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Transforms and Partial Differential Equations 2018

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SUBJECT NAME : Transforms and Partial Differential Equations

SUBJECT CODE : MA6351

MATERIAL NAME : University Questions

REGULATION : R2013

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UPDATED ON : April-May 2018

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Unit – I (Partial Differential Equations)

Formation of PDE and Standard Types of PDE

1. Find the partial differential equation of all planes which are at a constant distance ‘ a ’

from the origin. (A/M 2010),(A/M 2017)

Text Book Page No.: 1.25

2. Form the PDE by eliminating the arbitrary function from

2 2 2, 0x y z ax by cz . (N/D 2010),(M/J 2012)


3. Form the partial differential equation by eliminating arbitrary functions f and from

( ) ( )z f x ct x ct . (A/M 2011)


4. Form the PDE by eliminating the arbitrary functions ‘ f ’ and ‘ g ’ from

2 2( ) ( )z x f y y g x . (N/D 2013)


5. Form the PDE by eliminating the arbitrary function from the relation

2 12 logz y f y

x

. (M/J 2014)




6. Find the singular solution of 2 2z px qy p q . (N/D 2014),(N/D 2017)


7. Solve 2 2z px qy p q . (N/D 2009),(A/M 2015)


8. Find the singular integral of 2 21z px qy p q .

(N/D 2011),(M/J 2013),(N/D 2013),(M/J 2016)


9. Find the singular integral of 2 2z px qy p pq q .

(N/D 2012),(A/M 2017),(A/M 2018)


10. Solve 1p q qz (A/M 2010)


11. Solve 2 2 2 2p q x y (A/M 2010)


12. Solve 2 2 2 2 2z p q x y . (N/D 2011)


13. Solve 2 2 2 2 2x p y q z . (M/J 2014)


14. Obtain the complete solution of 2 2 2 2 2 2p x y q x z . (A/M 2015)




PDE of Lagrange’s Linear Equation

1. Solve the partial differential equation ( ) ( )mz ny p nx z q y mx .(A/M 2011)


2. Solve the partial differential equation 2 2 2x y z p y z x q z x y .

(N/D 2010),(M/J 2012)


3. Solve ( ) ( ) ( )x y z p y z x q z x y . (N/D 2011),(N/D 2014),(A/M 2018)


4. Solve 2 2 2 2 2 2x y z p y z x q z x y . (N/D 2011),(M/J 2013),(A/M 2017)


5. Solve ( 2 ) (2 )x z p z y q y x . (N/D 2012),(N/D 2017)


6. Solve ( ) ( ) ( )( )y xz p yz x q x y x y . (N/D 2009)


7. Solve 2 20y z p xyq xz . (N/D 2013)


8. Solve 2 2 2x yz p y zx q z xy . (A/M 2010),(A/M 2015),(M/J 2016)


9. Solve the Lagrange’s equation 22 2x z p xz y q x y . (M/J 2014)


10. Find the general solution of 2 22z yz y p xy zx q xy zx .(A/M 2017)



Homogeneous Linear Partial Differential Equation

1. Solve 2 2 22 sinh( )

x yD DD D z x y e

. (N/D 2009)


2. Solve 2 22 x yD DD D z e xy . (N/D 2017)

3. Solve 2 2 22

x yD DD D z x y e

. (A/M 2017)

4. Solve 2 2 2 42 2 3

x yD DD D z x y e

. (N/D 2013)


5. Solve 3 2 2 34 4 cos 2D D D DD D z x y . (N/D 2010),(M/J 2012)


6. Solve 3 2 37 6 sin 2D DD D z x y . (M/J 2013)


7. Solve 3 2 37 6 sin 2D DD D z x y . (N/D 2014)


8. Solve 2 2 24 5 sin( 2 )

x yD DD D z x y e

. (A/M 2018)


9. Solve 3 2 2 22 2 3

xD D D z e x y . (N/D 2011),(M/J 2016)


10. Solve 2 3 22

x yD DD z x y e

. (N/D 2014)


11. Solve 2 23 4 cos 2D DD D z x y xy . (N/D 2012)




12. Solve 3 2 37 6 cos 2D DD D z x y x . (N/D 2011)


13. Solve: 2 2 23 2 (2 4 )

x yD DD D z x e

. (A/M 2015)


14. Solve 2 26 cosD DD D z y x . (M/J 2013),(M/J 2014),(N/D 2016),(A/M 2018)


15. Solve 2 25 6 sinD DD D z y x . (N/D 2017)

Non Homogeneous Linear Partial Differential Equation

1. Solve 2 2 22 3 3 2

x yD DD D D D z e

. (N/D 2011)


2. Solve 2 22 4

x yD DD D z e

. (N/D 2012)


3. Solve 2 22 2 2 sin( 2 )D DD D D D z x y . (A/M 2010),(M/J 2016)


4. Solve 2 23 3 7D D D D z xy . (N/D 2009)


5. Solve 2 22 6 3

yD DD D D D z xe . (N/D 2010),(M/J 2012)


6. Solve 2 23 2 2 2 sin(2 )D DD D D D z x y x y .

Text Book Page No.: 1.192 (A/M 2011),(A/M 2017)



Unit – II (Fourier Series)

Fourier Series in the interval (0,2ℓ)

1. Expand ( ) 2f x x x as Fourier series in 0,2 and hence deduce that the

sum of 2 2 2 2

1 1 1 1...

1 2 3 4 . (A/M 2011)


2. Find the Fourier series of 2

( )f x x in 0,2 of periodicity 2 .

(M/J 2012)


3. Find the Fourier series of period 2 for the function ( ) cosf x x x in 0 2x .

(A/M 2017)

4. Find the Fourier series expansion of for 0

( )2 for 2

x xf x

x x

. Also, deduce

that 2

2 2 2

1 1 1...

1 3 5 8

. (N/D 2010)


5. Find the Fourier Series Expansion of 1 for 0

( )2 for 2

xf x

x

. (N/D 2013)


6. Find the Fourier series for a function , 0

( )0, 2

x xf x

x

in 0,2 .(N/D 2017)

7. Find the Fourier expansion of , 0 1

( )2 , 1 2

x xf x

x x

. Also, deduce

2

2 2 2

1 1 1...

1 3 5 8

. (N/D 2012)


8. Find the Fourier series for 2( ) 2f x x x in the interval 0 2x . (A/M 2010)




9. Obtain the Fourier series of periodicity 3 for 2( ) 2f x x x in 0 3x .

(N/D 2011)


Fourier Series in the interval (-ℓ,ℓ)

1. Find the Fourier series of ( )f x x in x . (M/J 2016)

2. Find the Fourier series of 2x in x . Hence deduce the value of

21

1

n n

and

4

4 4 4

1 1 1...

1 2 3 90

. (N/D 2014),(M/J 2013),(N/D 2016),(A/M 2018)


3. Obtain the Fourier series of ( ) sinf x x x in , . (N/D 2011)


4. Obtain the Fourier series to represent the function ( )f x x , x and

deduce

2

21

1

82 1n n

. (M/J 2012)


5. Find the Fourier series of ( ) sinf x x in x of periodicity 2 . (A/M 2015)


6. Find the Fourier series expansion of ( ) cosf x x in x . (M/J 2016)

7. Obtain the Fourier series of the periodic function defined by

0( )

0

xf x

x x

. Deduce that

2

2 2 2

1 1 1...

1 3 5 8

. (N/D 2009)




8. Obtain the Fourier series for the function ( )f x given by1 , 0

( )1 , 0

x xf x

x x

.

Hence deduce that2

2 2 2

1 1 1...

1 3 5 8

. (A/M 2011)


9. Find the Fourier series expansion of 1, 0

( )1, 0

x xf x

x x

. (N/D 2013)


10. Find the Fourier series of the function0, 0

( )sin , 0

xf x

x x

and hence evaluate

1 1 1...

1.3 3.5 5.7 . (N/D 2011)


11. Expand

21 , - 0

( )2

1 , 0

xx

f xx

x

as a full range Fourier series in the interval

, . Hence deduce that 2

2 2 2

1 1 1...

1 3 5 8

. (M/J 2014)


12. Expand 2( )f x x x as a Fourier series in L x L and using this series find the

root mean square value of ( )f x in the interval. (N/D 2009)


13. Find the Fourier series expansion of 2( )f x x x in , . (N/D 2012),(N/D 2017)


14. Find the Fourier series expansion of 2( ) 1f x x in the interval 1,1 .

(N/D 2010)




Half Range Fourier Series

1. Find the half-range cosine series for ( )f x x in 0, . Hence deduce the value of

2 2 2

1 1 1...

1 3 5 . (N/D 2017)

2. Find the half range cosine series of the function ( ) ( )f x x x in the interval

0 x . Hence deduce that4

4 4 4

1 1 1...

1 2 3 90

. (A/M 2010),(A/M 2018)


3. Find the half-range Fourier cosine series of 2

( )f x x in the interval (0, ) .

Hence find the sum of the series 4 4 4

1 1 1...

1 2 3 . (M/J 2012)


4. Obtain the Fourier cosine series of 2

1 , 0 1x x and hence show that

2

2 2 2

1 1 1...

1 2 3 6

. (M/J 2013),(N/D 2014)


5. Obtain the half range cosine series for ( )f x x in 0, . (N/D 2010),(N/D 2012)


6. Find the Fourier half-range cosine series of , 0 1

( )2 - , 1 2

x xf x

x x

. (A/M 2017)

7. Find the half-range sine series of , 0 / 2

( )- , / 2

x xf x

x x

. Hence deduce the sum

of the series 2

1

1

(2 1)n n

. (A/M 2015)


8. Obtain the Fourier cosine series expansion of sinx x in 0, and hence find the value

of 2 2 2 2

1 ...1.3 3.5 5.7 7.9

. (N/D 2011)




9. Find the half-range sine series of 2( ) 4f x x x in the interval 0,4 . Hence deduce

the value of the series 3 3 3 3

1 1 1 1...

1 3 5 7 . (M/J 2014)


10. Obtain the Fourier cosine series expansion of ( )f x x in 0 4x . Hence deduce the

value of 4 4 4

1 1 1...

1 3 5 . (N/D 2014)


11. Find the half range sine series of 2( )f x x x in 0, . (N/D 2013)


12. Obtain the sine series for

in 02

( )

in 2

x x

f x

x x

. (A/M 2011)


13. Obtain the Fourier cosine series for

in 02

( )

in 2

kx x

f x

k x x

. (M/J 2013)


Complex Form of Fourier Series

1. Find the complex form of the Fourier series of ( )ax

f x e , x .(A/M 2010)



f x e in, x . (N/D 2016)


f x e in, x . (A/M 2017)



4. Find the complex form of the Fourier series of ( )x

f x e in 1 1x .

(N/D 2009),(A/M 2015)


5. Find the complex form of Fourier series of cosax in , , where " "a is not an

integer. (M/J 2013)


6. Expand ( ) sinf x x as a complex form Fourier series in , . (M/J 2014)


Harmonic Analysis

1. Compute upto first harmonics of the Fourier series of ( )f x given by the following table

x 0 T/6 T/3 T/2 2T/3 5T/6 T

( )f x 1.98 1.30 1.05 1.30 – 0.88 – 0.25 1.98

(N/D 2009),(N/D 2011)


2. Find the Fourier series as far as the second harmonic to represent the function ( )f x

with the period 6, given in the following table.

(N/D 2009),(N/D 2010),(M/J 2012),(N/D 2012),(N/D 2016),(A/M 2017)

x 0 1 2 3 4 5

( )f x 9 18 24 28 26 20


3. Find the Fourier cosine series up to third harmonic to represent the function given by the

following data: (M/J 2016)

x 0 1 2 3 4 5 ( )f x 9 18 24 28 26 20

4. Find the Fourier series up to second harmonic for ( )y f x from the following values.

x: 0 π/3 2 π/3 π 4π/3 5 π/3 2 π

y: 1.0 1.4 1.9 1.7 1.5 1.2 1.0

(A/M 2011),(N/D 2013),(M/J 2014),(N/D 2014),(A/M 2015),(N/D 2017),(A/M 2018)




5. Calculate the first 3 harmonics of the Fourier of ( )f x from the following data

x : 0 30 60 90 120 150 180 210 240 270 300 330

( ) :f x 1.8 1.1 0.3 0.16 0.5 1.3 2.16 1.25 1.3 1.52 1.76 2.0

(N/D 2011)


Unit – III (Application of Partial Differential Equation)

One Dimensional Wave Equation with No Velocity

1. A string is stretched and fastened to points at a distance apart. Motion is started by

displacing the string in the form sinx

y a

, 0 x , from which it is released

at time 0t . Find the displacement at any time t . (M/J 2014)


2. A tightly stretched string with fixed end points 0x and x is initially in a position

given by 3

0( ,0) sin

xy x y

. It is released from rest from this position. Find the

expression for the displacement at any time t . (N/D 2012)


3. A uniform string is stretched and fastened to two points ‘ ’ apart. Motion is started by

displacing the string into the form of the curve ( )y kx x and then released from

this position at time 0t . Derive the expression for the displacement of any point of

the string at a distance x from one end at time t .


(A/M 2011),(N/D 2011),(N/D 2013),(A/M 2015),(N/D 2017),(A/M 2018)

4. A tightly stretched string of length ‘ ’ has its ends fastened at 0x and x . The

mid – point of the string is then taken to height ‘b’ and released from rest in that

position. Find the lateral displacement of a point of the string at time ‘t’ from the

instant of release. (A/M 2010)


5. A tightly stretched string of length 2 is fastened at both ends. The midpoint of the

string is displaced by a distance ‘b’ transversely and the string is released from rest in



this position. Find an expression for the transverse displacement of the string at any

time during the subsequent motion. (N/D 2010),(A/M 2017)


One Dimensional Wave Equation with Velocity

1. A tightly stretched string with fixed end points 0x and x is initially at rest in its

equilibrium position. If it is set vibrating giving each point a initial velocity 3 ( )x x ,

find the displacement. (N/D 2009)


2. A tightly stretched string between the fixed end points 0x and x is initially at

rest in its equilibrium position. If each of its points is given a velocity ( )kx x , find

the displacement ( , )y x t of the string. (M/J 2013)


3. A tightly stretched string of length ‘ ’ is initially at rest in its equilibrium position and

each of its points is given the velocity 3

0

0

sint

y xV

t

. Find the displacement

( , )y x t . (N/D 2011),(N/D 2014)


4. Find the displacement of a string stretched between two fixed points at a distance of

2 apart when the string is initially at rest in equilibrium position and points of the

string are given initial velocities v where

in 0,

2in ,2

x

vx

, x being the

distance measured from one end.

One Dimensional Heat Equation with Both Ends Are

Change to Zero Temperature

1. Find the solution to the equation 2

2

2

u ua

t x

that satisfies the conditions

(0,t) 0, ( ,t) 0u u , for 0t and , 0 / 2

( ,0), / 2

x xu x

x x

.

(N/D 2013),(A/M 2015)




2. A rod, 30 cm long has its ends A and B kept at 20⁰C and 80⁰C respectively, until steady

state conditions prevail. The temperature at each end is then suddenly reduced to 0⁰C

and kept so. Find the resulting temperature function is a regular function ( , )u x t taking

0x at A. (N/D 2009)


One Dimensional Heat Equation with Both Ends are Change

to Non Zero Temperature

1. A bar of 10 cm long, with insulated sides has its ends A and B maintained at temperatures

50 C and 100 C respectively, until steady-state conditions prevail. The temperature at

A is suddenly raised to 90 C and at B is lowered to 60 C . Find the temperature

distribution in the bar thereafter. (N/D 2011)(AUT)


2. A bar 10 cm long with insulated sides, has its ends A and B kept at 20 ˚C and 40 ˚C

respectively until steady state conditions prevail. The temperature at A is then suddenly

raised to 50 ˚C and at the same instant that at B is lowered to 10 ˚C. Find the subsequent

temperature at any point of the bar at any time. (A/M 2018)


3. The ends A and B of a rod 40 cm long have their temperatures kept at 0˚C and 80˚C

respectively, until steady state condition prevails. The temperature of the end B is then

suddenly reduced to 40˚C and kept so, while that of the end A is kept at 0˚C. Find the

subsequent temperature distribution ( , )u x t in the rod. (M/J 2012)


4. A rod of length cm has its ends A and B kept at 0˚C and 100˚C respectively, until

steady state conditions prevail. If the temperature at B is suddenly reduced to 0˚C and

maintained at 0˚C, find the temperature distribution ( , )u x t at a distance x from A at

any time t . (N/D 2017)

Two Dimensional Heat Equation

1. A rectangular plate with insulated surface is 10 cm wide and so long compared to its

width that may be considered infinite in length without introducing appreciable error.

The temperature at short edge 0y is given by20 for 0 5

20(10 ) for 5 10

x xu

x x

and



the other three edges are kept at 0°C. Find the steady state temperature at any point in

the plate. (A/M 2010),(M/J 2013)


2. A rectangular plate with insulated surface is 20 cm wide and so long compared to its

width that it may be considered infinite in length without introducing an appreciable

error. If the temperature of the short edge 0x is given by

10 for 0 10

10(20 ) for 10 20

y yu

y y

and the two long edges as well as the other short

edge are kept at 0°C. Find the steady state temperature distribution in the plate.

(A/M 2011),(A/M 2017)


3. A square plate is bounded by the lines 0, 0, 20x y x and 20y . Its faces are

insulated. The temperature along the upper horizontal edge is given by

( ,20) (20 ),u x x x 0 20x while the other two edges are kept at 0 C . Find

the steady state temperature distribution in the plate.

Text Book Page No.: 3.83 (N/D 2010),(N/D 2011),(N/D 2016)

4. A square plate is bounded by the lines 0, , 0x x a y and y b . Its surfaces are

insulated and the temperature along y b is kept at 100˚C. While the temperature

along other three edges are at 0˚C. Find the steady-state temperature at any point in

the plate. (N/D 2014)


5. Find the steady state temperature distribution in a rectangular plate of sides a and b

insulated at the lateral surfaces and satisfying the boundary conditions:

(0, ) ( , ) 0, for 0 ;u y u a y y b

( , ) 0 and ( ,0) ( ), for 0 .u x b u x x a x x a (N/D 2012)




6. A long rectangular plate with insulated surface is cm wide. If the temperature along

one short edge ( 0)y is 2( ,0) ( )u x k x x degrees, for 0 x , while the other

two long edges 0x and x as well as the other short edge are kept at 0˚C , find

the steady state temperature function ( , )u x y . (M/J 2012),(M/J 2016)


7. An infinitely long rectangular plate with insulated surfaces is 10cm wide. The two long

edges and one short edge are kept at 0˚C, while the other short edge 0x is kept at

temperature

20 , 0 5

20 10 , 5 10

u y y

u y y

. Find the steady state temperature

distribution in the plate. (M/J 2014)


Unit – IV (Fourier Transform)

Fourier Transform with Deduction

1. Find the Fourier transform of1 1

( )0 1

x if xf x

if x

and hence find the value of

4

4

0

sin

tdt

t

. (N/D 2009),(A/M 2010),(N/D 2011),(N/D 2012),(N/D 2016),(N/D 2017)


2. Find the Fourier transform of1 1

( )0 1

x if xf x

if x

and hence find the value of i)

2

2

0

sin

tdt

t

ii) 4

4

0

sin

tdt

t

. (N/D 2014),(M/J 2016)


3. Find the Fourier transform of ( )f x given by 1 for

( )0 for 0

x af x

x a

and using

Parseval’s identity prove that

2

0

sin

2

tdt

t

. (A/M 2011)




4. Find the Fourier transform of1 for

( )0 for

x af x

x a

and hence evaluate0

sin xdx

x

.

Using Parseval’s identity, prove that 2

2

0

sin

2

tdt

t

.

Text Book Page No.: 4.20 (M/J 2013),(A/M 2015),(A/M 2017)

5. Find the Fourier transform of

21 1( )

0 1

x if xf x

if x

. Hence evaluate

3

0

cos sincos

2

x x x xdx

x

. (A/M 2011),(N/D 2013),(A/M 2018)


6. Show that the Fourier transform of

2 2 , ( )

0, 0

a x x af x

x a

is

3

2 sin cos2

as as as

s

. Hence deduce that3

0

sin cos

4

t t tdt

t

. Using

Perserval’s identity show that

2

3

0

sin cos

15

t t tdt

t

. (N/D 2011),(N/D 2012)


Integration using Parseval’s Identity

1. Evaluate

22 2

0

dx

x a

using Parseval’s identity. (M/J 2013),(N/D 2013),(M/J 2014)


2. Using Parseval’s identity evaluate

2

22 2

0

xdx

x a

. (M/J 2014)


3. Find the Fourier sine and cosine transforms of a function @. Using Parseval’s identity,

evaluate: (N/D 2017)



(i)

22

0 1

dx

x

and (ii)

2

22

0 1

x dx

x

4. Evaluate 2 2

0 4 25

dx

x x

using transform methods. (N/D 2009)


5. Evaluate 2 2

0 1 4

dx

x x

using Fourier transforms. (A/M 2017)


6. Using Fourier cosine transform method, evaluate 2 2 2 2

0

dt

a t b t

.(A/M 2010)


7. Evaluate 2 2 2 2

0

dx

x a x b

using Fourier cosine transforms of ax

e and bx

e .

(N/D 2010),(A/M 2011),(N/D 2014)


Fourier Transform of Exponential Function&

Self Reciprocal Problems

1. Find the Fourier sine transform of axe and hence evaluate Fourier cosine transforms of

axxe

and sinax

e ax . (N/D 2011)


2. Find the Fourier cosine and sine transforms of ( ) , 0ax

f x e a and hence deduce

the integrals 2 2

0

cos sxds

a s

and 2 2

0

sins sxds

a s

. (N/D 2012),(M/J 2016)


3. Find the Fourier sine transformation of ax

e

x

where 0a . (N/D 2011),(N/D 2016)




4. Find the function whose Fourier Sine Transform is 0as

ea

s

. (N/D 2013)


5. Show that the Fourier transform of

2

2

x

e

is

2

2

s

e

.

Text Book Page No.: 4.28 (A/M 2010),(M/J 2013),(M/J 2016),(A/M 2018)

6. Find the Fourier cosine transform of 2 2

, 0a x

e a . Hence show that the function

2

2

x

e

is self-reciprocal. (N/D 2012),(A/M 2017),(A/M 2018)


7. Find the Fourier transform of 2 2

, 0a x

e a . Hence show that the function

2

2

x

e

is self-

reciprocal under the Fourier transform. (N/D 2014),(A/M 2015),(N/D 2016)


8. Show that

2

2

x

e

is a self reciprocal with respect to Fourier transform. (N/D 2011)


9. Find the Fourier transform of 1

( )f xx

. (M/J 2014)


10. Prove that1

xis self reciprocal under Fourier sine and cosine transforms.(N/D 2009)


11. Find Fourier sine and cosine transform of 1nx

and hence prove 1

x is self reciprocal

under Fourier sine and cosine transforms. (M/J 2012),(A/M 2015)


Fourier Transform of General Function& Derivations

1. Find the Fourier sine and cosine transform of sin , 0

( )0,

x x af x

x a

. (A/M 2010)




2. Find the Fourier integral representation of ( )f x defined as

0 for 0

1( ) for 0

2

for 0x

x

f x x

e x

.

Text Book Page No.: 4.33 (N/D 2010),(M/J 2012)

3. Find the Fourier sine transform of

, 0 1

( ) 2 , 1 2

0, 2

x x

f x x x

x

. (N/D 2010),(A/M 2011)


4. Solve for ( )f x from the integral equation 0

1, 0 1

( )sin 2, 1 2

0, 2

s

f x sx dx s

s

.

Text Book Page No.: 4.39 (M/J 2014)

5. Derive the Parseval’s identity for Fourier Transforms. (N/D 2010),(M/J 2012)


6. State and prove convolution theorem for Fourier transforms. (N/D 2011),(M/J 2012)


7. Verify the convolution theorem under Fourier Transform, for2

( ) ( ) ex

f x g x .

(M/J 2013)

Unit – V (Z - Transforms)

Simple problems on Z - transforms

1. Find ( 1)( 2)Z n n n . (M/J 2012)


2. Find the Z – transform of 1

( 1)( 2)n n . (N/D 2013)





( 1)n n, for 1n . (N/D 2014),(M/J 2016)


4. Find the Z-transform of 2 3

( 1)( 2)

n

n n

. (A/M 2017),(N/D 2017)

5. Find cosZ n and sinZ n . (N/D 2014)


6. Find the Z – transform of cosn and sinn . Hence deduce the Z – transforms of

cos 1n and sinn

a n . (N/D 2010)


7. Find cosZ n and hence deduce cos2

nZ

. (M/J 2013),(M/J 2016)



sin4

n

and cos2 4

n

. (N/D 2012)


9. Find the Z – transforms of cosn

a n and sinat

e bt . (A/M 2011)

Text Book Page No.: 5.21; 5.31

10. Find the Z – transforms of cosn

r n and cosat

e bt

. (M/J 2014)


11. Find cosn

Z r n and sinn

Z r n . (A/M 2015)


12. Find sinn

Z na n . (A/M 2010)

Text Book Page No.: -----

13. If ( ) ( )Z f n F z , find ( )Z f n k and ( )Z f n k . (N/D 2011)




14. State and prove the second shifting property of Z-transform. (M/J 2013)


Inverse Z - transform by Partial Fraction

1. Find the inverse Z – transform of 2

10

3 2

z

z z . (N/D 2009)


2. Find the inverse Z -transform of 3

2( 1) ( 2)

z

z z by method of partial fraction.(N/D

2017)

3. Find the inverse Z – transform of

3

3

20

2 4

z z

z z

. (N/D 2009)


4. Find 2

1

2

2

( 1)( 1)

z z zZ

z z

and 1

( 1)( 2)

zZ

z z

. (A/M 2010)


5. Find 3

1

2

4

(2 1) ( 1)

zZ

z z

, by the method of partial fractions. (A/M 2017)

6. Find the inverse Z-transform of 2

2( 1)( 1)

z z

z z

, using partial fraction. (N/D 2014)


7. Evaluate 31

5Z z

for 5z . (N/D 2011)


Inverse Z - transform by Residue Theorem



1. Find the inverse Z-transform of 2

3

( 5)( 2)

z z

z z

using residue theorem.

2. Find the inverse Z – transform of 3

( 1)

( 1)

z z

z

by residue method. (N/D 2010)


3. Evaluate

3

1

2

9

3 1 2

zZ

z z

, using calculus of residues. (N/D 2016)

4. Using residue method, find 1

22 2

zZ

z z

. (A/M 2015),(M/J 2016)


Inverse Z - transform by Convolution Theorem

1. Using convolution theorem, find the 1Z

of

2

4 3

z

z z . (N/D 2009)


2. Using convolution theorem, find inverse Z – transform of 2

( 1)( 3)

z

z z .

(A/M 2011),(N/D 2013) Text Book Page No.: 5.76

3. Using convolution theorem, find the inverse Z – transform of 2

2( )

z

z a. (N/D 2012)


4. Using convolution theorem, find the inverse Z – transform of 2

1

2( )

zZ

z a

.

(M/J 2016)

5. Using convolution theorem, find 2

1

( )( )

zZ

z a z b

.

Text Book Page No.: 5.75 (N/D 2011),(M/J 2013),(M/J 2014),(N/D 2016)

6. Using Convolution theorem, find the inverse Z – transform of 2

8

(2 1)(4 1)

z

z z .

(M/J 2012)




7. Using Convolution theorem, find 2

1 8

(2 1)(4 1)

zZ

z z

. (A/M 2017),(A/M 2018)


8. Using Convolution theorem, find

2

1

1 / 2 1 / 4

zZ

z z

. (A/M 2015),(N/D 2017)


9. Using convolution theorem, find the inverse Z – transform of

3

4

z

z

. (A/M 2010)


Formation and Solution of Difference Equation

1. Form the difference equation from the relation .3n

ny a b . (N/D 2010)


2. Derive the difference equation from ( 3)n

ny A Bn . (A/M 2011)


3. Form the difference equation form ( ) 2n

y n A Bn . (N/D 2013)


4. Form the difference equation of second order by eliminating the arbitrary constants A

and B from ( 2)n

ny A Bn . (N/D 2011)


5. Using Z-transform solve: 2 13 10 0

n n ny y y

, 0

1y and 10y .

(M/J 2013),(M/J 2014)


6. Using Z-transform, solve 2 13 2 0

n n nu u u

given that 0

0u , 11u .

(N/D 2014)




7. Using Z-transform, solve the difference equation ( 2) 3 ( 1) 2 ( ) 0x n x n x n

given that (0) 0x , (1) 1x . (A/M 2015)


8. Solve: 2 1

4 4 0n n n

y y y ,

01y ,

10y , using Z-transform. (A/M 2018)

9. Solve the equation 2 1

6 9 2n

n n nu u u

given

0 10u u .

Text Book Page No.: 5.91 (N/D 2009),(N/D 2012),(N/D 2016)

10. Using Z-transforms, solve 2 1

7 12 2n

n n ny y y

, given that

0 10y y .

(A/M 2017),(N/D 2017)

11. Solve by Z – transform 2 1

2 2n

n n nu u u

with

02u and

11u .(A/M 2010)


12. Solve2 1

4 3 2n

n n ny y y

with

00y and

11y , using Z – transform.(N/D 2010)


13. Solve: 2 1

4 3 3n

n n nu u u

given that

0 10, 1u u . (N/D 2011)


14. Solve 2 1

3 2 4n

n n nu u u

, given that

0 10, 1u u . (M/J 2014)


15. Solve the difference equation 22

n ny y

, given that 0

0y and 10y by using

Z -transforms. (M/J 2016)

16. Solve ( 2) ( ) 1, (0) (1) 0y k y k y y ,using Z-transform. (M/J 2012)


17. Using Z-transform solve the difference equation 2 12

n n ny y y n

given

0 10y y . (N/D 2013)




18. Solve2

2 .n

n ny y n

, using Z-transform. (M/J 2012)


19. Using Z-transform, solve2 1

4 5 24 8n n n

y y y n given that

03y and

15y .

(N/D 2011)


20. Solve the difference equation ( 3) 3 ( 1) 2 ( ) 0y n y n y n , given that (0) 4y ,

(1) 0y and (2) 8y . (A/M 2011),(N/D 2012)


Textbook for Reference:

“TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS”

Edition: 3rd Edition

Publication: Sri Hariganesh Publications Author: C. Ganesan

Mobile: 9841168917, 8939331876

To buy the book visit www.hariganesh.com/textbook

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Unit I (Partial Differential Equations) - hariganesh.comhariganesh.com/pdf/University Questions/uq_r13-m3.pdf · SUBJECT NAME : Transforms and Partial Differential Equations SUBJECT