Anna University Coimbatore 2010 B.E Electronics & Communication Engineering Transforms and partial differential equations- / - Question Paper
TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS- NOV / DEC 2010
ANNA UNIVERSITY OF TECHNOLOGY, COIMBATORE B.E. I B.TECH. DEGREE EXAMINATIONS : NOV / DEC 2010 REGULATIONS : 2008 THIRD SEMESTER 080100008 - TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS (COMMON TO CIVIL / EEE / EIE / ICE / ECE / BIOMEDICAL / BIOTECH / AERO / AUTO / CSE / IT I MECHANICAL / CHEMICAL / FT / TT / TC )
PART - A
ANSWER ALL QUESTIONS
1. State the conditions for f(x) to have Fourier series expansion.
2. Write ao ,a in the expansion of x+x3 as Fourier Series in (-tt, 7r).
3. Expand f(x)=1 in a sine series in 0<x< n
4. Find Root Mean Square value of the function f(x) = x in the interval (0,/).
5. Define Fourier Transform Pair.
2 x
6. Find Fourier Cosine transform of e
7. If F(S) is the Fourier Transform of/O) show that the Fourier Transform of eiax f(x) is F(S + a).
8. State Parsevals Identity for Fourier Transform.
9. Eliminate the arbitrary constants a & b from z = (x2 + a)(y2 + b).
10. Form the PDE by eliminating the functions from z = f{x +1) + g(x -1).
11. Find the complete integral q = 2px.
12. Solve (D3-3DD'2+2D'3)z=0.
13. Find the nature of PDE 4uxx+4uxy+ uyy + 2ux- uy=0.
14. What are the various solutions of one dimensional Wave Equation?
15. A string is stretched and fastened to two points T apart. Motion is started by
7ZX
displacing the string into the form y=y0Sin()from which it is released at time t=0.
Formulate this problem as the boundary value problem.
16. A rod of length 20cm whose one end is kept at 30C and the other end is kept at 70C is maintained so until steady state prevails. Find the steady state temperature.
17. Find Z[e~an].
_ z
18. Prove that Z[n] ~-7
u u
19. Prove that Z\f{n +1)] = zF{z) - zf (0)
20. State Initial and Final value theorem on Z- transform.
PART -B
(5x12 = 60 Marks)
ANSWER ANY FIVE QUESTIONS
21 (a). If = find the Fourier Series of the period 2x in the interval (0,2;r).
,111 n
Hence deduce that 1 ~r + 7~ + - ~~r (8)
3 5 7 4 v '
(b). Find the Fourier expansion of f(x) = x in the interval ( 7t,?r) (4)
12 2 ii J a -x \x\<a
22. Show that the Fourier Transform of /(*)=) is
0 otherwise
sin as - as cos as
Hence deduce that
Using Parsevals Identity show that J
dt = -
15
23.(a) Solve (mz-ny)p + (nx-lz)q = ly-mx
(6)
(6)
(b) Solve (D3 +D2D'-DD'-D";>)z=eA' Cos2Y
24. A string of length I is initially at rest in its equilibrium position and motion is
\cx ,0<x<l/2
V =
started by giving each of its points is given a velocity
c(l-x), l/2<x<l
Find the displacement function y(x,t).
25 (a) Evaluate z
+7z + 10
(6)
(b) Using z-transforms solve u(n+2) - 5u(n+1) + 6u(n)=4 given that u(0)=0, u(1)=1
(6)
26(a) Find the constant term and the coefficient of the first sine and cosine terms in
the Fourier expansion of, y=f(x) as given in the following table:- (6)
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
f(x) |
9 |
18 |
24 |
28 |
26 |
20 |
9 |
26(b) Find the Fourier transform of fix) =
JC < ]
0 otherwise
oo . 4
r sin jc
-dx
(6)
hence find the value of
27. A metal bar 30cm long has its ends A and B kept at 20C and 80C respectively,
until steady state conditions prevail. The temperature at each end is then suddenly reduced to 0C and kept so. Find the resulting temperature u(x,t) taking x=0 at A.
28(a) Solve p( 1 + q) = qz
(6)
(6)
(b) Using Convolution theorem, evaluate
4
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| Earning: Approval pending. |
