m5 upto nov 2012

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Register Number

SATHYABAMA UNIVERSITY(Established under section 3 of UGC Act,1956)

Course & Branch :B.E/B.Tech - AERO/AUTO/CIVIL/CSE/E&C/ECE/EEE/ETCE/M&P/MECH/BTE/CHEM

Title of the Paper :Applied Numerical Methods Max. Marks:80

Sub. Code :516501-6C0079-SMTX1011 (2007-08-09-10)

Time : 3 Hours

Date :17/11/2012 Session :FN_______________________________________________________________________________________________________________________________

PART - A (10 x 2 = 20)

Answer ALL the Questions

1. Give the normal equations to fit a curve of the formy = ax + bx2

by the method of least square.

2. Prove that 2 with usual notation.3. Give the order of the error in Simpsons one third rule.

4. Write down the Newtons forward interpolation formula.

5. Define transcendental equation.

6. Name two indirect methods to solve the system of linearequations.

7. What are the disadvantages of Taylors series method?

8. Give the modified Eulers method formula to solve the

differential equation.

9. Classify the partial differential equation 02

2

2

2

y

uy

x

ux ,

x > 0, y > 0.10. How many conditions are required to solve the Laplace equation?

PARTB (5 x 12 = 60)


11. (a) Fit a curve of the form y = ax2+bx+c to the following data.

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x 10 20 30 40 50 60 70

y 157 179 210 252 302 361 400

(b) Prove that2

112

22 .

(or)

12. (a) Convert the equationbxa

xy

with a linear form and hence

determine the values of a and b by the method of moments for

the following data.

X 8 10 15 20 30 40

y 13 14 16 18 19 21

(b) Prove that (i)

hD

e

(ii) 41

2

2

13. (a) Find the polynomial that passes through the points (0, 12)

(1,0) (3, 6) (4, 12) and hence findy atx = 2.

(b) Using Trapezoidal rule, find 2

0

sin

dxe x .

(or)

14. (a) Find the first derivative at x = 17 from the following data

x 15 17 19 21 23 25Y 3.873 4.123 4.359 4.583 4.796 5.00

(b) Solve 22

4 nyynn

15. (a) By Newton Raphson method, find the negative root of the

equation 035213 xx .(b) Solve the following system by Crouts method x + y + z = 9,

2x3y + 4z = 13, 3x + 4y + 5z = 40.(or)

16. (a) Find all the roots of the equation x32x25x+6 = 0 by

Graeffes root squaring method.

(b) Solve the system of equations by Gauss-Seidel method

10x5y2z = 3, 4x10y + 3z =3, x + 6y + 10z =3.

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17. (a) Using Taylors series method find y(1.1) given that 31

xydx

dy ,

y(1) = 1.

(b) Given 2yxdx

dy , y(0) = 0, y(0.2) = 0.02, y(0.4) = 0.0795,

y(0.6) = 0.1762. Find y(0.8) using Milnes predictor-Correctormethod.

(or)

18. (a) Using Runge-Kutta method of fourth order find y when x=1.1

given thatx

ydx

dyx

1 , y(1) = 1.01.

(b) Given that )1(2 yxdx

dy , y(1) = 1, y(1.1) = 1.233,

y(1.2)=1.5485, y(1.3) = 1.9789. Find y(1.4) by Adams method.

19. (a) Solve the equation2

2

x

u

t

u

subject to the conditions u(0,t) =

u(5,t) = 0 and u(x,0) = x2(25x2). Take h = 1 and tabulate the

values of u up to t = 3 sec.

(b) Solve2

2

2

2

xu

tu

given that u(0,t) =u(4,t)=0.

2

)4()0,(

xxxu

and ut(x, 0)=0. Take h = 1 and find the solution

up to 5 step in t-direction.

(or)

20. Solve 02 u in the following square region with the boundary

conditions as shown in the figure.200 100

100

200

300

400

400 300

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Register Number


Course & Branch :B.E/B.Tech - AERO/AUTO/CSE/CIVIL/CSE/E&C/ECE/EEE/ETCE/M&P/MECH/AERO/BTE/CHEM


Sub. Code :515501-526501-511505-6C0079 (2006-07-08-09)

Date :23/04/2012 Time : 3 Hours

Session :AN______________________________________________________________________________________________________________________

PART - A (10 x 2 = 20)


1. What is the Principle of least square?

2. Write down the normal equations to be used for finding a and b,

when fitting a straight line y = ax + b by the method of moments.

3. Write the Newtons forward derivative formula forx = x0.

4. What is the order of the error in Simpsons formula?

5. State the order of convergence and convergence condition for

Newtons Rap son method.

6. What is the convergence condition for solving a system of

equations by Gauss-Seidel method?

7. Write Taylors series formula to solvey/= f(x, y) withy(x0) = y0.

8. Explain multi-step method. Give examples.

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9. Write down the standard five point formula to solve the equation

.02

2

2

2

y

u

x

u

10. Name the two methods that are used to solve one dimensionalheat equation.

PARTB (5 x 12 = 60)

Answer All the Questions

11. By the method of least squares, fit a straight line and a parabola

to the data:

x 1 2 3 4

y 0.30 0.64 1.32 5.40(or)

12. (a) Prove that .2

2x

x

xx e

e

Eee

E

(b) Evaluate .)3)(2)(1( xxxx

13. The table below gives the velocity v of a moving particle at time t

seconds. Find the distance covered by the particle in 12 seconds

and also the acceleration at t = 2 seconds

t 0 2 4 6 8 10 12

v 4 6 16 34 60 94 136

(or)

14. (a) Using Simpsons th83 rule evaluate

6

0

21 xdx by dividing the

range into 6 equal parts. (b) Find the value of cos (1.74) from the

following table:

x 1.70 1.74 1.78 1.82 1.86

Sin x 0.9916 0.9857 0.9781 0.9691 0.9584

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15. (a) Solve the equation cos x = 3x1 by Regula Falsi method. (8)

(b) Evaluate 12 to 4 decimal places by Newton Rap sonsmethod. (4)

(or)

16. Solve the following system of equations by Gauss-Seidelmethod 10x5y2z = 3

4x10y + 3z = -3

x + 6y + 10z = -3

17. (a) Solve 1)0(),(log10 yyxdx

dyby Eulers modified

method and find the value of y(0.2) (4)

(b) Using Runge Kutta method of fourth order, find y(0.8) taking

h = 0.1 correct to 4 decimal places if y/ = yx2, y(0.6) = 1.7379.

(8)

(or)

18. Given ,2773.1)2.0(,1169.1)1.0(,1)0(,2 yyyyyx

dx

dy

find y(0.3) by Runge Kutta method of order four and y(0.4)

using Adam Bash forth method.

19. Find the solution 2

2

x

u

t

u

, subject to u(x, 0) = sin x;

0 x 1; u(0, t) = u(1, t) = 0 using Schmidt method.

(or)20. Solve the Poisson equation 2u = -10 (x2 + y2 + 10) over the

square mesh with sidesx = 0, y = 0, x = 3, y = 3 with u = 0 on

the boundary and mesh length of 1 unit.

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Course & Branch :B.E/B.TechCommon to ALL Branches


Sub. Code :6C0079(2006-07-08-09) Time : 3 HoursDate :01/11/2011 Session :FN_______________________________________________________________________________________________________________________________

PART - A (10 x 2 = 20)


1. Write down Newtons forward and backward difference

formulae.

2. Evaluate

10

(1x) (12x) (1 10x) by taking h = 1.

3. Form the divided difference table for

x: 1 3 6 11

y: 4 32 224 1344

4. Using Simpsons rule find.

4

0

4321 6.5409.20,39.7,72.2,1 eandeeeegivendxe ox

5. Why Gauss Seidel iteration is a method of successive

corrections?

6. State the condition for convergences of Jacobis Iteration methodfor solving a System of simultaneous algebraic equations.

7. State the Adams-Bashforth PredictorCorrector formula.

8. Find y(0.1) given y/= 1)0(),(2

1 yyx by modified Eulers method.

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9. Write an explicit formula to solve the wave equations.

10. For what points of x and y, the equation x.fxx + y.fyy = 0, x > 0,

y > 0 is elliptic.

PARTB (5 x 12 = 60)


11. (a) Fit a curve of the form Y = abx to the data.

(b) Find the sixth term of the sequences 8,12,19,29,42.(or)

12. By the method of moments fit a straight line and a parabola to the

following data.

x: 1 2 3 4

y: 1.7 1.8 2.3 3.2

13. (a) Using Lagranges interpolation formula, find y (10) from the

following table:

x: 5 6 9 11

y: 12 13 14 16

(b) Given

x: 1 2 3 4 5 6 7 8

f(x): 1 8 27 64 125 216 343 512

Estimate f(7.5). Use Newtons formula.

(or)

x: 1 2 3 4 5 6

y: 151 100 61 50 20 8

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14. (a) A river is 80 meters wide. The depth d in meters at a

distance x meters from one bank is given by the following

table.

Calculate the area of cross-section of the river using Simpsonsrd

3

1

rule.

x: 0 10 20 30 40 50 60 70 80

d: 0 4 7 9 12 15 14 8 3

(b) Evaluate 1

0

21 x

dxusing Trapezoidal rule with h = 0.2. Hence

determine the value of.

15. (a) Using Gauss Jordan method, solve the following system of

equations.

2xy + 3z = 8

-x + 2y + z = 4

3x + y4z = 0

(b) Solve for a positive root of x cos x = 0 by Regula Falsi

method.(or)

16. (a) Find the root between 1 and 2 of 2x23x6 = 0 by Newton-

Rapson method correct to five decimal places.

(b) Solve by Gauss-Jacobi method, the following equations.

4x1 + x2 + x3 = 6

x1 + 4x2 + x3 = 6x1 + x2 + 4x3 = 6

17. Given 0)0(,1)0(,0/

2

2

yyydx

dyx

dx

ydfind the value of

y(0.1) by using Range-Kutta method of fourth order.

(or)

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18. Using Milnes method find y (4.4) given 5xy / + y22 = 0. Given

y(4) = 1, y(4.1) = 1.0049, y(4.2) = 1.0097 and y(4.3) = 1.0143.

19. Solve the elliptic equation Uxx + Uyy = 0 for the following square

nesh with boundary values as shown.

(or)

20. Using BenderSchmidt scheme, solve

2

2

x

u

t

u

subject to u(0,t) = 0, u(1,t) = 0. and u(x,0) = sin x,0 < x < 1.

1000 1000 1000 1000

2000

2000

u1 u2 500

0u3 u4

1000 500 0 0

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Register Number


Course & Branch :B.E/B.Tech - CSE/ECE/EEE/EIE/ETCE/E&C/

MECH/M&P/AERO/CIVIL/AUTO/CHEM/BTE

Title of the Paper :Applied Numerical Methods Max. Marks :80

Sub. Code :511505-501-6C0079 Time : 3 Hours

Date :25/04/2011 Session :AN______________________________________________________________________________________________________________________

PART - A (10 x 2 = 20)

Answer ALL the Questions1. State the principle of least squares.

2. What is the relation between E and ?

3. Write the formula for Simpsons one-third rule.

4. Solve yn+2yn+1 + 8yn = 0.

5. Write the iterative formula for calculating Nby Newton-Raphson method.

6. What is the other name of Regula-Falsi method?

7. What is the formula for Modified Eulers method of finding thesolution of a first order differential equation?

8. Write the formula for Milnes predicator corrector formula.

9. Classify the equation uxx + 2uxy + 4uyy = 0.

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10. What is Bender-Schmidt recurrence equation?

PARTB (5 x 12 = 60)


11. (a) Fit a parabola to the following data using the method of least

squares.

x 1 2 3 4 5y 2 3 5 8 10

(b) Fit a curve of the formy = axb to the data given below using

the method of least squares

x 1 2 3 4 5 6

y 1200 900 600 200 110 50

(or)

12. (a) Prove that hD = log(1+

) = -log(1 -

).

(b) From the data given below, find the value of x when y = 13.5

x 93.0 96.2 100.0 104.2 108.7

y 11.38 12.80 14.70 17.07 19.91

13. (a) The following table gives the density of saturated water for

various temperatures of saturated stream.

Temperature C 100 150 200 250 300

Density hg/m3 958 917 865 799 712

Find by interpolation the density when the temperature is 130

degree and when the temperature is 275 degree.

(b) Evaluate

6

0

21 x

dx

by using

(i) Trapezoidal rule

(ii) Simpsons 3

1rule and

(iii) Simpsons 8

3rule

(or)

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14. (a) A rod is rotating a plane. The following table gives the angle

(radians) through which the rod has turned for various values of

time t seconds.

t 0 0.2 0.4 0.6 0.8 1.0

0 0.12 0.49 1.12 2.02 3.20

Calculate the angular velocity of the rod when t = 0.6 sec.

(b) Solve yn+24yn+1 + 3yn = 3n.

15. (a) Solve the following system by Gauss Jordan method:

2x1 + x2 + x3 = 10,

3x1 + 2x2 + 3x3 = 18,

x1 + 4x2 + 9x3 = 16

(b) Solve the following Gauss-jocobi method and Gauss seidel

method:

3x1x2x3 = 1,

3x1 + 6x2 + 2x3 = 0,

3x1 + 3x2 + 7x3 = 4

(or)

16. (a) Find the real positive root of3x cosx1 = 0by Newtonsmethod correct to 4 decimal places.

(b) Find the Regula-Falsi method the root of the equationxlog10x

1.2 = 0 correct to three places of decimals.

17. Solve

22

xydx

dy

with y(0) = 1(a) Use Taylor series at x = 0.2 and x = 0.4 and

(b) Use Runge-Kutta method of order 4 at x = 0.6.

(or)

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18. Determine the value of y(0.4) using Milnes method given

,2yxydx

dy y(0) = 1 y(0.1) = 1.1167, y (0.2) = 1.2767,

y(0.3) = 1.5023

19. (a) Solve 2u = -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.

(b) Derive Crank-Nicholson implicit formula for one dimensional

heat equation.

(or)

20. (a) Solve ut = uxx, u(0, t) = u(5, t) = 0, u(x, 0) = x2(25-x2) in therange taking h = 1 and up to 5 seconds using Bender-Schmidt

explicit formula.

(b) Derive the finite difference scheme for the hyperbolic

equation.

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Register Number

SATHYABAMA UNIVERSITY(Established under section 3 of UGC Act, 1956)

Course & Branch: B.E/B.Tech-CSE/ECE/EEE/ETCE/CIVIL/E&C/

MECH/M&P/AERO/AUTO/BTE/CHEM-P-EEE

Title of the Paper: Applied Numerical Methods Max. Marks: 80

Sub. Code: 505-501-6CPT0011-6C0079 Time: 3 Hours

Date: 08/11/2010 Session: FN______________________________________________________________________________________________________________________

PART - A (10 X 2 = 20)


1. Write down the normal equations to fit a quadratic curve by least

square method.

2. Show that 111 with usual notation.

3. State Lagranges interpolation formula.

4. Form the difference equation by eliminating a and b from the

relation .2x

x bxay

5. Define order of convergence.

6. Differentiate direct and indirect method for solving system of

linear equations.

7. Giveny = x + y, y(0) = 1 findy(0.1) by Taylor series.

8. State the Milnes predictor-corrector formula.

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9. Classify .0,0,0 yxyfxf yyxx

10. State Bender-Schmidt recurrence formula.

PARTB (5 x 12 = 60)


11. Fit a second degree parabola to the following

X 1 2 3 4 5

Y 5 12 26 60 97

also estimate y at x = 3.5.

(or)

12. Prove that (a).

41

2

1 22

(b) .sinh1log1log1 hD

13. (a) Solve .3244 12 nn

nnn yyy

(b) From the data given below, find the number of students

whose weight is between 60 and 70

Weight in lbs 0-40 40-60 60-80 80-100 100-120

No. of students 250 120 100 70 50

(or)

14. Dividing the range into 10 equal parts, find the approximate

value of 2.5

4

log xdxe by

(a) Trapezoidal rule (b) Simpsons rule.

15. Using Relaxation method,

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solve 10x 2y 2z = -6; -x + 10y z = -7; -x y + 10z = -8

correct to 3 decimal places.

(or)

16. (a) Use Regula Falsi method to find a positive root of the

following equationx-cos x = 0 correct to 3 decimal places. (5)

(b) Solve the system of equations using Gauss-Jordan method

2xy + z = 0.3; -4x + 3y2z = -1.4; 3x8y + 3z = 0.1. (7)

17. (a) Find y(0.05) and y(0.1), given that y = x + y, y(0) = 1 by

using Modified Eulers method.

(b) Compute y(0.8) given

02 xydy

dy

, y(0.6) = 1.7379by using R-K method of 4th order.

(or)

18. Solve y = 0.5(x + y) using Milnes predictor-corrector method

forx = 2 given the initial valuex = 0, y = 2 the values of y forx

=0.5, 1 and 1.5 should be computed by Taylor series expansion.

19. Solve uxx + uyy = 0 over the square mesh of side 4 unitssatisfying the following boundary conditions u(0, y) = 0; u(4, y)

= 8 + 2 y; u(x, 0) = 0.5 x2; u(x, 4) = x2; for0 x 4 and0 y

4.

(or)

20. Solve ut= uxx subject u(0, t) = 0, u(5, t) = 0 and u(x, 0) = x2(25

x2), 0 < x < 5 and 0 < t < 5 by taking h = 1 and k = 0.5.

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Course & Branch :B.E/B.Tech - AERO/BTE/CHEM/CIVIL/CSE/

E&C/ECE/EEE/ETCE/M&P/MECH

Title of the Paper :Applied Numerical methods Max. Marks :80Sub. Code :411505/413501/414501/415501/511505/513501/514501

/515501/516505/6C0079 Time : 3 Hours

Date :21/04/2010 Session :AN______________________________________________________________________________________________________________________

PART - A (10 x 2 = 20)


1. What is the principle of least square?

2. What is the relation between and E?

3. Write Gregory-Newton forward interpolation formula.

4. Solve yn+25yn+1 + 6yn = 0.

5. What is the convergence condition for Newton-Raphson method?

6. What is the formula for Regula falsi method?

7. Write Milnes predictor and corrector formula.

8. Write formula for modified Euler method.

9. Classify the equation x2fxx +(1y2) fyy = 0.

10. Write an explicit formula to solve 2

2

x

u

t

u

numerically.

PARTB (5 x 12 = 60)


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11. (a) The observations in the following table fit a law y=axb. Using

method of groups average find a and b from the following table:

x 10 20 30 40 50 60 70 80

y 1.06 1.33 1.52 1.68 1.81 1.91 2.01 2.11

(b) Find a straight line using method of least squares to the

following data:

x 1 2 3 4 5

y 14 27 40 55 68

(or)

12. (a) Fit a straight line to the following data using method of

moments:x 1 2 3 4

y 16 19 23 26

(b) From the following data, estimate the number of persons

earning weekly wages between 60 and 70 rupees.

Wage(in Rs) Below 40 40-60 60-80 80-100 100-120

No. of Persons

(in thousands)

250 120 100 70 50

13. (a) Using Lagranges formula find f(x) from the following data:

x 0 1 4 5

f(x) 4 3 24 39

(b) From the following data, find at x = 43 and x = 84:

x 40 50 60 70 80 90 184 204 226 250 276 304

(or)

14. (a) Evaluate

4.1

2.0

logsin dxexx xby Simpsons 3

1rule.

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(b) Solve yn+22yn+1 + yn = 2n n2.

15. (a) Find the root of the equation sin x = 1 + x3 between (-2, -1) to

3 decimal places by Newtons Raphson method.

(b) Find the solution of the following system of equations bycrouts method:

x + 3y + 8z = 4;

x + 4y + 3z = -2;

x + 3y + 4z = 1.

(or)

16. (a) Using Gauss-Jordan method solve the following equations:

10x + y + z = 12;

2x + 10y + z = 13;

x + y + 5z = 7.

(b) Solve the equations using relation method:

9xy + 2z = 9;

x + 10y2z = 15;

2x2y13z = -17.

17. By applying the fourth Runge-Kutta method find y(0.2) from

y = y x, y(0) = 2 taking h = 0.1.

(or)

18. Using Adams method find y(0.4) given

,2

'xy

y

y(0.1) = 1.01, y(0.2) = 1.022, y(0.3) = 1.023.

19. (a) Evaluate the function u(x, y) satisfying 2u = 0 at the lattice

points given the boundary values as follows:

1000 1000 1000 1000

2000 u1 u2 500

2000 u3 u4 0

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1000 500 0 0

(b) Using Crank Nicholsons 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 .4

1

h (or)

20. (a) Solve 4uxx = utt with the boundary conditions u(0, t) = 0 and

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).

(b) Solve uxx = 32ut taking h = 0.25 for t > 0, 0 < x < 1 and u(x, 0)

= 0, u(0, t) = 0, u(1, t) = t.

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Register Number


Course & Branch :B.E/B.TechCommon to ALL Branches

Title of the Paper :Numerical MethodsSub. Code :501-411505-511505-6C0079

Time : 3 Hours Max. Marks :80

Date :04/11/2009 Session :FN______________________________________________________________________________________________________________________

PART - A (10 x 2 = 20)


1. What are the methods used for fitting curve?

2. Write down the normal equations to be used for finding a, b and

c, when fitting a parabolay = a + bx + cx2 by the method of

Least squares.

3. Define Numerical differentiation.

4. What is the order of the error in Trapezoidal rule?

5. If f(x) is continuous in [a, b], then under what condition the

iteration methodx = f(x) has a solution in [a, b].

6. Solve the system of equationsx2y = 0 and 2x + y = 5 by gauss

Jordan method.

7. Using Eulers method, find .1)0(,)2.0(2 yxy

dx

dyify

8. Explain single step method. Give example.

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9. Write down the diagonal five point formula to solve the equation

.02

2

2

2

y

u

x

u

10. Write down the Crank-Nicolson formula to solve ut= uxx.

PARTB (5 x 12 = 60)


11. By the method of moments, fit a straight line and a parabola to

the data:

x 1 2 3 4

y 1.7 1.8 2.3 3.2(or)

12. (a) Evaluate 2(cos 2x). (4)

(b) Explain the difference between .22

x

xx

Eu

uandu

E

(8)

13. Following are data from the stream table. Using Newtonsformula, find the pressure of the stream for a temp of 142

Temp c 140 150 160 170 180

Pressure Kfg/cm2 3.685 4.854 6.302 8.076 10.225

(or)

14. By dividing the range into 10 equal parts, evaluate

0

sin xdxby

Simpsons rd3

1rule and Trapezoidal rule.

15. Solve the equation x3 + 2x2 + 10x 20 = 0 by Regula Falsi

method.

(or)

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16. Solve 27x + 6yz = 85

6x + 15y + 2z = 72

x + y + 54z = 110

by Gauss seidal method of iteration.

17. (a) Using Taylors series method, find y when x = 1.1 from

).4.(1)1(,31

cesdecimalplayxydx

dy (4)

(b) Using Runge-Kutta method of 4th order, solve 22

22

xy

xy

dx

dy

giveny (0) = 1 at x = 0.2. Take h = 0.2. (8)(or)

18. Solve

,1)0(,2

12

/

yyx

y find y(0.1), y(0.2), y(0.3) by

Eulers method and y(0.4) by Milnes Predictor-Corrector

method.

19. Solve by Crank Nicolson method ,22

x

u

t

u

0 < x < 1, t > 0,

u(x, 0) = 100 x(1x), u(0, t) = u(1, t) = 0 for two steps with h =

0.25 and k = 0.0625.

(or)

20. Evaluate the pivotal values of the following equation taking h = 1

upto one half of the period of the oscillation 2

2

2

2

16 t

u

x

u

given u(0,t) = u(5,t) = 0, 0)0,(

x

t

uand u(x, 0) = x2(5x).

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Course & Branch: B.E/B.Tech -AERO/BTE/CHEM/CIVIL/CSE/E&C/ECE/EEE/ETCE/M&P/MECH

Title of the paper: Applied Numerical MethodsSemester: V Max.Marks: 80

Sub.Code: 501-505(2003-2004-2005)- 6C0079 Time: 3 Hours

Date: 22-04-2009 Session: AN

PARTA (10 x 2 = 20)


1. State the principle of least squares.

2. Express 3x32x2 + 7x6 in factorial polynomial.

3. Write the Newtons back ward interpolation formula.

4. Can we use Lagranges interpolation formula when the intervals

are equal?

5. Write the condition for convergence for GaussSeidal method

of solving system of simultaneous equation.

6. Write the order of convergence of NewtonRaphson method.

7. Write the formula for the RangeKutta method at 2nd order.

8. Write the predictor formula for Adams Bash forth method.

9. Classify the equation: Uxx + 2Uxy + Uyy = 0.

10. Write down standard five point formula in solving Laplace

equation over a region.

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PARTB (5 x 12 = 60)


11. (a) Fit the curve y = axb to the following data by method of least

squares.

x 10 20 30 40 50 60 70 80

y 1.06 1.33 1.52 1.68 1.81 1.91 2.01 2.11

(b) Fit a straight line y = a + bx to the following data by method

of moments.

x 1 3 5 7 9

y 1.5 2.8 4.0 4.7 6.0(or)

12. (a) Fit a straight line y = ax + b to the data given below by

method of least squares and find the value of y at x = 2.5.

x 0 1 2 3 4

y 1 1.8 3.3 4.5 6.3

(b) Prove that

41

2

22

13. (a) Using Newtons forward interpolation formula, find the

number of students whose weight is between 60 and 70.

Weight 0-40 40-60 60-80 80-100 100-120

No. of Students 250 120 100 70 50

(b) Find the first derivative (dy)/(dx) 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

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(or)

14. (a) Find y(10), from the data given below by Lagranges

interpolation formula.

x 5 6 9 11y 12 13 14 16

(b) Evaluate 6

0

21/ xdx by (i) trapezoidal rule, (ii)

Simpson 1/3 Rule.

15. (a) Find the real positive root of 3xcos x1 = 0 by NewtonRaphson method.

(b) Solve by Crouts method.

x + y + z = 3

2xy3z = 16

3x + yz = -3

(or)

16. (a) Find all the roots of the equation x39x2 + 18x6 = 0 by

graeffes root square method.

(b) Solve following system of equation by Gauss Seidel

method:

10x5y2z = 3

4x10y + 3z = -3

x + 6y + 10z = -3

17. (a) Find y(0.1) and y(0.2) using Eulers modified formula:dy/dx = x2 + y2, y(0) = 1

(b) Find the value of y(0.4) using Milnes Predictor and

corrector

method given,

y' = xy + y2, y(0) = 1

y(0.1) = 1.1167

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y(0.2) = 1.2767

y(0.3) = 1.5023

(or)

18. (a) Find y (0.2) correct to 4 decimal places if y' =xy

xy

;

y(0)=1 by 4

th

order RangeKutta method.(b) Using tailor series method find y (0.1), y(0.2) and y(0.3),

given y' = x2y; y(0) = 1.

19. Using crank Nicholcon method solve Uxx = 16ut, 0 < x < 1,

t > 0.

Given u (x,0) = 0

u (0,t) = 0

u(1,t) = 100tFind the value of U for one time step, taking h =

(or)

20. Evaluate the pivotal values of the following equation taking h = 1

and upto one half of the period of the oscillation

16 uxx = utt

given u (0,t) = 0

u (5,t) = 0u (x,0) = x2 (5x)

and ut (x,0) = 0.

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Course & Branch: B.E/B.TechCommon to ALL Branches

Title of the paper: Applied Numerical Methods

Semester: V Max. Marks: 80Sub.Code: 11505/501 (2003/2004/2005)/ 6C0079 Time: 3 Hours

Date: 01-11-2008 Session: FN

PARTA (10 x 2 = 20)


1. Write down the normal equations to be used for finding a, b and c

when fitting a parabola y = ax2 + bx + c by the method of leastsquare.

2. Define the operators E and D.

3. State Newtons backward difference interpolation formula.

4. Solveyx+32yx+2yx+1 + 2yx = 0.

5. Explain briefly Gauss Jordon iteration to solve simultaneous

equation.

6. Write down the different methods for solving simultaneous

equation.

7. State the Milnes predictor-corrector formula.

8. Explain briefly modified Eulers method.

9. Name at least two numerical methods that are used to solve one

dimensional diffusion equation.

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10. Write down the standard five-point formula for solving laplace

equation.

PARTB (5 x 12 = 60)


11. Fit a curve of the formy = ax + c to the following data:

X 2 4 6 8 10

Y 5.0 6.6 13.9 25.3 35.6

(or)

12. (a) Find the missing term in the following

X 1 2 3 4 5 6 7

Y 2 4 8 - 32 64 128

(b) By the method of moments, fit a straight line to the data:

X 1 2 3 4

Y 1.7 1.8 2.3 3.2

13. Using Lagranges interpolation formula, fit a polynomial to the

data:

X 0 1 3 4

Y -12 0 6 12

Also find y at x = 2.

(or)

14. (a) Evaluate

5.1

0

5.1

1

2 dxdye yxby trapezoidal rule with

h = k = 0.5.

(b) Computedxe

x

4

0by Simpsons 3

1rule with 8 subdivisions.

15. Solve the following system by Gauss-Seidal method:

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28x + 4yz = 32, x + 3y + 10z = 24, 2x + 17y + 4z = 35.

(or)

16. (a) Find a root of f(x) = x35x + 3 = 0, using Newton-Raphson

method.

(b) Solve for a positive root of x cos x = 0 by the method ofFalse position.

17. Compute y(0.2), given,

22

22

xy

xy

dx

dy

y(0) = 1, by Runge-

Kutta method of fourth order, taking h = 0.2.

(or)

18. Given y/ = y-x2, y(0) = 1, y(0.2) = 1.218, y(0.4) = 1.4682,y(0.6) = 7.7379, estimate y(0.8) by Adam-Bashforth method

19. Use Crank-Nicholson scheme to solve ,162

2

t

u

x

u

0 < x < 1

and t > 0 given u(x, 0) = 0, u(0, t) = 0 and u(1, t) = 100t. compute

u(x, t) for one time step, taking .41x

(or)

20. Solve the Poissons equation

1||,1||,1002

2

2

2

yx

y

u

x

ugiven that u = 0 on the

boundary of square. Take .2

1

h

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