TPDE Impor. Ques

8/14/2019 TPDE Impor. Ques

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n

181301:Transforms and Partial Differential Equations

Important 2 marks Questions

1. Find the Fourier constants bn for xsinx in ( ,) .

2. What you mean by Harmonic Analysis?

3.Write the Fourier transforms pair.

4. If F(s) is the Fourier transform of f (x) then F[f (x) cos ax]=1[F(s + a) + F(s a)]

2

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

z2 xy,

x

= 0.

z6. Find the root mean square value of

7. If the Fourier series for the function

f (x) =x2

0f (x) =

in the interval

0


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.

n

2

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

steady state conditions prevail. Find the steady state temperature in the rod.

18. Find the z-transform of n.

19. Form the difference equation from y = a + b3n

20. What are the possible solutions of one dimensional wave equation?

16 marks

11. Find the Fourier series for f (x) = cosx in the interval ( ,) .

12 Find the Fourier series for

l x,

f (x) = 0,

0 a > 0

sint

sint

2

Hence deduce that (i)

0 tdt = 2

(ii) 0 t

dt = 2

16 Find the Fourier sine transform ofsinx,

f (x) = 0,

0


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2

(or)

17 State and prove convolution theorem for Fourier transforms.

18 Find the Fourier transform of ea x

and hence deduce that

F[xea x ]=i 2 2as (s

2+ a

2)

2

19 Solve z = px + qy + 1+ p2

+ q2

20Solve: (D 4DD+ 4D2)z=xy + e2x+y

Form a partial differential equation by eliminating arbitrary functions from

Solve: (x2 +y2 +yz)p+ (x2 +y2 xz)q=z(x +y).A tightly stretched string with fixed end points x=0 and x=l is initially at rest in itsequilibrium position. If it is set vibrating string giving each point a velocity x(l-x),

8l3 1

(2n 1)x

(2n 1)a t

show that displacementy(x,t) =

4

(2

sin

1)4

sin a n=1 n l l A bar 10 cm long with insulated sides, has its ends A and B kept at 200 C and 400C

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

suddenly raised to 500C and at the same time instant that at B is lowered to 100C and

maintained thereafter. Find the subsequent temp. distribution in the bar.

Find

Find

Z (sin at)

z2

Z 1

and

3z

Z (cosat)

(z 5)(z + 2)Z

1 8z 2

using Convolution theorem.Find

(2z 1)(4z +1)

Solve the difference equation y(k + 2) 4y(k +1)+ 4y(k )= 0 given y(0)= 1,y(1)= 0

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

L +xf (x) =

L x

in (L,0)

in (0,L)Hence deduce that

1+

112 32

1 2

+2

+K = .5 8

2 Find the Fourier series for f (x) = cosx in the interval ( ,) .

3 Find the complex form of Fourier series of f (x) = eax (


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ax n

t

sinh a e = (1)

a + in

einx and hence prove that

(1)n

= . a

2+ n

2 asinh a

2 n + a

2

4 Find the Fourier series expansion defined in (0 , T ) by means of the table of

values given below. Find the series up to the first harmonic.

t-Sec 0 T/6 T/3 T/2 2T/3 5T/6 T

A -

amp

1.98 1.30 1.05 1.30 -0.88 -0.25 1.98

5 Find the Fourier sine transform of

sinx,

f (x) = 0,

0 1

that 0 t

dt = 3

8. Find the Fourier transform of f(x) if f (x) =1,

x < aHence deduce that

sin t

0, x > a > 0

dt =0 2

9. Solve:

10 Solve:

p2y(1+x2 )= qx2

x(z2 y2 )p+y(x2 z2 )q=z(y2 x2 )

11 Solve: (D2 +DD 6D2 )z=x2y + e3x+y

12 Solve z = px + qy + 1+ p2 + q2


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u

13 A string of length l is fastend at both ends. The mid point of the string is

taken to a height b and then released from in that position.Find the expression

for the transverse displacement of the string at any time during the subsequent

motion

14. An infinitely long rectangular plate with insulated surface is 10 cm wide.

The two long edges and one short edge are kept at zero temperature given by

10y,=

20(10y),the plate

for

for

0 y 5

5 y 10Find the steady state temperature distribution in

15 State and prove convolution theorem on z-transforms

(ii) FindZsin

n and Zcosn

2

2 Z 1

8z 2 using Convolution theorem

16) Find (2z 1)(4z +1)

17. Solve the difference equation

y (k + 2)+ 6y (k +1)+ 9y (k ) = 0 given y (0) =1,y (1) = 0

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