University of Delhi 2010-2nd Year M.Tech Information Technology 1st nd sem NANOSCIENCE AND NANOTECHNOLOGY UNIVERSITY - Question Paper

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6992 Ynur Roll \ a

M.Tech, / II Sem. j

NANOSCIENCE AND NANOTECHNOLOGY Pappr NSNT-202 Computational Methods Time 3 hours Maximum Marks 38

(Wnte wur Roll No on tht top imntrdtatelv on m apt of this question paper I

Attempt any four questions AU questions carry equal marks

1 (<?) Explain Newton-Raphson method of hnding a

root of the equation /(jc)=0 Show that it has a second order convergence Use the method to iind root of the equation x sin*-1 *0=0, starting with x--1 *0 and use three iterations only 5i/2

(6) Explain Ramanujans iterative method which can be used to determine the smallest root of the

equation f(x)=0. Use this method to find the

smallest root of*³(w¹+1 Ir6=0 4

forward formula from the following data (for

x and e^r): (1 . 2 7181), (1 05 2 8577); (1 10 3 0042), (1 15 3 1582), (1 20 3 3201), (1 25

3 4903) and (1 30 3 6693) 5i/>

(") Define the operators A, V, <5, E and ft and show:

(i) fiE=En

(h) ²=l+j<5² ₄

3 (a) Use Gauss Quadrature method based on intervals of unequal width to calculate a definite integral

Calculate the weights and abscissae for n2. Obtain the first four Legendre polynomials using recurrence relation

(n + l)P_n+1{u)~(2n+ l)uP_n{uynP_n(u).

Take P₀(u)~l and P_x(u)~u 6

(h) Use lour pomt Gauss-Legendre formula to

C i

evaluate the integral J ~== dx

Abscissae and weights are

, = 0-33998 = 0*65215 w, = 0 86114 iv, = 0*34785

4 (a) Discuss the finite difference method in a two

point linear boundary value problem m an ordinary differential equation and estimate the error involved in this method

Use second order Runge-Kutta formula to cly

evaluate ^x+y\ here y(0)=l; for *=0*3,

take h=0*1 5i/2

(b) Describe with equation Adams-Moulton method (Predietor-Correetor tormula) to find the solution of a differential equation 4

5 (a) Obtain the expressions tor the best value of the

slope and intercept and their standard errors using least square method 5i/2

(b) Discuss different types ot errors m an experimental data, standard error m a single and a set of observations 4

6 Attempt any two ol the following 9in

(f) Obtain Trapezoidal formula for the evaluation of a definite integral Work out the error involved in this method

(u) Obtain the electron-density equation for a crystal in terms of scattering amplitude and location of the atoms in a crystal using Fourier transforms

PT.O

(m) Write the admittance and current matrix of the following circuit and find the voltage vector using matrix operations