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- Numerical Methods - NM
- 2018
- PYQ
**Biju Patnaik University of Technology Rourkela Odisha - BPUT**- Chemical Engineering
- B.Tech
**613 Views**- 5 Offline Downloads
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Registration No. Total number of pages : 04 B.Tech. PCE6I101 6th Semester Regular Examination 2017-18 NUMERICAL METHODS & MATLAB BRANCH : CHEM Time : 3 Hours Max Marks : 100 Q.CODE : C141 Answer Part-A which is compulsory and any four from Part-B. The figures in the right-hand margin indicate marks. Assume suitable notations and any missing data wherever necessary. Answer all parts of a question at a place. Q1. (a) (b) (c) (d) Part – A (Answer all the questions) Answer the following questions : _____ is used to denote the process of finding the valuesinside the interval(X0, Xn). i. Interpolation ii. Extrapolation iii. Iterative iv. Polynomial equation Lagrange's interpolation formula is used to compute the values for _____ intervals. i. Equal ii. Unequal iii. Open iv. Closed Romberg's method is also known as _____. i. Trapezoidal rule ii. Simpson's (1/3)rd Rule iii. Simpson's (3/8)th Rule iv. Rombergs Integration In Simpson's 1/3rd rule the number of intervals must be _____. i. A multiple of 3 ii. A multiple of 6 iii. Odd iv. Even (e) (f) The Eigenvalues of are i. 37,5,-19 ii. -37,-5,19 iii. 7,-3,2 iv. 37,-5,3 The Eigen values of a 4×4 matrix [A] are given as 2,-3, 13, and 7. The det(A) is _____. i. 546 ii. 19 iii. 25 iv. Cannot be determined (2 x 10)

(g) (h) y(x+h) = y(x) + h f(x,y) is referred as _____ method. i. Euler ii. Modified Euler iii. Taylor's Series iv. Runge-Kutta The power method for approximating Eigen value is _____ method. i. Iterative ii. Point-wise iii. Direct iv. Indirect (i) (j) Q2. (a) (b) (c) (d) The partial differential equation is classified as i. Elliptic ii. Parabolic iii. Hyperbolic iv. None of these A partial differential equation requires i. Exactly one independent variable ii. Two or more independent variables iii. More than one dependent variable iv. Equal number of dependent and independent variables Answer the following questions : If Y(Xi) =Yi, i=0, 1,2, …., n write down the formula for the cubic spline polynomial Y(X) valid in Xi-1≤X≤Xi. What is interpolation? What is the difference between interpolation and extrapolation? State Forward divided difference formula for finding F’(x) and f’’(x). The table given below reveals the velocity v of a body during the time t specified. Find its acceleration at t=1.1. T(in sec) V(in m/s) 1.0 43.1 1.1 47.7 1.2 52.1 1.3 56.4 1.4 60.8 (e) (f) Define Discrete Fourier Transform and algebraic form of FFT. (g) (h) (i) What is the need of numerical solution for differential equations? “Multistep methods are not self-starting”. Justify. State the condition of the equation Au xx + Buyy + Cuyy +Du x +Euy + Fu =G where A, B, C, D, E, F, G are functions of x and y to be (i) elliptic (ii) parabolic (iii) hyperbolic. Write down Adam-Bashforth predictor formula. (j) 3 7 4 4 . Find a QR factorization of a matrix (2 x 10)

Q3. (a) Part – B (Answer any four questions) The following table gives some relationship between steam pressure and temperature. Find the pressure at temperature 372 using piecewise linear interpolation. T(K) P(kPa) (b) 361 154.9 367 167.9 (a) 399 244.2 0 2 2 5 4 8 0.2 0.12 0.4 0.49 0.6 1.12 (03) 6 14 Find the values of f’’(0.2), f’’(0.6), f’’(1.0) from the following data using appropriate initial values based on finite difference and Richardson’s extrapolation method. x F(x) Q4. 387 212.5 Find the second derivative at x=4, using the following data: x y (c) 378 191.0 (04) 0.8 2.02 (08) 1.0 3.20 (05) 3 Compute I= tan xdx , using Simpson’s rule with h=/6, /12, /24 and then by 0 (b) Romberg’s method. Using Hermite’s interpolation formula estimate the value of ln3.2 from the following data x 3.0 3.5 4.0 Q5. (a) F(x)=lnx 1.09861 1.25276 1.38629 (10) F’(x)=1/x 0.33333 0.28571 0.25 Find the dominant Eigen value of the following matrix by power method and compare with Rayleigh’s quotient method. (10) 2 1 0 A= 1 2 1 0 1 2 (b) Q6. (a) (b) The differential equation = − satisfied by y(0)=1, y(0.2)=1.1218, y(0.4)=1.4282,y(0.6)=1.7379.Compute y(0.8) by Milne’s predictor-corrector method. 1 1 0 Find the QR factorization of the matrix 1 0 1 using Gram Schmidt process. 0 1 1 Compute 4-point DFT of the following sequence using DIT and DIF algorithms. X(n)={0,1,2,3} (05) (10) (05)

Find the numerically smallest Eigen value of the matrix A by finding A-1 and without finding A-1 given that one of the Eigen values of A is -20. Q7. (15) 15 4 3 A= 10 12 6 20 4 2 Q8. Q9. Solve 25uxx-utt=0 for u with the boundary conditions u(0,t)=0, u(5,t)=0 and the initial conditions ut(x,o)=0 and u(x,0)=2x for 0<x<2.5 u(x,0)=10-2x for 2.5<x<5, taking h=1.(for four time steps) (a) Given (b) 2 f f , t x 2 Subject to f(0,t)=f(5,t)=0, f(x,0)=x2(25-x2). Find f in the range taking h=1 and up to 5 seconds. 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 is 1 unit. (15) (05) (10)

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