sri ramakrishna institute of technology (an …

1

SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY

(AN AUTONOMOUS INSTITUTION)

COIMBATORE- 641010.

Question and Answer bank

Course code UICM004

Course name Numerical Methods

Semester IV

Programme B. E – (Civil & EEE)

Regulations R – 2017

PART A

SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS

S. No Question and Answer

1. How can we find an initial approximation to the root of ( ) ? Solution: Using intermediate value theorem, we find an interval (a, b) which contains the root of the equation ( ) . This implies that ( ) ( ) . Any point in this interval (including the end points) can be taken as an initial approximation to the root of ( ) .

2. When does the fixed-point iteration method converge? Solution:The method converges when | ( )| , for all in the interval ( ). We normally check this condition at .

3. When does the Newton-Raphson method fail?

Solution: The method may fail when the initial approximation is far away from the exact

root α

4. Compare Newton - Raphson method and Fixed point iteration method.

Solution: Newton - Raphson method is a special case of fixed point iteration method.

Fixed point iteration method converges linearly. Newton - Raphson method x quadratically

convergent

5. When does the Gauss elimination method fail?

Solution:Gauss eliination method fails when any one of the pivots is zero or it is a very small

number, as the elimination progresses. If a pivot is zero, then division by it gives over flow error,

since division by zero is not defined. If a pivot is a very small number then division by it introduces

large round off errors and the Solution may contain lagre errors.

6. Describe the principle involved in the Gauss-Jordan method for finding the inverse of a square

matrix .

Solution: We start with the augumented matrix of with the identity matrix of the same

order. When the Gauss-Jordan elimination procedure using elementary row transformation is

completed, we obtain

[ | [ | since .

7. When does we use the power method?

2

Solution: We use the power method to find the largest eigenvalue in magnitude and the

corresponding eigenvector of a matrix .

8. When can we expect faster convergence in power method?

Soltion:To apply power method, we assume | | | | | |. Faster convergence is

obtained when | | | |. That is, the leading eigenvalue in magnitude is much larger than

the remaining eigenvalues in magnitudes.

9. Find the eigenvalues of the matrix (

).

Solution: Characteristic equation

Eigenvalues are -2 and 3

10. Compare Gauss elimination method and Gauss Jordan method.

Solution:

Gauss elimination method Gauss Jordan method

Coefficient matrix is

transformed into upper

trangular matrix

Coefficient matrix is transformed into unit

matrix

We obtain the solution by

backward substitution

No need for substitution method.

11. Find an iterative formula to find √ where is a positive number.

Solution: Let √ Then ( ) and ( )

( )

( )

( )

( )

(

)

12. Locate the negative root of approximately.

Solution: Let ( ) . Then

( ) ( ) ( ) ( ) ( ) ( ).

Roots lies between -2 and -3 and closer to -2 since | ( )| | ( )|.

13. State the difference between direct and iterative methods for solving system of equations.

Solution:

Direct method Iterative method

It gives exact value It gives only approximate solutions

This method determine all the roots at

the same time

This method determine only one roots

at a time

Simple, take less time Time consuming and labourious

14. What are the normal equations in fitting a straight line ?

Solution: ∑ ∑ ∑ ∑ ∑

15. What is the condition for convergence of Gauss seidel method of iteraton?

Solution: The coefficient matrix should be diagonall dominant.

16. Establish an iterative formula to find the reciprocal of a positive number by Newton

3

Raphson method.

Solution: Let

Then ( )

and ( )

( )

( )

(

)

( )

17. Find the dominant eigenvalue of [

] by power method.

Solution: Let ( ) be the initial eigenvalue.

Then ( ) (

); (

) (

)

(

) (

) Here

The dominant eigenvalue of is 5.3722.

18. Which direct methods do we use for solving the system of equations ?

Solution: ( )Gauss elimination method and Gauss Jordan method

19. What is principle of least squres ?

Solution: The technique of minimising the sum of squares of errors is known as least squares.

20. Does the power method give the sign of the largest eigenvalue?

Solution: No. Power method gives the largest eigenvalue in magnitude.

INTERPOLATION AND APPROXIMATION


21. If then find the value of .

0 3 9 -11 11

1 12 -2 0

2 10 -2

3 8

22. Give two uses of interpolating polnomials.

Solution: The first use to construct the function ( ) when it is not given explicitlyand only

values of ( ) and/or its certain order derivatives are given at a set of distinct points.The

second use is to perform the required operations which were intended for ( ), like

determination of roots, differentiation and integration etc. can be carried out using the

approximating polynomials ( ).

4

23. For performing interpolation for a given data, when do we use the Newton’s forward and

backward difference formulas?

Solution: We use the forward difference interpolation when we want to interpolate near

the top of the table and backward difference interpolation when we want to interpolate

near the bottom of the table.

24. Write the expression for the bound on the error in Lagrange interpolation.

Solution: The bound for the error in Lagrange interpolation is given by

| ( )|

( ) |( )( ) ( )|| ( )|

( ) [

|( )( ) ( )|] [

| ( )( )|]

25. What is the diadvantage of Lagrange interpolation?

Solution: Assume that we have determined the Lagrange interpolation polynomial of

degree based on the data values ( ( )) given at the ( )distinct

points . Suppose that to this given data, a new value ( ( )) at the distinct points

is added at the end of the table. If we wequire the Lagrange interpolating polynomial of

degree for this new data, then we need to compute all the Lagrange fundamental

polynomial again. The degree Lagrange polynomial obtained earlier is of no use. Ths s

the disadvantage of the Lagrange interpolation.

26. Give the relation between the divided differences and forward or backward differences.

Solution: (

.

27. If , write the error expression in the Newton’s forward difference formula.

Solution: The error expression is given by ( ) ( )( ) ( )

( ) ( )( )

28. If , write the error expression in the Newton’s backward difference formula. Solution: The error expression is given by

( ) ( )( ) ( )

( ) ( )( )

29. What do you understand by inverse interepolation?

Solution: When ( ) and for a given values of arguments and corresponding entries

a vlaue of is said to be estimated for given value of , the interpolation is known as

inverse interpolation. In other words, if the value of the independent variable is said to be

estimated for a given value of the dependent variable within the series, it is called inverse

interpolation.

30. Enunicate Stirling’s formula.

Solution: Stirling’s formula is applicable only when the values of the arguments are

equidistant. It is most suitable for interpolation near the middle of the tabulated values of

the set.

31. When we apply Bessel’s formula?

Solution: Bessel’s formula is applicable only when the values of the arguments are

equidistant. It is most suitable for interpolation near the middle of the tabulated values of

the set.

5

32. Make a comparative statement on Stirling’s and Bessel’s Formulae of interpolation .

Solution: Both the formulae are applcable equidistant arguments. Both are preferable for

interpolating values near the middle of the series. Stirlings used as in Newton-Gauss

formulae whereas Bessel used which is equal to (

)

33. Find the polynomial through the points ( ) ( ) ( ) by Lagrange’s interpolation

formula.

Solution:

34. Find the value of ( ) from the following data:

0 1 3

( ) 0 1 0

: ( )

( )

( )

35. Construct divided difference table for the following data:

0 2 3 5 6

( ) 1 19 55 241 415

Solution:

( ) ( ( )) ( ( )) ( ( )) ( ( ))

0 1 9 9 2 0

2 19 36 19 2

3 55 93 27

5 241 174

6 415

36. Find the second divided difference of ( )

, using the points .

Solution: [ ( ) ( )

(

)

[

[ [ [

6

37. Construct the forward difference table for the data:

-1 0 1 2

( ) -8 3 1 12

Solution:

( ) ( ) ( ) ( )

-1 -8 11 -13 26

0 3 -2 13

1 1 11

2 12

38. Can we decide the degree of the polynomial that a data represent by writing the forward or

backward difference table? Explain it.

Solution: Given a table of values, we can determine the degree of the forward / backward

difference polynomial using difference table. The column of the difference table

contains the forward/ backward differences. If the vaklues of these differences are same,

then the ( ) and higher order differences are zero. Hence, the given data represents a

degree polynomial.

39. Find the polynomial which takes the values

0 1 2

1 2 1

( )

( )

( )( )

where ( )

( )

40. Find ( ( )) if ( ) and .

Solution: ( ( )) ( ( ) ( )) ( ) ( ) ( ) ( )

7

NUMERICAL DIFFERENTIATION AND INTEGRATION


41. Given the data ( ( )) at equispaced points where is the

step length, write the formula to compute ( ) using the Newton’s forward difference

formula.

Solution: ( )

[

]


step length, write the formula to compute ( ) using the Newton’s forward difference

formula.

Solution: ( )

[

]


step length, write the formula to compute ( ) using the Newton’s backward difference

formula.

Solution: ( )

[

]


step length, write the formula to compute ( ) using the Newton’s backward difference

formula.

Solution: ( )

[

]

45. What is the order of the trapezium rule for integrating ∫ ( )

? What is the expression

for the error term?

Solution: The order of the trapezium rule is 1. The expression for the error term is

Error= ( )

( )

( ) where

46. When does the trapezium rule for integrating ∫ ( )

gives exact results?

Solution: The trapezium rule gives exact results when ( ) is a polynomial of degree .

47. What is the restriction in the number of nodal points, required for using the trapezium rule

for integrating ∫ ( )

?

Solution: There is no restriction in the number of nodal points, required for using the

trapezium rule.

48. What is the geometric representation of the trapezoidal rule for integrating ∫ ( )

?

Solution: Geometricall, the right hand side of the trapezoidal rule is the sum of areas of the

trapezoids with width , and ordinates ( )and ( ), . This sum is an

approximation to the area under the curve ( ) above the and the ordinates

and .

49. Which one is more reliable, Simpson’s rule or trapezoidal rule?

8

Solution: Simpson’s rule

50. Give the bound on the error of the trapezoidal rule.

Solution: | |

( )

where |

( )|

51. When does the Simpson’s 1/3 rule for integrating ∫ ( )

gives exact results?

Solution: The Simpson’s 1/3 rule gives exact results when ( ) is a polynomial of degree

.

52. What is the restriction in the number of nodal points, required for using the Simpson’s 1/3

rule for integrating ∫ ( )

?

Solution: The number of nodal points must be odd for using the Simpson’s 1/3 rule or the

number of sub interval must be even.

53. Using Simpson’s rule, find ∫

given that

.

Solution:

∫

[( ) ( )

54. Give the bound on the error of the Simpson’s 1/3 rule.

Solution: | | ( )

where |

( )( )|

55. What is the restriction in the number of nodal points, required for using the Simpson’s 3/8

rule for integrating ∫ ( )

?

Solution: The number of sub intervals must be evendivisible by 3.

56. What are the disadvantages of the Simpson’s 3/8 rule compared with the Simpson’s 1/3 rule? Solution: The disadvantages are the following: (i) The number of subintervals must be divisible by 3. (ii) It is of the same order as the Simpson’s 1/3 rule, which only requires that the number of nodal points must be odd. (iii) The error constant c in the case of Simpson’s 3/8 rule is c = 3/80, which is much larger than the error constant c = 1/90, in the case of Simpson’s 1/3 rule. Therefore, the error in the case of the Simpson’s 3/8 rule is larger than the error in the case Simpson 1/3 rule.

57. Explain why we need the Romberg method. Solution: In order to obtain accurate results, we compute the integrals by trapezium or Simpson’s rules for a number of values of step lengths, each time reducing the step length. We stop the computation, when convergence is attained. Convergence may be obtained after computing the value of the integral with a number of step lengths. While computing the value of the integral with a particular step length, the values of the integral obtained earlier by using larger step lengths were not used. Further, convergence may be slow. Romberg method is a powerful tool which uses the method of extrapolation. Romberg method uses these computed values of the integrals obtained with various step lengths, to refine the solution such that the new values are of higher order. That is, as if they are obtained using a higher order method than the order of the method used.

58. An integral I is evaluated by the trapezium rule with step lengths and . Write the Romberg method for improving the accuracy of the value of the integral. Solution: Let ( ) ( ) denote the values of the integral evaluated using the step

9

lengths and . The required Romberg approximation is given by ( ( ) ( ))

59. Write the error term in the Gauss two point rule for evaluating the integral ∫ ( )

.

Solution: the error term in the Gauss two point rule is given by

( )

( )( )

60. Write the error term in the Gauss three point rule for evaluating the integral ∫ ( )

.

Solution: the error term in the Gauss two point rule is given by

( )

( )( )

NUMERICAL SOLUTION TO ORDINARY DIFFERENTIAL EQUATION


61. Define truncation error of a single step method for the solution of the initial value problem

( ) ( ) .

Solution: A single step method method for the solution of the initial value problem is

( ) Exact solution ( ) satisfes the equation ( )

( ) ( ) where is called the truncation error or

discretization error. It is dfined as ( ) ( ) ( ).

62. Write the truncation error of the Euler’s method.

Solution: The truncation error of the Euler’s method is

( ) .

63. Write the bound on the error of the Euler’s method.

Solution: | | | ( )|

64. What is the disadvantage of the Talor series method?

Solution: Talor series method requires the computation of higher order derivatives. The

number of partial derivatives to be computed increases as the order of the derivative of y

increases. Therefore we find computation of higher order derivatives is very difficult.

65. Write the bound on the error of the Talor series method.

Solution: | |

( ) where |

( )( )|

66. What is the order of modified Euler’s method?

Solution: The order of modified Euler’s method is two.

67. Write the modified Euler’s method for solving the first order initial value problem.

Solution: (

( ))

68. Write Adams-Baseforth predictor-corrector method for solving the initial value problem

( ) ( )

Solution: Predictor

[ ]

10

Corrector

[ ]

69. What is the order and error term of Adams-Baseforth predictor-corrector method for solving

the initial value problem ( ) ( ) ?

Solution: order Adams-Baseforth predictor-corrector method is four.

( )( )

( )( )

70. Write the truncation error of the Runge-Kutta method of fourth order.

71. What is the order of Milne’s predictor-corrector method for solving the initial value problem

( ) ( ) ?

Solution: The order Milne’s predictor-corrector method is four.

( )( )

( )( )

72. How many prior values are required to predict the next value in Adams-Baseforth Moulton

predictor-corrector method?

Solution: The Adams-Baseforth Moulton predictor-corrector method requires four starting

values . That is we require the values .

73. How many prior values are required to predict the next value in Milne’s predictor-corrector

method?

Solution: The Milne’s predictor-corrector method requires four starting values

. That is we require the values .

74. Solve

( ) for by Euler’s method.

Solution: ( ) ( ) ( ) ( )( )

.

75. Compare Taylor series method and Runge-Kutta method.

Solution: Runge-Kutta method gives quick convergence to the solutions of the differential

equations than Taylor series method.

76. Using modified Euler’s method find ( ) when ( )

Solution: ( )

(

( ))

77. In the derivation of fourth order Runge-kutta method, why it is called fourth order?

Solution: The fourth order Runge-kutta method agree with Talor series solution upto the

terms of . Hence it is called fourth order Runge-kutta method.

78. Solve

( ) for by Euler’s method.

Solution: ( ) ( ) ( ) .

11

79. What are the advantages of Runge-kutta method over Taylor’s series method?

Solution: The Runge-kutta methods are designed to give greater accuracy and they possess

the advantage of requiring only the function values at some selected points on the sub-

interval.

80. Where the Taylor’s series method of solving differential equation is powerful?

Solution: It is possible to find the successive derivatives in a very easy manner, then only

Taylor series method is powerful.

NUMERICAL SOLUTION TO PARTIAL DIFFERENTIAL EQUATIONS


81. Finite difference methods when applied to Laplace equation or Poisson equation give rise to

a system of algebraic equations . Name the types of methods that are available for

solving these systems.

Solution:( )Direct methods like Gauss elimination method or Gauss-Jordan method can be

used when the system of equations is small.

( ) Iterative methods like Gauss-Jacobi method or Gauss-Seidel method can be used

when the system of equations is large.

82. What is the importance of classification of partial differential equation?

Solution When a method converges, it implies that the errors in the numerical solutions

as Suppose that a method is of order ( ) Then, if we reduce the step length by a

factor, say 2, and re-compute the numerical solution using the step length

, then the error

becomes [(

)

] [ ( )]

. That is, the errors in the numerical solutions are reduced by a

factor of 4. This can easily be checked at the common points between the two meshes.

83. Write the general linear second order partial differential equation in two variables.

Solution: where are

functions of .

84. When is the linear second order partial differential equation

called an elliptic or hyperbolic or parabolic equation?

Solution: The given linear second order partial differential equation is called ( ) an elliptic

equation when , ( ) a hyperbolic equation when and ( )a

parabolic equation when .

85. Classify the partial differential equation .

Solution: The given pde is parabolic since .

86. Classify the partial differential equation

Solution: The given pde is elliptic since .

87. Classify the partial differential equation

Solution: The given pde is hyperbolic since

12

.

88. What is the order and truncation error of the Schmidt method?

Solution The order of the method is ( ). For a fixed value of λ, the method behaves like an ( ) method. The truncation error of the method is given by

T.E =

[( )

]

89. What is the condition of stability for the Schmidt method? Solution: Schmidt method is stable when the mesh ratio parameter λ satisfies the condition λ ≤ 1/2.

90. What is the order and truncation error of the standard five point formula for the solution of

Laplace’s equation ,with uniform mesh spacing ?

Solution:

(

)

91. What is the order and truncation error of the diagonal five point formula for the solution of

Laplace’s equation ,with uniform mesh spacing ?

Solution:

(

)

92. When do we normally use the diagonal five point formula while finding the solution of

Laplace or Poisson equation?

Solution: We use the diagonal five point formula to obtain initial approximations for the

solutions to start an iterative procedure like Liebmann iteration.

93. When do we use the Liebmann method?

Solution: We use the Liebmann method to compute the solution of the difference

equations for the Laplace’s equation or the Poisson equation. The initial approximations are

obtained by judiciously using the standard five point formula

[ ] or the diagonal five point formula

[

] after setting the values of one or two variables as zero. If in

some problems, these two formulas cannot be used, then we set the values of required

number of variables as zero.

94. What is the condition of convergence for the system of equations obtained, when we apply

finite difference methods for Laplace’s or Poisson equation?

Solution: A sufficient condition for convergence of the system of equations is that the

coefficient matrix A of the system of equations , is diagonally dominant. This

implies that convergence may be obtained even if is not diagonally dominant.

95. Write the one dimensional heat conduction equation and the associated conditions.

Solution: The heat conduction equation is given by .

Initial condition: ( ) ( ) .

Boundary conditions: ( ) ( ) ( ) ( )

96. Write the Schmidt method for solving the one dimensional heat conduction equation.

Solution: The Schmidt method for solving the one dimensional heat conduction equation

is given by

13

( ) where

is the mesh

ratio parameter and and are the stel lengths in the and directions respectively.

97. Write the Bender-Schmidt method for solving the one dimensional heat conduction equation.

Solution: The Bender-Schmidt method for solving the one dimensional heat conduction

equation is given by

( ). This method is a particular case of the Schmidt method in which

we use the value

.

98. Write the Crank-Nicholson method for solving the one dimensional heat conduction

equation.

Solution: The Bender-Schmidt method for solving the one dimensional heat conduction

equation is given by

( )

( )

99. What is the order of the the Crank-Nicolson method for solving the heat conduction equation? Solution: The order of the Crank-Nicolson method is ( ).

100. Write an explicit method for solving the one dimensional wave equation.

Solution:

1

PART B

SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS

S. No Question Marks

1. (a)Derive the Newton’s method for finding the root of a positive number

where . Hence compute

correct to four decimal places, assuming the

initial condition as .

(b)Find the smallest positive root of the equation , using fixed point

iteration method.

10+6

2. (a)Using Newton-Raphson method solve x g x with x .

(b) Solve the equation for the positive root by iteration method.

10+6

3. (a) Find a positive root of √ i by iteration method.

(b)Perf rm f ur iterati f the Newt ’ meth d t fi d the ma e t p itive r t

of the equation ( )

10+6

4. (a)Solve e x by the method of iteration.

(b)Solve the system of equations by Gauss elimination method.

x y z

x y z

x y z

8+8

5. (a)Find a real root of the equation x x correct to four decimal places by

iteration method.

(b) Solve the system of equations by Gauss elimnation method.

x y z

8+8

6. Solve the system of equations by Gauss Jordan method.

16

2

7. Solve the system of equations by Gauss Jordan method.

16

8. Solve the system by Gauss - Seidel method.

16

9. Solve the following system of equations by Gauss - Seidel method

16

10. Determine the numerically largest eigenvalue and the corresponding eigenvector of

the following matrix, using power method.

[

]

16

11. Find the dominant eigenvalue and the corresponding eigenvector of

[

] by power method.

16

12. Using the Jacobi method find all the eigenvalues and the corresponding eigenvector of

[ √

√ √

√

].

16

13. Find all the eigenvalues and the corresponding eigenvector of

[

] by the Jacobi method

16

14. A chemical company, wishing to study the effect of extraction time on the efficency of

an extraction operation, obtained the data shown in the following table.

extraction time

minutes ( )

27 45 41 19 3 39 19 49 15 31

16

3

Efficiency ( ) 57 64 80 46 62 72 5 77 57 68

Fit a straight line to the given data by the method of least squares.

15. By the method of least squares, find the straight line that best fit the following data:

1 2 3 4 5

( ) 14 27 40 55 68

16

INTERPOLATION AND APPROXIMATION


16. From the following table of half-yearly premium for policies maturing at different

ages, estimate the premium for policies maturing at age, estimate the premium for

policies maturing at age 46 and 63.

Age 45 50 55 60 65

Premium 114.84 96.16 83.32 74.48 68.48

16

17. The population of a town is as follows:

Year 1941 1951 1961 1971 1981 1991

Population in

lakhs

20 24 29 36 46 51

Estimate the population increase during the period 1946 to 1976

16

18. The following data are taken from the steam table.

emp 140 150 160 170 180

3.685 4.854 6.302 8.076 10.225

Find the pressure at temperature and

16

19. From the following data , find at and

40 50 60 70 80 90

184 204 226 250 276 304

Also express in terms of .

16

20. From the data given below, find the number of students whose weight is between 60

and 70.

16

4

Weight in

lbs.

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

No. of

students

250 120 100 70 50

21. From the following table , estimate correct to five decimal places using (i)

Stirling’s formula (ii) Bessel’s formula. Also find at .

0.61 0.62 0.63 0.64 0.65 0.66 0.67

1.8404

31

1.8589

28

1.8776

10

1.8964

81

1.9155

41

1.93 7

92

1.9542

37

16

22. The following table gives the values of the probability integral ( )

√ ∫

for certain values of . Find the value of this integral when using (i)

Stirling’s formula (ii) Bessel’s formula.

.51 0. 52 0. 53 0.54 0.55 0.56 0.57

( )

0.529

2437

0.537

8987

0.546

4641

0.554

9392

0.563

3233

0.571

6157

0.579

8158

16

23. Given the following table , find ( ), by using (i) Stirling’s formula (ii) Bessel’s

formula.

20 30 40 50

( ) 512 439 46 243

16

24. Using Newton’s divided difference formula, find the values of ( ) ( ) and ( )

given the following table.

4 5 7 10 11 13

( ) 48 100 294 900 1210 2028

16

25. The following table gives same relation between steam pressure and temperature.

Find the pressure at temperature

154.9 167.9 191 212.5 244.2

16

26. Find the age corresponding to the annuity value 13.6 from the given table

Age(x) 30 35 40 45 50

Annuity value(y) 15.9 14 9 14.1 13.3 12.5

16

5

27. Use Lagrange’s formula to fit a polynomial to the data

-1 0 2 3

-8 3 1 12

And hence find ( )

16

28. Using Lagrange’s interpolation formula, find the equation of the cubic curve that

passes through the points ( ) ( ) ( ) and ( )

16

29. Given g g g and

g . Find the value of g using Newton’s divided difference

formula.

16

30. From the data given below, find the value of when

93 96.2 100 104.2 108.7

11.38 12.8 14.7 17.07 19.91

16

NUMERICAL DIFFERENTIATION AND INTEGRATION

S.

No

Question Marks

31. The population of a certain town is given below. Find the rate of growth of the population in

1931, 1941, 1961 and 1971.

Year 1931 1941 1951 1961 1971

Population

in thousands

40.62 60.80 79.95 103.56 132.65

16

32. Find the first two derivatives of

at and , for the given table:

50 51 52 53 54 55 56

3.6840 3.70 4 3.7325 3.7563 3.7798 3.8030 3.8259

16

33. The table below gives the results of an observation: is the observed temperature in

degrees centigrade of a vessel of cooling water: is the time in minutes from the beginning

of observation:

16

6

1 3 5 7 9

85.3 74.5 67 60.5 54.3

Find the approximation rate of cooling at and .

34. A rod is rotating in a plane. The following table gives the angle (in radians) through which

the rod has turned for various values of time (seconds). Calculate the angular velocity and

angular acceleration of the rod at seconds.

0 0.2 0.4 0.6 0.8 1

0 0.12 0.49 1.12 2.02 3.2

16

35. Evaluate ∫

using Trapezoidal rule with . Hence obtain an approximate value of

. Can you use other formula in this case.

16

36. Evaluate ∫

by using ( ) Trapezoidal rule ( )Simpson’s rule. Verify your answer with

actual integration

16

37. Evaluate ∫

by using ( ) Trapezoidal rule ( )Simpson’s rule. Also check up by

direct integration.

16

38. By dividing the range into ten equal parts, evaluate ∫ i

by Trapezoidal rule and

Simpson’s rule. Verify your answer with integration.

16

39. The table below gives the velocity of a moving particle at time seconds. Find the distance

covered by the particle in 12 seconds and also the acceleration at seconds.

0 2 4 6 8 10 12

4 6 16 34 60 94 136

16

40. A river is 80 meters wide. The depth ’ in meters at a distance meters from one bank is

given by the following table. Calculate the area of cross section of the river using Simpson’s

rule.

0 10 20 30 40 50 60 70 80

0 4 7 9 12 15 14 8 3

16

41. Evaluate the integral ∫

using the Gauss three point formula. Compare with the exact

solution.

16

42. Evaluate the integral ∫ ∫

using trapezoidal rule with . 16

7

43. Evaluate the integral ∫ ∫

using Simpson’s rule with . 16

44. Evaluate ∫ ∫

by using trapezoidal rule taking . 16

45. Using Romberg’s Method evaluate

1

01 x

dxcorrect to three decimal places.

16

NUMERICAL SOLUTION TO ORDINARY DIFFERENTIAL EQUATION


46. Using Taylor series method, compute ( ) ( ) correct to four decimal places

given

and ( )

16

47. Using Taylor series method, compute ( ) ( ) correct to four decimal places

given

and ( )

16

48. Solve the equation

( ) using Modified Euler’s Method and tabulate

the solution at and . Compare your results with the exact solution.

16

49. Using Modified Euler’s Method, find ( ) ( ) ( ) given

( )

16

50. Using Modified Euler’s Method, find ( ) ( ) given

( ) 16

51. Using Runge - kutta Method of th order, solve

with ( ) at 16

52. Find ( ) given that ( ) by using Runge-Kutta method of

fourth order. Take

16

53. Using Runge-Kutta method of fourth order, find ( ) for the initial value problem

( )

16

54. Apply fourth order Runge-Kutta method to determine ( ) with from

( ) .

16

55. Determine the value of ( ) using Milne’s method given

( ) use

Taylor series method to get the values of ( ) ( ) ( ).

16

56. Using Milne’s method find y( ) given xy y , y( ) y( )

y( ) y( )

16

8

57. Using Milne’s method find y( ) if y(x)is the solution of

x y, given

y( ) y( ) y( ) y( )

16

58. Determine the value of y( ) using Adam’s method given

y( ) y( ) y( ) ( )

16

59. Using Adam’s method find y( ) if y(x)is the solution of

, given

y( ) y( ) y( ) y( )

16

60. Using Adam’s method find y( ) if y(x)is the solution of

x y, given

y( ) y( ) y( ) y( )

16

NUMERICAL SOLUTION TO PARTIAL DIFFERENTIAL EQUATIONS


61. Using finite difference method, find ( ) ( ) ( ) satisfying the

differential equation

subject to the boundary condition ( ) ( )

16

62. Solve ( ) ( ) by finite difference method. 16

63. Solve with the boundary conditions ( ) ( ) 16

64. Using finite difference method, compute ( ) given ( ) ( ) ( ) subdividing the interval into ( ) 4 equal parts ( ) 2 equal parts.

16

65. Solve the equation subject to the condition ( ) ( ) ( ) using Crank – Nicolson method.

16

66. Solve taking for and ( ) ( )

( )

16

67. Solve with boundary conditions ( ) ( ) and the initial

condition ( ) ( ) ( – ) taking h = 1 and k = ½ (for four time

steps).

16

68. Solve over the square mesh of side 4 units; satisfying the following

boundary conditions:

( ) ( ) for

( ) ( ) for

( ) ( ) for

( ) ( ) for

16

9

69. Solve ( ) over the square mesh with sides

with on the boundary and mesh length one unit.

16

70. Given

( ) ( ) ( ) ( ). Find in the range taking

and upto 5 seconds.

16

71. Using Crank-Nicholson method , solve

subject to ( ) ( ) and

( ) ( ) taking and

and ( )

and

.

16

72. Solve by Crank-Nicholson method,

, given that

( ) i ( ) and ( ) . Compute for one time step taking

.

16

73. Evaluate the pivotal values of the following equation taking and upto one half

of the period of the oscillation given ( ) ( ) ( )

( ) and ( ) .

16

74. Solve the Poisson equation given that

( ) ( ) ( ) ( ) and

.

16

75. Solve in | | | | with

and ( ) ( )

( ) ( ) for

( ) ( ) for

( ) ( ) for

( ) ( ) for

16

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