FIFTH SEMESTER B.TECH. (ENGINEERING) DEGREEEXAMINATION, DECEMBER 2009

ME/AM 04 50l-COMPUTATIONAL METHODS IN ENGINEERING

1. (a) What is the order of convergence ofNewton-Raphson method? Write down the range for p forwhich Stirling's formula gives most accurate result.

(b) If g(x) is continuous in [a, bJ, then under what condition the iterative method x = g (x) has aunique solution in [a, b] ? Mention the importance of Graeffe's root squaring method.

(c) Compare Gauss-Jacobbi and Gauss-Seidel methods for solving linear systems of the formAX = B. What are the constraints in Crout's reduction method

(d) What is the order of interpolating polynomial could be constructed, if n sets of are given? Listdown the uses of relaxation method.

(e) Using Langrange's interpolation formula find the equation of straight line passing throughthe points (1, 1), (-2, 3). Give an example for numerical differentiation.

(f) What is the geometrical meaning of trapezoidal rule? Make a note on errors of integrationformulae.

(g) How do you apply Runge-Kutta method of order from to solve? Make a brief note on finitedifference method.

(h) Give two disadvantages of Taylor's series method. Explain Milne's predictor corrector formulaeprocedure.

2. (a) By method of Regula Falsi, find a positive toot of xex = cos x.(b) Given the following pairs of values of x and y :

x 1 2 4 8 10y 0 1 5 21 27

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3. (a) Solve:

(i) From the following system by Gauss-Seidel method :-

30x - 2y + 3z = 75 ; 2x + 2y + 18z = 30, x + 17y - 2z = 48.

(ii) By the method of least square fit a straight line to the following data :-

x 5 10 15 20 25 30 35

y 15 20 24 30 37 42 53

(b) Solve 4uxx = utt, subject to the conditions u(O, t) = 0 = u (4, t) ; ut(x, 0) = 0 and u (x, 0) =x (4 - x). Take h = 1 and obtain solutions up to 5 times steps.

1Use Simpson's 3" rule for the first six intervals and trapezoidal rule for the test interval to

2.1evaluate J j(x)dx. Also use Trapezoidal rule for the ptinterval and Simpson's rule for

0.7

2.1the rest of the intervals to evaluate f j(x )dx. COmnlent on the obtain values by comparing

0.7

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1.5 2

(ii) Evaluate J e-x dx, using the 3 points Gaussian quadrants.0.2

(b) Use Newton's forward difference formula to construct interpolating polynomial of degree one,two, and three from the data = f{- 0.75) = 0.0718125, f{ - 0.5) = - 0.02475, f{-0.25) = 0.3349375,f{10) = 1.101 to approximate f{- ?).

(15 marks)

(6 marks)

(b) Given y' = x2 - y, Y (0) = 1 and the starting values y (0.1) = 0.90516, y(0.2) 0.82127 evaluatey(0.3) using 4th order Range-Kentta method and y(O.4) using Adams-Bash forth method.

(9 marks)

(c) Consider 2nd order Initial value problem y" - 2y' + 2y = e21 sin t; with y(O) = - 0.4 andy' (0) = - 0.6, using (i) Taylor's series approximation find y(O.1) ; (ii) Using 4th order R-Kmethod, find y(0.2).

(7 marks)

(d) Using Adam's Bash-forth method find y(4.4) given 5xyl + y2 = 2 ; y(4) = 1 ; y(4.1l = 1.0049,y(4.2.) = 1.0097 and y(4.3) = 1.0143.

(8 marks)[4 x 15 = 60 marks]

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