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

Regulations 2010

Third Semester

Common to all branches

181301 - Transforms and Partial Differential Equations

Time : Three Hours Maximum : 100 Marks

Answer ALL Questions

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)