Anna University Chennai 2012-3rd Sem B.E Computer Science and Engineering ./B.Tech , /E (ester, )-TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS - Question Paper
Monday, 25 February 2013 09:00Web
B.E./B.Tech. DEGREE EXAMINATION, MAY/JUNE 2012
Third Semester
Common to all branches
MA 2211/181301/MA 31/10177 MA 301/MA 1201/080100008-TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
(Regulation 2008)
Time : 3 hours
Maximum : 100 marks
ans all ques..
PART A-(10*2 = 20 marks)
1. obtain the constant term in the expansion of (cos^2)x as a fourier series in the interval (-p,p).
2. describe 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. obtain the Fourier sine transform of f(x) = e^-ax, a>0.
5. From the partial differential formula by eliminating the arbitrary function from z^2-xy = f(x/z).
6. Solve (D^2-7DD+6D^2)z=0.
7. What is the basic difference ranging from the solutions of 1 dimensional wave formula and 1 dimensional heat formula with respect to the time?
8. Write down the partial differential formula that represents steady state heat flow in 2 dimensions and name the variables involved.
9. obtain the z-transform of x(n) = {¦(a^n/n! for n=0@0,otherwise)¦
10.Solve Yn+1-2Yn = 0given Y0 = 3.
PART B-(5*16=80 marks)
11. (a) (i) obtain the fourier series of f(x)= (p-x)^2 in (0,2p) of periodicity 2p.
(ii) find the Fourier series to represent the function f(x)=|x|, -p
(ii) obtain the Fourier series upto 2nd harmonic for the subsequent data for y with period 6.
X: 0 one two three four 5
Y: nine 18 24 28 26 20
12. (a) (i) Derive the Parsevals identity for Fourier Transforms.
(ii) obtain the Fourier integral representation of f(x) described as
F(x)={¦(0 for x<0@1/2 forx=0@e^(-x) for x>0)¦
Or
(b) (i) State and prove convolution theorem on fourier transform.
(ii) obtain the fourier sine and cosine transform of x^(n-1) and hence prove 1/vx is self reciprocal under fourier sine and cosine transforms.
13. (a) (i) Form the PDE by eliminating the arbitrary function ? from ?(x^2+y^2+z^2,ax+by+cz)=0.
(ii) Solve the partial differential formula
X^2(y-z)p+y^2(z-x)q=z^2(x-y).
Or
(b) (i) Solve the formula )D^3+(D^2)D-4DD^2-4D^3)z=cos(2x+y) .
14. (a) The ends A and B of a rod 40 cm long have their temperatures kept at 0 degree Celsius and eighty degree Celsius respectively, until steady state condition prevails. The temperature of the end B is then suddenly decreased to 40 degree Celsius and kept so, while that of the end A is kept at zero degree Celsius. obtain the following temperature distribution u(x,t) in the rod.
Or
(b) A long rectangular plate with insulated surface is l cm wide. If the temperature along 1 short edge (y=0) is u(x,0) = k(lx-x^2) degrees, for 0
(ii) Using convolution theorem, obtain the inverse Z-transform of 8z^2/(2z-1)(4z-1).
Or
(b) (i) Solve the difference formula y(k+2)+y(k)=1, y(0)=y(1)=0, using Z-transform.
(ii) Solve Yn+2 + Yn = 2n.n, using Z-transform.
0
| Earning: Approval pending. |
