Anna University Questions - Fmcetfmcet.in/AUTO/MA6351_auque.pdf · 5. Form the partial differential equation by el8iminating the constants a and b from z x a y b 2 2 2 2 . 6. Solve

MA6351-TRANSFORMS AND PARTIAL

DIFFERENTIAL EQUATIONS

Anna University

Questions

Department of Mathematics

FATIMA MICHAEL

COLLEGE OF ENGINEERING &

TECHNOLOGY

MADURAI – 625 020, Tamilnadu, India

B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2010

Third Semester

Civil Engineering

MA2211 – TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

(Common to all Branches of B.E./B.Tech)

(Regulation 2008)

Time : Three hours Maximum : 100 marks

Answer ALL Questions

PART A – (10 x 2 = 20 marks)

1. Write the conditions for a function ( )f x to satisfy for the existence of a Fourier series.

2. If2

2

21

( 1)4 cos

3

n

n

x nxn

, deduce that

2

2 2 2

1 1 1...

1 2 3 6

.

3. Find the Fourier cosine transform of , 0ax

e x .

4. If ( )F s is the Fourier transform of ( )f x , show that ( ) ( )ias

F f x a e F s .

5. Form the partial differential equation by el8iminating the constants a and b from

2 2 2 2z x a y b .

6. Solve the partial differential equation pq x .

7. 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

. If it is released from rest in this position, write the boundary

conditions.

8. Write all three possible solutions of steady state two – dimensional heat equation.

9. Find the Z – transform of sin2

n.

10. Find the difference equation generated by 2n

n ny a b .

PART B – (5 x 16 = 80 marks)

11. (a) (i) Find the Fourier series for 2( ) 2f x x x in the interval 0 2x .

(ii) 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

.

Or

(b) (i) Find the complex form of the Fourier series of ( )ax

f x e , x .

(ii) Find the first two harmonics of the Fourier series from the following table:

x: 0 1 2 3 4 5

y: 9 18 24 28 26 20

12. (a) (i) Find the Fourier transform of1 1

( )0 1

x if xf x

if x

. Hence deduce the value of

4

4

0

sin

tdt

t

. (10)

(ii) Show that the Fourier transform of

2

2

x

e

is

2

2

s

e

. (6)

Or

(b) (i) Find the Fourier sine and cosine transform of sin , 0

( )0,

x x af x

x a

.

(ii) Using Fourier cosine transform method, evaluate 2 2 2 2

0

dt

a t b t

.

13. (a) Solve:

(i) 2 2 2x yz p y zx q z xy (8)

(ii) 1p q qz (4)

(iii) 2 2 2 2p q x y (4)

Or

(b) (i) Find the partial differential equation of all planes which are at a constant distance ‘ a ’

from the origin.

(ii) Solve 2 22 2 2 sin( 2 )D DD D D D z x y where D

x

and Dy

.

14. (a) 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.

Or

(ii) 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.

15) (a) (i) Solve by Z – transform 2 12 2

n

n n nu u u

with 0

2u and 11u .

(ii) Using convolution theorem, find the inverse Z – transform of

3

4

z

z

.

Or

(b) (i) Find 2

1

2

2

( 1)( 1)

z z zZ

z z

and 1

( 1)( 2)

zZ

z z

. (6 + 4)

(ii) Find sinn

Z na n . (6)


Third Semester



(Regulation 2008)



PART A – (10 x 2 = 20 marks)

1. Give the expression for the Fourier Series co-efficient nb for the function ( )f x defined in

(((( ))))2, 2−−−− .

2. Without finding the values of 0 , na a and nb , the Fourier coefficients of Fourier series, for the

function 2( )F x x==== in the interval (((( ))))0,ππππ find the value of (((( ))))

22 20

12 n nn

aa b

∞∞∞∞

====

+ ++ ++ ++ +

∑∑∑∑ .

3. State and prove the change of scale property of Fourier Transform.

4. If ( )cF s is the Fourier cosine transform of ( )f x , prove that the Fourier cosine transform of

( )f ax is 1

c

sF

a a

.

5. Form the partial differential equation by eliminating the arbitrary constants a and b from

(((( )))) (((( ))))2 2z x a y b= + += + += + += + + .

6. Solve the equation (((( ))))30D D z′′′′− =− =− =− = .

7. A rod 40 cm long with insulated sides has its ends A and B kept at 20⁰C and 60⁰C respectively.

Find the steady state temperature at a location 15 cm from A .

8. Write down the three possible solutions of Laplace equation in two dimensions.

9. Find the Z – transform ofna .

10. What advantage is gained when Z – transform is used to solve difference equation?

PART B – (5 x 16 = 80 marks)

11. (a) (i) Expand (((( ))))( ) 2f x x xππππ= −= −= −= − as Fourier series in (((( ))))0, 2ππππ and hence deduce that the

sum of 2 2 2 2

1 1 1 1...

1 2 3 4+ + + ++ + + ++ + + ++ + + + .

(ii) Obtain the Fourier series for the function ( )f x given by

1 , 0( )

1 , 0

x xf x

x x

ππππππππ

− − < <− − < <− − < <− − < <==== + < <+ < <+ < <+ < <

. Hence deduce that

2

2 2 2

1 1 1...

1 3 5 8ππππ+ + + =+ + + =+ + + =+ + + = .

Or

(b) (i) Obtain the sine series for

in 02( )

in 2

x xf x

x x

≤ ≤≤ ≤≤ ≤≤ ≤==== − ≤ ≤− ≤ ≤− ≤ ≤− ≤ ≤

��

��

.

(ii) 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

12. (a) (i) Find the Fourier transform of

21 1( )

0 1

x if xf x

if x

− <− <− <− <==== >>>>

. Hence evaluate

30

cos sincos

2x x x x

dxx

∞∞∞∞ −−−− ∫∫∫∫ .

(ii) 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

sin2

tdt

tππππ∞∞∞∞

==== ∫∫∫∫ .

Or

(b) (i) Find the Fourier sine transform of

, 0 1

( ) 2 , 1 2

0, 2

x x

f x x x

x

< << << << <= − < <= − < <= − < <= − < < >>>>

.

(ii) Evaluate (((( )))) (((( ))))2 2 2 20

dx

x a x b

∞∞∞∞

+ ++ ++ ++ +∫∫∫∫ using Fourier cosine transforms of axe−−−−

and bxe−−−−

.

13. (a) (i) Form the partial differential equation by eliminating arbitrary functions f and φφφφ from

( ) ( )z f x ct x ctφφφφ= + + −= + + −= + + −= + + − .

(ii) Solve the partial differential equation ( ) ( )mz ny p nx z q y mx− + − = −− + − = −− + − = −− + − = −� �� .

Or

(b) (i) Solve (((( )))) (((( ))))2 2 sin 2 3x yD D z e x y−−−−′′′′− = +− = +− = +− = + .

(ii) Solve (((( ))))2 23 2 2 2 sin(2 )D DD D D D z x y x y′ ′ ′′ ′ ′′ ′ ′′ ′ ′− + + − = + + +− + + − = + + +− + + − = + + +− + + − = + + + .

14. (a) 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 .

Or

(b) 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.

15) (a) (i) Using convolution theorem, find inverse Z – transform of

2

( 1)( 3)z

z z− −− −− −− −.

(ii) Find the Z – transforms of cosna nθθθθ and sinate bt−−−−.

Or

(b) (i) Solve the difference equation ( 3) 3 ( 1) 2 ( ) 0y n y n y n+ − + + =+ − + + =+ − + + =+ − + + = , given that (0) 4y ==== ,

(1) 0y ==== and (2) 8y ==== .

(ii) Derive the difference equation from (((( )))) ( 3)nny A Bn= + −= + −= + −= + − .

B.E./B.Tech. DEGREE EXAMINATION, MAY/JUNE 2012

Third Semester


(Common to all Branches)

(Regulation 2008)



PART A – (10 x 2 = 20 marks)

1. Find the constant term in the expansion of 2cos x as a Fourier series in the interval (((( )))),π ππ ππ ππ π−−−− .

2. Define Root Mean square value of a function ( )f x over the interval (((( )))),a b .

3. What is the Fourier transform of ( )f x a−−−− , if the Fourier transform of ( )f x is ( )F s ?

4. Find the Fourier sine transform of ( ) , 0axf x e a−−−−= >= >= >= > .

5. Form the partial differential equation by eliminating the arbitrary function from

2 xz xy f

z − =− =− =− =

.

6. Solve (((( ))))2 27 6 0D DD D z′ ′′ ′′ ′′ ′− + =− + =− + =− + = .

7. What is the basic difference between the solution of one dimensional wave equation and one

dimensional heat equation with respect to the time?

8. Write down the partial differential equation that represents steady state heat flow in two

dimensions and name the variables involved.

9. Find the Z – transform of for 0

( ) !0 otherwise

nan

x n n

≥≥≥≥====

.

10. Solve 1 2 0n ny y++++ − =− =− =− = given 0 3y ==== .

PART B – (5 x 16 = 80 marks)

11. * (a) (i) Find the Fourier series of (((( ))))2( )f x xππππ= −= −= −= − in (((( ))))0, 2ππππ of periodicity 2ππππ .

(ii) Obtain the Fourier series to represent the function ( )f x x==== , xπ ππ ππ ππ π− < <− < <− < <− < < and

deduce (((( ))))

2

21

182 1n n

ππππ∞∞∞∞

====

====−−−−

∑∑∑∑ .

Or

(b) (i) Find the half-range Fourier cosine series of (((( ))))2( )f x xππππ= −= −= −= − in the interval (0, )ππππ .

Hence find the sum of the series 4 4 4

1 1 1...

1 2 3+ + + + ∞+ + + + ∞+ + + + ∞+ + + + ∞ .

(ii) Find the Fourier series up to second harmonic for the following data for y with period 6.

x: 0 1 2 3 4 5

y: 9 18 24 28 26 20

12. (a) (i) Derive the Parseval’s identity for Fourier Transforms.

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

0 for

1( ) for 0

2 for 0x

x a

f x x

e x−−−−

<<<<= == == == = >>>>

.

Or

(b) (i) State and prove convolution theorem on Fourier transform.

(ii) Find Fourier sine and cosine transform of 1nx −−−−

and hence prove 1

x is self reciprocal

under Fourier sine and cosine transforms.

13. (a) (i) Form the PDE by eliminating the arbitrary functionsφφφφ from

2 2 2( , ) 0x y z ax by czφφφφ + + + + =+ + + + =+ + + + =+ + + + = .

(ii) Solve the partial differential equation 2 2 2( ) ( ) ( )x y z p y z x q z x y− + − = −− + − = −− + − = −− + − = − .

Or

(b) (i) Solve the equation (((( )))) (((( ))))3 2 2 34 4 cos 2D D D DD D z x y′ ′ ′′ ′ ′′ ′ ′′ ′ ′+ − − = ++ − − = ++ − − = ++ − − = + .

(ii) Solve 2 22 6 3 yD DD D D D z xe′ ′ ′′ ′ ′′ ′ ′′ ′ ′ − − + + =− − + + =− − + + =− − + + = .

14. (a) 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.

Or

(b) 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 .

15) (a) (i) Find [[[[ ]]]]( 1)( 2)Z n n n− −− −− −− − .

(ii) Using Convolution theorem, find the inverse Z – transform of

28(2 1)(4 1)

zz z− −− −− −− −

.

Or

(b) (i) Solve the difference equation ( 2) ( ) 1, (0) (1) 0y k y k y y+ + = = =+ + = = =+ + = = =+ + = = = ,using Z-transform.

(ii) Solve 2 2 .nn ny y n++++ + =+ =+ =+ = , using Z-transform.

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2009

Third Semester

Civil Engineering


(Common to all branches)

(Regulation 2008)


Answer ALL questions

PART A – (10 x 2 = 20 marks)

1. State the sufficient condition for a function ( )f x to be expressed as a Fourier series.

2. Obtain the first term of the Fourier series for the function 2( ) , f x x x .

3. Find the Fourier transform of,

( )0, and

ikxe a x bf x

x a x b

.

4. Find the Fourier sine transform of1

x.

5. Find the partial differential equation of all planes cutting equal intercepts from the x and y axes.

6. Solve 3 22 0D D D z .

7. Classify the partial differential equation2

24

u u

x t

.

8. Write down all possible solutions of one dimensional wave equation.

9. If2

( )1 1 3

2 4 4

zF z

z z z

, find (0)f .

10. Find the Z – transform of for 0

( ) !

0 otherwise

na

nx n n

.

PART B – (5 x 16 = 80 marks)

11. (a) (i) 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

.

(ii) 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

Or

(b) (i) 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.

(ii) Find the complex form of the Fourier series of ( )x

f x e in 1 1x .

12. (a) (i) 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

.

(ii) Evaluate 2 2

0 4 25

dx

x x

using transform methods.

Or

(b) (i) Find the Fourier cosine transform of 2x

e

.

(ii) Prove that1

xis self reciprocal under Fourier sine and cosine transforms.

13. (a) 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.

Or

(b) 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.

14. (a) (i) Find the inverse Z – transform of 2

10

3 2

z

z z .

(ii) Solve the equation 2 16 9 2

n

n n nu u u

given 0 1

0u u .

Or

(b) (i) Using convolution theorem, find the 1Z

of

2

4 3

z

z z .

(ii) Find the inverse Z – transform of

3

3

20

2 4

z z

z z

.

15. (a) (i) Solve 2 2z px qy p q .

(ii) Solve 2 2 22 sinh( )

x yD DD D z x y e

.

Or

(b) (i) Solve ( ) ( ) ( )( )y xz p yz x q x y x y .

(ii) Solve 2 23 3 7D D D D z xy .

B.E./B.Tech. DEGREE EXAMINATION, November/December 2010

Regulations 2008

Third Semester

Common to all branches

MA2211 Transforms and Partial Differential Equations

Time : Three Hours Maximum : 100 Marks


PART A – (10 x 2 = 20 Marks)


2. Find the root mean square value of 2( )f x x==== in (((( ))))0, �� .

3. Write the Fourier transform pair.



2 xz xy f

z − =− =− =− =

.

6. Find the particular integral of (((( ))))2 22 x yD DD D z e −−−−′ ′′ ′′ ′′ ′− + =− + =− + =− + = .

7. Write down the three possible solutions of one dimensional heat equation.

8. Give three possible solutions of two dimensional steady state heat flow equation.

9. Define the unit step sequence. Write its Z – transform.

10. Form a difference equation by eliminating the arbitrary constant A from .3nny A==== .

Part B – (5 x 16 = 80 Marks)

11. (a) (i) Find the Fourier series expansion of for 0

( )2 for 2

x xf x

x x

πππππ π ππ π ππ π ππ π π

≤ ≤≤ ≤≤ ≤≤ ≤==== − ≤ ≤− ≤ ≤− ≤ ≤− ≤ ≤

. Also,

deduce that

2

2 2 2

1 1 1...

1 3 5 8ππππ+ + + ∞ =+ + + ∞ =+ + + ∞ =+ + + ∞ = . (10)

(ii) Find the Fourier series expansion of 2( ) 1f x x= −= −= −= − in the interval (((( ))))1,1−−−− . (6)

OR

(b) (i) Obtain the half range cosine series for ( )f x x==== in (((( ))))0,ππππ .

(ii) 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.

x 0 1 2 3 4 5

( )f x 9 18 24 28 26 2 0



0 for 0

1( ) for 0

2 for 0x

x

f x x

e x−−−−

<<<<= == == == = >>>>

.

OR


, 0 1

( ) 2 , 1 2

0, 2

x x

f x x x

x

< << << << <= − < <= − < <= − < <= − < < >>>>

.

(ii) Evaluate (((( )))) (((( ))))2 2 2 20

dx

x a x b

∞∞∞∞


and bxe−−−−

.

13. (a) (i) Form the PDE by eliminating the arbitrary function φφφφ from

(((( ))))2 2 2 , 0x y z ax by czφφφφ + + + + =+ + + + =+ + + + =+ + + + = .

(ii) Solve the partial differential equation (((( )))) (((( )))) (((( ))))2 2 2x y z p y z x q z x y− + − = −− + − = −− + − = −− + − = − .

OR

(b) (i) Solve the equation (((( ))))3 2 2 34 4 cos 2D D D DD D z x y′ ′ ′′ ′ ′′ ′ ′′ ′ ′ + − − = ++ − − = ++ − − = ++ − − = + .


14. (a) 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.

OR

(b) 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 , 0 20u x x x x= − < <= − < <= − < <= − < < while the other two edges are kept at 0⁰C. Fine the steady

state temperature distribution in the plate.

15. (a) (i) Find the Z – transform of cos nθθθθ and sin nθθθθ . Hence deduce the Z – transforms of

(((( ))))cos 1n θθθθ++++ and sinna nθθθθ . (10)

(ii) Find the inverse Z – transform of 3

( 1)( 1)z zz

++++−−−−

by residue method. (6)

OR

(b) (i) Form the difference equation from the relation .3nny a b= += += += + .

(ii) Solve 2 14 3 2nn n ny y y+ ++ ++ ++ ++ + =+ + =+ + =+ + = with 0 0y ==== and 1 1y ==== , using Z – transform.


Regulations 2008

Third Semester





PART A – (10 x 2 = 20 Marks)

1) State the Dirichlet’s conditions for the existence of the Fourier expansion of ( )f x , in the

interval 0,2 .

2) Find the root mean square value of the function ( )f x x in 0, l .

3) Write the Fourier transform pair. 4) State Parseval’s identity on Fourier transform. 5) Find the PDE of the family of spheres having their centers on the z – axis.

6) Solve the equation 3

0D D z .

7) In the wave equation2 2

2

2 2

y yc

t x

, what does 2

c stand for?

8) A plate is bounded by the lines 0, 0,x y x l and y l . Its faces are insulated. The edge

coinciding with x – axis is kept at100 C . The edge coinciding with y – axis is kept at 50 C . The

other two edges are kept at 0 C . Write the boundary conditions that are needed for solving two dimensional heat flow equation.

9) Find the Z – transform of1

!n.

10) Form a difference equation by eliminating arbitrary constants from 12

n

nU A

.

Part B – (5 x 16 = 80 Marks)

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

ii) Obtain the Fourier series of ( ) sinf x x x in , .

Or

b) i) 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

.

ii) The following table gives the variations of a periodic current over a period T

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

Find the fundamental and first harmonics of ( )f x to express ( )f x in a

Fourier series in the form 0

1 1( ) cos sin

2

af x a b , where

2 x

T

.

12) a) i) Show that

2

2

x

e

is a self reciprocal with respect to Fourier transform.

ii) Find the Fourier transform of the function1 , 1

( )0, 1

x if xf x

if x

and

hence find the value of4

4

0

sin tdt

t

.

Or

b) i) Find the Fourier sine transform of axe and hence evaluate Fourier cosine

transforms of axxe

and sinax

e ax .

ii) State and prove convolution theorem for Fourier transforms.

13) a) i) Find the singular integral of 2 21z px qy p q .

ii) Solve the partial differential equation ( ) ( ) ( )x y z p y z x q z x y .

Or

b) i) Solve 3 2 2 22 2 3

xD D D z e x y .

ii) Solve 2 2 22 3 3 2

x yD DD D D D z e

.

14) a) A tightly stretched string of length ‘l’ is initially at rest in its equilibrium position and each of

its points is given the velocity 3

0sin

xV

l

. Find the displacement ( , )y x t .

Or

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

steady state temperature distribution in the plate.

15) a) i) If ( ) ( )Z f n F z , find ( )Z f n k and ( )Z f n k .

ii) Evaluate 31

5Z z

for 5z .

Or

b) i) Solve: 2 1

4 3 3n

n n nu u u

given that

0 10, 1u u .

ii) Form the difference equation of second order by eliminating the arbitrary

constants A and B from ( 2)n

ny A Bn .

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2011

Regulations 2010

Third Semester


181301 - Transforms and Partial Differential Equations



PART A – (10 x 2 = 20 Marks)

1) Find the sum of the Fourier series for, 0 1

( )2, 1 2

x xf x

x

≤ <≤ <≤ <≤ <==== < << << << <

at 1x ==== .

2) The cosine series for ( ) sinf x x x==== for 0 x ππππ< << << << < is given as

22

1 ( 1)sin 1 cos 2

2 1

n

n

x x xn

∞∞∞∞

====

−−−−= − −= − −= − −= − −−−−−∑∑∑∑ . Deduce that

1 1 11 2 ...

1.3 3.5 5.7 2ππππ + − + − =+ − + − =+ − + − =+ − + − =

.

3) Define Fourier transformation pair.

4) Find the Fourier Sine transform of1x

.

5) Form the p.d.e form ( ) ( )z f x t g x t= + + −= + + −= + + −= + + − .

6) Find the complete integral of 2q px==== .

7) State the governing equation for one dimensional heat equation and necessary to solve the

problem.

8) Write the boundary conditions for the following problem. A rectangular plate is bounded by the

line 0, 0,x y x a= = == = == = == = = and y b==== . Its surfaces are insulated. The temperature along 0x ==== and

0y ==== are kept at 0 C�� and the others at100 C�� .

9) Find Z – transformation of!

nan

.

10) Find(((( ))))

12

1

zZ

z−−−−

−−−− .

Part B – (5 x 16 = 80 Marks)

11) a) Calculate the first 3 harmonics of the Fourier of ( )f x from the following data: (16)

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

Or

b) Find the Fourier series of the function0, 0

( )sin , 0

xf x

x x

ππππππππ

− ≤ ≤− ≤ ≤− ≤ ≤− ≤ ≤==== ≤ ≤≤ ≤≤ ≤≤ ≤

and hence evaluate

1 1 1...

1.3 3.5 5.7+ + ++ + ++ + ++ + + . (16)

12) a) Show that the Fourier transform of

2 2 , ( )

0, 0

a x x af x

x a

− ≤− ≤− ≤− ≤==== > >> >> >> >

is

3

2 sin cos2

as as assππππ

−−−−

. Hence deduce that 30

sin cos4

t t tdt

tππππ∞∞∞∞ −−−− ====∫∫∫∫ . Using Perserval’s

identity show that

2

30

sin cos15

t t tdt

tππππ∞∞∞∞ −−−− ====

∫∫∫∫ . (16)

Or

b) i) Find Fourier cosine transformation of 2xe−−−−

. (8)

ii) Find the Fourier sine transformation of

axex

−−−−

where 0a >>>> . (8)

13) a) i) Solve (((( )))) (((( )))) (((( ))))2 2 2 2 2 2 0x y z p y z x q z x y− + − − − =− + − − − =− + − − − =− + − − − = . (8)

ii) Solve (((( ))))2 2 2 2 2z p q x y+ = ++ = ++ = ++ = + . (8)

Or

b) Solve (((( )))) (((( ))))3 2 37 6 cos 2D DD D z x y x′ ′′ ′′ ′′ ′− − = + +− − = + +− − = + +− − = + + . (16)

14) a) A string is stretched and fastened to two points 0x ==== and x l==== apart. Motion is started by

displacing the string into the form (((( ))))2y k lx x= −= −= −= − from which it is released at time 0t ==== . Find

the displacement of any point on the string at a distance of x from one end at time t . (16)

Or

b) 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. (16)

15) a) Using Z-transform, solve 2 14 5 24 8n n ny y y n+ ++ ++ ++ ++ − = −+ − = −+ − = −+ − = − given that 0 3y ==== and 1 5y = −= −= −= − .

(16)

Or

b) i) State and prove convolution theorem on Z-transformation. Find

21

( )( )z

Zz a z b

−−−− − −− −− −− −

. (10)

ii) If

2

4

2 5 14( )

( 1)z z

U zz+ ++ ++ ++ +====

−−−−, evaluate 2u and 3u . (6)

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Third Semester



(Regulation 2008)



PART A – (10 x 2 = 20 marks)

1. Give the expression for the Fourier Series co-efficient nb for the function ( )f x defined in

(((( ))))2, 2−−−− .

2. Without finding the values of 0 , na a and nb , the Fourier coefficients of Fourier series, for the

function 2( )F x x==== in the interval (((( ))))0,ππππ find the value of (((( ))))

22 20

12 n nn

aa b

∞∞∞∞

====

+ ++ ++ ++ +

∑∑∑∑ .

3. State and prove the change of scale property of Fourier Transform.

4. If ( )cF s is the Fourier cosine transform of ( )f x , prove that the Fourier cosine transform of

( )f ax is 1

c

sF

a a

.

5. Form the partial differential equation by eliminating the arbitrary constants a and b from

(((( )))) (((( ))))2 2z x a y b= + += + += + += + + .

6. Solve the equation (((( ))))30D D z′′′′− =− =− =− = .

7. A rod 40 cm long with insulated sides has its ends A and B kept at 20⁰C and 60⁰C respectively.

Find the steady state temperature at a location 15 cm from A .

8. Write down the three possible solutions of Laplace equation in two dimensions.

9. Find the Z – transform ofna .

10. What advantage is gained when Z – transform is used to solve difference equation?

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PART B – (5 x 16 = 80 marks)

11. (a) (i) Expand (((( ))))( ) 2f x x xππππ= −= −= −= − as Fourier series in (((( ))))0, 2ππππ and hence deduce that the

sum of 2 2 2 2

1 1 1 1...

1 2 3 4+ + + ++ + + ++ + + ++ + + + .

(ii) Obtain the Fourier series for the function ( )f x given by

1 , 0( )

1 , 0

x xf x

x x

ππππππππ

− − < <− − < <− − < <− − < <==== + < <+ < <+ < <+ < <

. Hence deduce that

2

2 2 2

1 1 1...

1 3 5 8ππππ+ + + =+ + + =+ + + =+ + + = .

Or

(b) (i) Obtain the sine series for

in 02( )

in 2

x xf x

x x

≤ ≤≤ ≤≤ ≤≤ ≤==== − ≤ ≤− ≤ ≤− ≤ ≤− ≤ ≤

��

��

.

(ii) 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

12. (a) (i) Find the Fourier transform of

21 1( )

0 1

x if xf x

if x

− <− <− <− <==== >>>>

. Hence evaluate

30

cos sincos

2x x x x

dxx

∞∞∞∞ −−−− ∫∫∫∫ .

(ii) 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

sin2

tdt

tππππ∞∞∞∞

==== ∫∫∫∫ .

Or


, 0 1

( ) 2 , 1 2

0, 2

x x

f x x x

x

< << << << <= − < <= − < <= − < <= − < < >>>>

.


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(ii) Evaluate (((( )))) (((( ))))2 2 2 20

dx

x a x b

∞∞∞∞


and bxe−−−−

.

13. (a) (i) Form the partial differential equation by eliminating arbitrary functions f and φφφφ from

( ) ( )z f x ct x ctφφφφ= + + −= + + −= + + −= + + − .

(ii) Solve the partial differential equation ( ) ( )mz ny p nx z q y mx− + − = −− + − = −− + − = −− + − = −� �� .

Or

(b) (i) Solve (((( )))) (((( ))))2 2 sin 2 3x yD D z e x y−−−−′′′′− = +− = +− = +− = + .

(ii) Solve (((( ))))2 23 2 2 2 sin(2 )D DD D D D z x y x y′ ′ ′′ ′ ′′ ′ ′′ ′ ′− + + − = + + +− + + − = + + +− + + − = + + +− + + − = + + + .

14. (a) 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 .

Or

(b) 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.

15) (a) (i) Using convolution theorem, find inverse Z – transform of

2

( 1)( 3)z

z z− −− −− −− −.

(ii) Find the Z – transforms of cosna nθθθθ and sinate bt−−−−.

Or

(b) (i) Solve the difference equation ( 3) 3 ( 1) 2 ( ) 0y n y n y n+ − + + =+ − + + =+ − + + =+ − + + = , given that (0) 4y ==== ,

(1) 0y ==== and (2) 8y ==== .

(ii) Derive the difference equation from (((( )))) ( 3)nny A Bn= + −= + −= + −= + − .


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Reg. No. :


Third Semester

Civil Engineering

MA2211 — TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS


(Regulation 2008)

Time: Three hours Maximum: 100 marks


PART A — (10 × 2 = 20 Marks)

1. Write the conditions for a function ( )xf to satisfy for the existence of a Fourier

series.

2. If ( )∑

∞

=

−+=

12

22 cos

14

3 n

n

nxn

xπ

, deduce that 6

...3

1

2

1

1

1 2

222

π=+++ .

3. Find the Fourier cosine transform of axe− , 0≥x .

4. If ( )sF is the Fourier transform of ( )xf , show that ( )( ) ( )sFeaxfF ias=− .

5. Form the partial differential equation by eliminating the constants a and b

from ( )( )2222 byaxz ++= .

6. Solve the partial differential equation xpq = .

7. A tightly stretched string with fixed end points 0=x and lx = is initially in a

position given by ( )

=l

xyxy

π30 sin0, . If it is released from rest in this

position, write the boundary conditions.

Question Paper Code: E3121


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E 3121 2

8. Write all three possible solutions of steady state two-dimensional heat

equation.

9. Find the Z-transform of 2

sinπn

.

10. Find the difference equation generated by nnn bay 2+= .

PART B — (5 × 16 = 80 Marks)

11. (a) (i) Find the Fourier series for ( ) 22 xxxf −= in the interval 20 << x .

(8)

(ii) Find the half range cosine series of the function ( ) ( )xxxf −= π in

the interval π<< x0 . Hence deduce that 90

...3

1

2

1

1

1 4

444

π=+++ . (8)

Or

(b) (i) Find the complex form of the Fourier series of ( ) axexf = ,

ππ <<− x . (8)

(ii) Find the first two harmonics of the Fourier series from the

following table: (8)

x : 0 1 2 3 4 5

y : 9 18 24 28 26 20

12. (a) (i) Find the Fourier transform of ( )

>

≤−=

1if0

1if1

x

xxxf . Hence

deduce that the value of ∫∞

0

4

4sindt

t

t. (10)

(ii) Show that the Fourier transform of 2

2x

e

−

is 2

2s

e

−

. (6)

Or

(b) (i) Find the Fourier sine and cosine transforms of

( )

>

<<=

ax

axxxf

,0

0,sin. (8)

(ii) Using Fourier cosine transform method, evaluate ( )( )∫∞

++0

2222 tbta

dt.

(8)


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E 3121 3

13. (a) Solve :

(i) ( ) ( ) xyzqzxypyzx −=−+− 222 (8)

(ii) ( ) qzqp =+1 (4)

(iii) 2222 yxqp +=+ . (4)

Or

(b) (i) Find the partial differential equation of all planes which are at a

constant distance ‘a’ from the origin. (8)

(ii) Solve ( ) ( )yxzDDDDDD 2sin222 22 +=′−−′+′+ where x

D∂∂

= and

yD

∂∂

=′ . (8)

14. (a) A tightly stretched string of length ‘l’ has its ends fastened at 0=x and

lx = . 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. (16)

Or

(b) 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 0=y is

given by ( )

≤≤−

≤≤=

105for1020

50for20

xx

xxu and the other three edges are

kept at 0°C. Find the steady state temperature at any point in the plate.

(16)

15. (a) (i) Solve by Z-transform nnnn uuu 22 12 =+− ++ with 20 =u and 11 =u .

(8)

(ii) Using convolution theorem, find the inverse Z-transform of 3

4

−zz

. (8)

Or

(b) (i) Find ( )

( )( )

2

1

2

2

1 1

z z z

Z

z z

− − +

+ − and

( )( )1

1 2

zZ

z z

−

− − . (6 + 4)

(ii) Find ( )θnnaZ n sin . (6)

—————————


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Regulations 2008

Third Semester





PART A – (10 x 2 = 20 Marks)


2. Find the root mean square value of 2( )f x x==== in (((( ))))0, �� .

3. Write the Fourier transform pair.



2 xz xy f

z − =− =− =− =

.

6. Find the particular integral of (((( ))))2 22 x yD DD D z e −−−−′ ′′ ′′ ′′ ′− + =− + =− + =− + = .

7. Write down the three possible solutions of one dimensional heat equation.

8. Give three possible solutions of two dimensional steady state heat flow equation.

9. Define the unit step sequence. Write its Z – transform.

10. Form a difference equation by eliminating the arbitrary constant A from .3nny A==== .


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Part B – (5 x 16 = 80 Marks)

11. (a) (i) Find the Fourier series expansion of for 0

( )2 for 2

x xf x

x x

πππππ π ππ π ππ π ππ π π

≤ ≤≤ ≤≤ ≤≤ ≤==== − ≤ ≤− ≤ ≤− ≤ ≤− ≤ ≤

. Also,

deduce that

2

2 2 2

1 1 1...

1 3 5 8ππππ+ + + ∞ =+ + + ∞ =+ + + ∞ =+ + + ∞ = . (10)

(ii) Find the Fourier series expansion of 2( ) 1f x x= −= −= −= − in the interval (((( ))))1,1−−−− . (6)

OR

(b) (i) Obtain the half range cosine series for ( )f x x==== in (((( ))))0,ππππ .

(ii) 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.

x 0 1 2 3 4 5

( )f x 9 18 24 28 26 2 0



0 for 0

1( ) for 0

2 for 0x

x

f x x

e x−−−−

<<<<= == == == = >>>>

.

OR


, 0 1

( ) 2 , 1 2

0, 2

x x

f x x x

x

< << << << <= − < <= − < <= − < <= − < < >>>>

.

(ii) Evaluate (((( )))) (((( ))))2 2 2 20

dx

x a x b

∞∞∞∞


and bxe−−−−

.

13. (a) (i) Form the PDE by eliminating the arbitrary function φφφφ from

(((( ))))2 2 2 , 0x y z ax by czφφφφ + + + + =+ + + + =+ + + + =+ + + + = .

(ii) Solve the partial differential equation (((( )))) (((( )))) (((( ))))2 2 2x y z p y z x q z x y− + − = −− + − = −− + − = −− + − = − .


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OR

(b) (i) Solve the equation (((( ))))3 2 2 34 4 cos 2D D D DD D z x y′ ′ ′′ ′ ′′ ′ ′′ ′ ′ + − − = ++ − − = ++ − − = ++ − − = + .


14. (a) 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.

OR

(b) 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 , 0 20u x x x x= − < <= − < <= − < <= − < < while the other two edges are kept at 0⁰C. Fine the steady

state temperature distribution in the plate.

15. (a) (i) Find the Z – transform of cos nθθθθ and sin nθθθθ . Hence deduce the Z – transforms of

(((( ))))cos 1n θθθθ++++ and sinna nθθθθ . (10)

(ii) Find the inverse Z – transform of 3

( 1)( 1)z zz

++++−−−−

by residue method. (6)

OR

(b) (i) Form the difference equation from the relation .3nny a b= += += += + .

(ii) Solve 2 14 3 2nn n ny y y+ ++ ++ ++ ++ + =+ + =+ + =+ + = with 0 0y ==== and 1 1y ==== , using Z – transform.


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