Indian Institute of Technology Kharagpur (IIT-K) 2010 M.Sc Mathematics Transform Calculus - Question Paper

Department of Mathematics
IIT Kharagpur
MA 20101 Transform Calculus End-Autumn 2010
Max-marks: 50 Time: 3hrs
No.of students: 550

Department of Mathematics

11 T Kharagpur MA 20101 Transform Calculus End - Autumn 2010 m Max. Marks : 50 Time : 3hrs.

No. of Students : 550

Instructions : (i) Answer ALL the questions. Provide answers to all parts of each question together, otherwise it will be ignored.

(ii)    L and L'¹ denote the Laplace and inverse Laplace transforms, respectively.

1 r⁰⁰    1 r

(iii)Use    F(u>) = / f{t)e^luJtdt or = f f(t)e~^luJtdt as Fourier transform of the

v27T J-oo    v27T /-oo

function f(t).

1.a)    State and prove the convolution theorem for Laplace transform.

b)    Find L~^l < --7= > in terms of error function.

11 + y/$ j

c)    Using Laplace transform solve the following differential equations (*) W +y* + ty = 0, y(0) = 2, y'(0) = 0.

(ii) y^>'+2y'+5y = e^-tsin(t), j/(0) = 0, y'(O) = 1.

2.    a) Using Laplace transform solve the following boundary value problem

du d²u

Si ^{= k}W^0<x<a'^t>0 subject to the boundary conditions: ix(0,t) 0 and u(a, t) 0, and initial condition

(7T \

1, where k > 0 is a constant.    [6 marks]

_e-aw

b)    Find /(t), if its Fourier sine transform is given by Fs{vj} = - and hence find

Fg¹1 |,    [3 marks]

c)    Find Fourier cosine transform of the function f(t), if

_ / ^cos(). if 0 <  < a, ^J{Z) ~\0, if t>a.

3.a) Using Laplace transform, solve

d²u

s? = a?'^I>0t>0'

with u(,0) = 0, fj(z_}0) = 0, x > 0; u(0,t) = F(f), t > 0 and the solution u(x,t) remains bounded as x 00.    [5 marks]

b)    Find the Fourier transform of exp(~3|t|) and hence find the value of the integral f cos(2)    _rn

J, T+9^dt    [3marksl

c)    Find Fourier transform of the function f(t), if

*u\ - S ^e~^2t> ^if t >

^JK) \ 0, ifi < 0.

Hence vising convolution theorem for Fourier transform find, F^-11 - 1. [4 marksl

((w + 2i)² J

4.a) If the Fourier transform of f(t) defined by F(u>) ~= / /(tjedi, then show

V27T J-do

that

(i)    Fourier transform of f{at) = F(~).

(ii)    Fourier transform of f(t a) e^lwaF(w).

(iii)    Fourier transform of f(t) cos (at) = \ {F(w - a) + F(w + a)}. [3x1 = 3 marks)

b) Solve the following boundary value problem in the half-plane y > 0 by using Fourier transform

d²u d²u

dtf ⁺ dtf =0,~oo<x<oo,y>0) u(x, 0) = f(x), OO < X < 00,

u(x, y) is bounded as y > oo, and u and both vanishes as \x\ > co.    [6 marks]

c) Show that the following differential equation

d²u d²u

dtf ⁺ W ^{= 0}-^0<a:<oo>^>0!

%(#, 0) = f(x), oo < x < oo, u(x, y) is bounded as y v oo, and u and both vanishes as \x\ > oo can be reduced to

d²<f> C?V _r,    _n

+ Qy2 =Q,-oo<x<oo,y>0;

4>(x, 0) = f(x), ~oo < X < oo,

where 4>(x, y) = u_y(x, y). Further using the solution of (46) (or solving independently), show that

1 f

-OO

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2

Attachment: indian-institute-of-technology-kharagpur--iit-k--2010-m-sc-mathematics-2601-2.pdf

Earning:   Approval pending.