11148S51B/12148S51B Revised June 2015 Students admitted from 2011 onwards

PRIST UNIVERSITY

11148S51B/12148S51B-NUMERICAL METHODS

(COMMON TO ECE,EEE,CIVIL,MECHANICAL)

SYLLABUSUnit I SOLUTION OF EQUATIONS AND EIGEN VALUE PROBLEMS

Linear interpolation methods (method of false position) - Newtons method - Statement of Fixed Point Theorem - Fixed pointer iteration x=g(x) method - Solution of linear system of Gaussian elimination and Gauss-Jordan methods - Iterative methods: Gauss Jacobi and Gauss Seidel methods- Inverse of a matrix by Gauss-Jordan method. Eigen value of a matrix by power methods.

Unit II INTERPOLATION AND APPROXIMATION

Lagrangian Polynomials - Divided difference - Interpolation with a cubic spline - Newton forward and backward difference formulae.

UNIT III NUMERICAL DIFFERENTIATION AND INTEGRATION

Derivatives from difference table - Divided difference and finite difference - Numerical integration by Trapezoidal and Simpsons 1/3 and 3/8 rules - Rombergs method - Two and three point Gaussian quadrature formulas - Double integrals using trapezoidal and Simpsons rules.

UNIT IV INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL

EQUATIONS

Single step Methods : Taylor Series and methods - Euler and Modified Euler methods - Fourth order Runge-Kutta method for solving first and second order equations - Multistep methods Milnes and Adams predictor and corrector methods.

UNIT V BOUNDARY VALUE PROBLEMS

Finite difference solution for the second order ordinary differential equations. Finite difference solution for one dimensional heat equation by implict and explict methods - one dimensional wave equation and two dimensional Laplace and Poisson equations.

TEXT BOOKS

1. Gerald, C.F, and Wheatley, P.O, Applied Numerical Analysis, Sixth Edition, Pearson Education Asia, New Delhi.2002.

2. Balagurusamy, E., Numerical Methods, Tata McGraw-Hill Pub. Co. Ltd., New Delhi, 1999.

REFERENCES

1. Kandasamy, P.Thilakavthy, K and Gunavathy, K. Numerical Methods, S.Chand and Co. New Delhi.1999PRIST UNIVERSITYNUMERICAL METHODS 11148S51B/12148S51B

(COMMON TO ECE,EEE,CIVIL,MECHANICAL)(Students admitted on 2011 onwards)

Unit ISOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS Part A (2 marks)

1. Write the iterative formula of Newton Raphson method.

2. What is the order of convergences of Newton- Raphson method if the multiplicity of the root is one?

3. What is the condition for applying the fixed point iteration method to find the real root of the equation x = f(x)?4. What is the order of convergence for fixed point iteration?

5. In what form is the coefficient matrix transformed into when Ax =B is solved by Gauss Elimination and Gauss Jordan methods.

6. For solving a linear system, compare Gaussian elimination method and Gauss Jordan method.

7. Write a sufficient condition for Gauss seidel method to converge.

8. Find the inverse of the coefficient matrix by Gauss Jordan elimination method

9. Determine the largest eigen values and the corresponding eigen vector of the matrix correct to two decimal places using power method.

10. Why Gauss Seidel method is a better method than Jacobis iterative method?

Part B

11.(i) Find the positive root between 0 & 1 of correct to two decimal places using Newton-Raphsons method.

(6)

(ii) Solve the system of equation using Gauss Jordan method.

(10)12.(i)Solve by the method of fixed point iteration

(8)

(ii) Using Gauss Jordan method, find the inverse of the matrix

(8)13.(i)Solve the equation for the positive root by iteration method.

(8)

(ii)Using Gauss Jordan method finds the inverse of A=

(8)

14.(i) Find an appropriate root of by false position method

(8)

(ii) Using Gauss- Seidel method, solve the following equations

(8)15.( i) Compute the real root of correct to 4 decimal places using the method of false position.

(8)

(ii) Solve the following system of equation by using Gauss Jacobi method.

(8)16. Find the numerically largest eigen value of by power method.

(16)UNIT II INTERPOLATION AND APPROXIMATIONPart A (2 marks)1. State Lagranges interpolation formula.

2. What advantage has the Lagranges formula over Newton ?

3. Give the inverse of Lagranges interpolation formula.

4. Form the divided difference table for the following data.

X2510

Y529109

5. State Newtons divided difference formula .6. State Newtons forward difference formula .

7. State Newtons backward difference formula .

8. Write Bessels central difference formula.

9. Write Stirlings central difference formula.

10. What is cubic spline?

Part B

11.(i) Find y(2) in the following table using Lagranges interpolation

(8) X:0134

Y:0181 256

(ii)Using Newtons divided difference formula find f(8) for x457101113

f(x)4810029490012102028

12.( i). Find the missing value by Newtons divided difference formula.

(8)

X : 12456F(x):14155-9

(ii).Find y when x =46 and x =63 from the following data .

(8) X : 45 50 55 60 65 Y : 114.84 96.16 83.32 74.48 68.48

13.( i). From the data given below, find the number of students whose weight is between 60 to 70 (8)Weight in Ibs:- 0-40

40-60

60-80

80-100

100-120 No.of Students:-250

120

100

70

50

(ii). From the following table find the value of tan(0.28)

(8)x: 0.10

0.15

0.20

0.25

0.30

y:0.1003

0.1511

0.2027

0.2553

0.3093

14.( i). From the following table estimate e0.644 correct five decimal places using Bessels formula. (8)x: 0.61 0.62 0.63 0.64 0.65 0.66

0.67

y:ex:1.840431 1.858928 1.877610 1.896481 1.915541 1.934792 1.954237

(ii). Given the following table, find y(35) by using strilings formula

(8)x:

20

30

40

50

y:

512

439

346

243

15.( i). Using stirling formula to find f(1.22)

(8)x:

1.0

1.1

1.2

1.3

1.4

y:0.841

0.891

0.932

0.963

0.985

(ii).From the following table, find y when x=0.5437 using Bessels formula

(8)x:0.510.52

0.53

0.54

0.55

0.56

0.57

y:0.52920.5379

0.5465

0.5549

0.5633

0.5716

0.5798

16.Obtain the cubic spline approximation for the function y = f(x) from the following data, given that

(16)x:-1012

y:-11335

UNIT III

NUMERICAL DIFFERENTIATION & INTEGRATION

Part A (2 marks)

1. Write the formula for , at using forward difference operator.

2. Using Newtons backward difference formula, write the formulas for first and second order derivatives at .

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

4. State Trapezoidal rule to evaluate

5. State Simpsons three eighth rules.

6. Evaluate by Trapezoidal rule dividing the range into 4 equal parts.

7. In order to evaluate by Simpsons rule as well as by Simpsons rule , what is the restriction on the number of intervals?

8. Using Simpsons rule find given

EMBED Equation.3

&

9. State trapezoidal rule for evaluating .

10. State Simpsons rule for evaluating .

PART- B

11. (i) find for following data

(10)

3.03.23.43.63.84.0

-14-10.032-5.296-0.2566.67214.

(ii) Evaluate the integralwith h=1/6using trapezoidal rule

(6)

12. (i) Find the first, second derivatives of at if

(10)

50515253545556

3.68403.70843.73253.75633.77983.80303.8259

(ii) Using trapezoidal rule, evaluate taking 8 intervals.

(6)

13. Dividing the range into 10 equal parts find the value of by

(16)

a) trapezoidal rule b) Simpsons rule and also check the result by direct integration

14. Evaluate using Rombergs method. Hence obtain an approximate value for.(16)15.) i) Evaluate with by trapezoidal rule (8) ii) Using Simpsons rule evaluate with

(8)

16.Evaluateusing trapezoidal and Simpsons rule also by actual integration.

(16)

UNIT IV

INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

Part A (2 marks)1. Write down the fourth order Taylor algorithm .

2. State the disadvantage of Taylor series method.

3. Write down the Euler algorithm to the differential equation.

4. State modified Euler algorithm to solve at.

5. Write the Runge- kutta algorithm of second order for solving.

6. Write down the Runge-kutta formula of fourth order.

7. Write Milnes predictor corrector formula.

8. write down Adams Bashforth predictor formula.

9. Compare Runge Kutta metheods and Predictor- corrector methods for solution of intial value problem.

10. How many prior values are required to predictor formula.?Part B 11. (i)Solve with use Taylor series find y at

(10) (ii) Using Eulers method find given by assuming .(6)12. Using Taylors method solve with find (16)

(1) (2), (3) .

13. Solve by modified Eulers method to find ,, (16)

14. By fourth order R-K method find with from (16)

15. Using Milnes method find if is the solution of given and

(16)16. Given . Evaluate by Adams Bash fourth method.

(16)Unit V BOUNDARY VALUE PROBLEMS IN ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS.

PART - A1. State the conditions for the equation.

where A,B,C,D,E,F,G are function of x and y to be (i)elliptic (ii)parabolic (iii)hyperbolic.

2. What is the classification of

3. Give an example of a parabolic equation.

4. What is the equation of one dimensional heat flow equation?

5. What type of equations can be solved by using crank-nickolsons difference formula?

6. Write a note on the stability and convergence of solutions of the difference equations corresponding

to the hyperbolic equation

7. Write the diagonal five-point formula to solve the Laplace equations

8. What is the purpose of Liebmanns process?

9. Define a difference quotient.

10. Write down the finite difference form of the equation.

PART B11. Solve taking h= 0.25 for t>0 ,0