x , 4 ) = x2 for 0 £ x £ 4

iii) u( x , 0 ) = 3x

8. Solve numerically, 4uxx = utt with the boundary conditions u(0,t)=0, u(4,t)=0 and the initial

conditions ut ( x, 0) = 0

and u(x, 0) = x(4 - x) , taking h=1 (for 4 time steps).

9. Solve 25uxx = utt for u at the pivotal points given

ì2x

, 0 £ x £ 2.5

u(0, t) = u(5, t) = 0, ut ( x, 0) = 0

for one half period of vibration.

and u( x, 0) = í

î10 - 2x , 2.5 £ x £ 5

10. Solve Ñ2u = 0

at the nodal points for the following square region given the boundary conditions.

clip_image0390 10 20 30

20

40

40

50

60 60 60 60