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MODEL EXAMINATION

TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATION

Year/Semester & Branch: II / III Common to all Branches

Max. Marks: 100 Time: 180 min

PART-A Answer ALL Questions (20X2=40)

1. Determine the value of n a in the Fourier series expansion of (((( )))) , x x f 3==== π ππ π π ππ π <<<<<<<<−−−− x . (A/M 08)

2. Define Root mean square of over the range (( )) b , a . (Tri-N/D 08)

3. If (((( )))) x 2 x f ==== in the interval (0,4), then find the value of a in the Fourierseries expansion. (Cbe-N/D 08)

4. State Dirichlet’s condition for Fourier series. (Tnl-N/D 08) (Nov 05)5. Let (((( )))) s F c be the Fourier cosine transform of (((( )))) x f . Prove that

(((( ))))[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]] a s F a s F1

ax cos x f F c c c −−−−++++++++==== . (Cbe-N/D 08)

6. State Fourier integral theorem. (A/M 08) (Cbe-N/D 08)

7. If (((( ))))[[[[ ]]]] (((( ))))S F x f F ==== then prove that (((( )))) (((( )))) a s F x f e F iax ++++==== (Tnl-N/D 08)

8. what is the sine transform of (( )) ax f if (((( )))) s f s is the Fourier sine transform of

(((( )))) x f .(Tri-N/D 08)9. Form the partial differential equation by eliminating arbitrary constants from

b ay x a z 2 2 ++++++++==== (Cbe-N/D 08)

10. Find the complete integral of (((( ))))(((( )))) 1q pqy px z ====++++−−−−−−−− (N/D 08).

11. Eliminate the function ‘f’ from 2 2 y x f z ++++==== (Tnl-N/D 08).12. Write the complete integral of y xq p ++++====++++ (Tnl-N/D 08)

13. A rod 50cm long with insulated sides has its ends A and B kept at C 200 and

C 70 0 respectively. Find the steady state temperature distribution of the rod.(Cbe-N/D 08)

14. Classify the partial differential equation 0u 2u 3u 4u 3 x y xy xx ====−−−−++++++++ .

(A/M 08)15. Write the initial conditions of the wave equation if the string has an initial

displacement but no initial velocity. (Tnl-N/D 08)16. List all the possible solutions of the one dimensional wave equation and state the

proper solution. (Tri-N/D 08)

17. Form the difference equation from n n 3 b a y ++++==== . (A/M 08)

18. Find {{{{ }}}} . n Z (A/M 08) (Tnl-N/D 08)

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19. If (( ))[[ ]] (( )) , Z F n f Z ==== What is (( ))[[ ]] k n f Z −−−− ? (Tri-N/D 08)

20. Find the Z- transform of ! n

1. (Tri-N/D 08)

PART-B (Answer ANY 5 questions) (5 X 12 = 60)

21. a) Obtain the Fourier series of (((( )))) x f of period 2 l and defined as follows

(((( ))))<<<<≤≤≤≤

≤≤≤≤<<<<−−−−====

l xl

l x xl x f

2

0

,0

,.Hence deduce

(((( ))))∑∑∑∑∞∞∞∞

==== ++++0 n 2

.1 n 2

1(Tnl-

N/D 08)

b) Find the half range sine series of ( ) 2 x x f = in (((( ))))π ππ π ,0 (Tnl-N/D 08)

22. Find the Fourier cosine transform of (((( )))) −−−−====0

1 2 x x f otherwise

x 10 <<<<<<<<. Hence

prove that ∫∫∫∫∞∞∞∞

====

−−−−

03

.163

2cos

cossin π ππ π dx

x

x

x x x(Tnl-N/D 08)

23. a) Solve (((( )))) (((( )))) (((( )))) y x zq x z y p z y x −−−−====−−−−++++−−−− .(Tri-N/D 08)

b) Solve y x z D 2 D D 3 D 2 2 ++++====′′′′++++′′′′++++ .(Tri-N/D 08)

24. If a string of length ‘ l ’ is initially at rest in its equilibrium position and each of

its points is given a velocity ‘ v ’ such that−−−−

====)( xl c

cxv

for

for

l xl

l x

<<<<<<<<

<<<<<<<<

2

20

.

Determine the displacement function (( )) t , x y at any time t .(Cbe –N/D 08)

25. a) Find the Z-transform of θ θθ θ n cos and θ θθ θ n sin . Hence find 2

n cos Z

π ππ π .

b) Solve n n n 2 y y ====−−−−++++ using Z-transform given 0 y y 10 ======== .

26. a) Obtain the Fourier series upto second harmonic from the datax : 0 3 / π ππ π 3 / 2π ππ π π ππ π 3 / 4π ππ π 3 / 5π ππ π π ππ π 2 (((( )))) x f : 0.8 0.6 0.4 0.7 0.9 1.1 0.8 (Cbe N/D

08)

b) Find the Fourier cosine transform of axe−−−−

. Hence deduce the value of

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(((( ))))(((( ))))∫∫∫∫∞∞∞∞

++++++++0 2 2 4 x1 x

dx.(Tri-N/D 08)

27. The ends A and B of a rod ‘ l ’cm long have their temperatures kept at C o

30

and C o

80 , until steady state conditions prevail. The temperature at the end B issuddenly reduced to C

o

60 and that of A is increased to C o

40 .Findtemperature distribution in the rod after time’ t ’.(Tnl-N/D 08)

28. a) Find(((( ))))(((( ))))++++−−−−

−−−−

1 z 41 z 2 z8

Z 2

1 using Convolution theorem. (Tnl-N/D 08)

b) Find the singular integral of pqqy px z ++++++++==== . (Cbe N/D 08)