This document contains practice questions from 5 units of the MA8353 Transforms and Partial Differential Equations course: Unit 1 covers solving partial differential equations using methods like Lagrange's method and Clairaut's method. Unit 2 covers Fourier series topics including half range Fourier series and harmonic analysis. Unit 3 applies partial differential equations to problems involving waves, heat, and vibrating strings. Unit 4 covers Fourier transforms, including finding transforms of basic functions and using properties like self-reciprocity. Unit 5 covers solving difference equations using Z-transforms and applying techniques like convolution and working with poles and residues for inverse transforms. The document provides a pool of important practice questions to help prepare for exams,

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Anna University Exams Nov / Dec 2018 – Regulation 2017
Rejinpaul.com Unique Important Questions – 3rd Semester BE/BTECH
MA8353 Transforms and Partial Differential Equations
UNIT -1 PARTIAL DIFFERENTIAL EQUATIONS
PART-B
1. Solve ).()()( 222222
xyzqxzypzyx 
2. Solve .1 22
qpqypxz 
3. Solve ).cos()( 23223
yxezDDDDDD yx
 
4. Form the partial differential equation by eliminating the arbitrary functions f and g in
).()( 22
xgyyfxz 
5. Solve : p2y(1+x2) = qx2
UNIT -2 FOURIER SERIES
PART – B
1. Find the H.R cosine series for the function f(x)= x(-x) in 0< x<  .Hence deduce that
90
.....
3
1
2
1
1
1 4
444


2. Find the Fourier series of f(x) = {
1 − 𝑥 , −𝜋 < 𝑥 < 0
1 + 𝑥 , 0 < 𝑥 < 𝜋
. Hence deduces
2
2 2 2
1 1 1
.....
1 3 5 8

    
3. Find the half range sine series f(x) = x( -x) in (0,  ) and Deduce that  ....
5
1
3
1
1
1
333
4. Compute the first two harmonics of the Fourier series of y=f(x) from the data
X 0 /
3
2/3  4/3 5/3 2
f(x) 1.0 1.4 1.9 1.7 1.5 1.2 1.0
5. Find the Fourier series for f(x) = 2
x in (- , ) and hence find (i)  ....
3
1
2
1
1
1
444
.(ii)
12
....
3
1
2
1
1
1 2
222


(iii) 6
....
3
1
2
1
1
1 2
222


UNIT -3 APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATION
PART – B
1.A string is tightly stretched and its ends are fastened at two points at x = 0, and x = l. The midpoint of the string is
displaced transversely through a small distance ‘b’and the string is released from rest in that position. Find an
expression for the transverse displacement of the string at any time during the subsequent motion?

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2. A tightly stretched string with fixed end points x=0 and x=l is initially at rest in its equilibrium position. If it
is
set vibrating by giving each point a velocity kx(l-x). Find the displacement of the string at any time.
3. The ends A and B of a rod l cm long have the temperature 400C and 900C until steady state prevails. The
temperature at A is suddenly raised to 900C and at the same time that at B is lowered to 400C. Find the
temperature distribution in the rod at time t. Also show that the temperature at the mid point of the rod
remains unaltered for all time, regardless of the material of the rod.
4. A rectangular plate with insulate surface is 10cm wide and so long compared to it’s with that it may be
considered
infinite in length without introducing appreciable error. Temperature at short edge y = 0 is given
u =





105)10(20
5020
xforx
xxfor
and all the other three edges are kept at 
0 c. Find the steady state
temperature
at any point in the plate.
UNIT 4 – FOURIER TRANSFORMS
PART-B
1. Show that Fourier Transform of











1,0
;
)(
22
xin
axinxa
xf is 




 
3
cossin22
s
asasas

. Hence deduce that
4
cossin
0
3





dt
t
ttt
. Using Parseval’s identity show that
15
cossin
2
0
3






 


dt
t
ttt
.
2. Evaluate (i)
  

0
2222
bxax
dx
(ii)
 
2
2
0
dx
x a


 (iii)
 
2
2
2
0
x dx
x a


 (iv)
  
2
2 2 2 2
0
x dx
x a x b

 
 using Fourier Transform.
3. Find the Fourier Transform of
xa
e

if a>0. Deduce that
 222
22
sa
as
ixeF
xa





 

.
4. Find the Fourier Transform of f(x) given by f(x) =
,
0, 0
a x for x a
for x a
  

 
Hence show that
(i)
2
sin
2
0









dt
t
t
(ii)
3
sin
0
4









dt
t
t
5. Find the Fourier Cosine Transform of
22xa
e
and hence find Fourier sine transform of x
22xa
e

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UNIT- 5 Z TRANSFORMS
PART- B
1. Using Z- transform solve 82454 12   nyyy nnn given that 0y =3 and 1y =-5.
2. Using convolution theorem evaluate inverse Z- transform of
2
8
(2 1)(4 1)
z
z z
 
 
  
3. Find   )(sincos  nandZnZ
4.(i) Find the inverse Z-transform of
𝑧(𝑧+1)
(𝑧−1)3
by Residue method.(ii)Find
2
1
2
( 2)
( 1)( 1)
z z z
Z
z z
   
 
  
by partial
fraction method.
5.State and prove intial and final value theorem also prove Convolution theorem

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Topic wise Important Questions
UNIT 1 PARTIAL DIFFERENTIAL EQUATIONS
1 LAGRANGES LINEAR PARTIAL DIFF EQUATION
2 CLAIRAUTS METHOD TO SOLVE PDE (TYPE-2)
3 SOLVING HOMOGENEOUS AND NON-HOMOGENEOUS PDE
4
FORMING PDE BY ELIMINATION ARBITARTY CONSTANTS AND
FUNCTION
UNIT 2 FOURIER SERIES
1 HALF RANGE FOURIER SERIES WITH PARSEVALS IDENTITY
2 HARMONIC ANALYSIS
3
FOURIER SERIES FOR -EVEN OR ODD FUNCTION WITH CONTINUOUS
POINT AND DISCONTINUOUS POINT
4 FOURIER SERIES FOR FULL RANGE
UNIT 3 APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATION
1
ONE DIMENSION WAVE EQUATION (GIVEN- INITIAL DISPLACEMENT OR
INITIAL VELOCITY )
2 ONE DIMENSIONAL HEAT EQUATION (TYPE-2 AND TYPE-3)
3
TWO DIMENSIONAL HEAT EQUATION ( INFINITE PLATE FOR HORIZONTAL
AND VERTICAL PLATE)
UNIT 4 FOURIER TRANSFORMS
1
Find Fourier Transform of











1,0
;
)(
22
xin
axinxa
xf
2 FIND FOURIER TRANSFORM FOR SELF –RECIPROCAL PROBLEMS
3
Find the Fourier Transform of f(x) given by f(x) =
,
0, 0
a x for x a
for x a
  

 

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4
Evaluate (i)
  

0
2222
bxax
dx
(ii)
 
2
2
0
dx
x a


 (iii)
 
2
2
2
0
x dx
x a


 (iv)
  
2
2 2 2 2
0
x dx
x a x b

 
 using Fourier Transform.
5
Find the Fourier Cosine Transform of
22xa
e
and hence find Fourier sine
transform of x
22xa
e
UNIT 5 Z TRANSFORMS
1 SOLVE DIFFERENCES EQUATION
2 CONVOLUTION PROBLEMS
3 CONVOLUTION STATEMENT AND PROOF
4
FIND INVERSE Z-TRANSFORM BY RESIDUE AND PARTIAL FRACTION
METHODS
5 PROBLEMS IN Z-TRANSFORM.
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