4lvtu Reg. No.
Itr
:
B.E.,ts.Tech. DEGREE E)LAMINATION, NO\,,EMBER"/DECEMBER 2012. Third Semester CiviJ Engineering I,l-A.22 UA4A 3tAtA t20l A/cK 201/t0i77 MA 301/080100008/0802I 0001/ IVI,AU 2I I,ET]\,IA 92] 1_ TRANSPORMS AN' PARiili" b;i;;ffiNT"lAI EQUATIONS (Common to all branches) (Regulation 2008) Time : Three hours Maximum : 100 marks Answer AI_L questions.
L
PARTA_(tO/2= 20marks) Find the co-efficient b, of the Fourier series for the function /(:c)=* sin in (-2, 2)
rc
.
2. 3. 4. 5.
Defrne Root Mean Square value ofa functron l(.jr) over the interval (d, b). Find the Fourier transform of e-"1, . a>0. State convolution theorem in Fouder transform
6. ?.
t)tD D' I l)z _0. jn"ulared An rod oflenerh 60 cor has its ends al ,4 and B msintaincd at 2O.C and 80"C respecti\ely. irnd rhe stpady srate solurion ofthe rod. A plate is bounded by the Jines x=0, y=0, y=l and y=l.Its fbces are
8.
Etminarp (hp arbiirarl luncrion
f ft.nr
.
=
/f{) \.:r,/
and form the pDE
Solre: tD-
insulated. Tha edge coinciding wir h r-a_tis rs kepr ai 100"C. The
edge comciding wrth J-axis is kept ar 50.C. The orher iwo pdge" are f."pi ri 6;F. w.ir"t-t" boundary confitions that are needed for solvilj G. ail"""i."ii
ir".t rfo*
equation.
9. 10.
Find the Z-transform of a". Solve ya-2y,=0, given that J(0)=2.
11. (a) (i)
(ii)
PARTB_(5x 16=80marks) Find the Fourier series expansion of l(.tj
Find the Fouder series expansion
=r+r2 h(_r,z).
(8)
of flr)={*, o=rar. 12- x. 1. x.2 ^"o
t,3,5!8 (b) (i) (ii) .
Or Obtain the halfrange cosine series for (8) /(r) =, in (0, z) . Firrd the Fourier series as far as the second harmolic to represent the function /(rc) with period 6, gi""" i" tfr" f"lf"*i"giutf"', (8)
n0t2S4S i(") 9 18 24 2a 26
20
12. (a) ljr
Find lhe Fourjpr transfo,m ol evaluate
(ii)
iI{
l"l. i l
,
;i
{;
t1l
and hence
(8)
71
Find thp Fourier rransform deduce that
'
,,.,
", ,,r,
{lt
l,
' t
s"
I'
x 'a>oJ
Hence
sin , -, cos ,
(8)
4
(b) (, (ii) 13. (a)
Or Find the Fourier cosine and sine transforms of f(x)=e",, cr>O and hence deduce the inversion formula. (8) Find the Fouder cosine transform of e "". o > 0 . Hence show t}rrt the function e is sel{-reciprocal. (g)
"/2
(i)
(D
Find the singular integral ol z= pr+qy+ p2 + pq+q,. (S) Solve the partial differential equation (t 22)p+(22-gq=f-x. (8)
Or
O) (i) Solve: (D'z+3DD'-4D,2)z=cos(2*+y)+xjt. (ii) Solve: (D2 DD,+2D)z=e2.*t +4. 14. (a) A tightly stretched stdng with fixed end points .t:0 and r=,
(S)
(8)
is
' \r,
initially in a position given Uv ytx, ol = y" s;n.J,il'1. It is released from
(b)
rest ftom this position. Find the expression for the alisplacement at any time I (16) Or Find the steady state temperature distdbution in a rectangular plate of sides o and b insulated at the lateral surfaces and satisfring the boundary conditions : a(0, J) = u(a, Y) :0, for 0 < y < b; u(r, b)=0 and u(:c,0)=x(a ,r), for O
15. (a) (i,
Find theZ-transform"of
(ii) Using
convolution
n
o
(8) , 1. t 4)J un6 "o"(la ",n,llj \2 4) theorem, find the inverse Z-trarsform
oi_
(8)
\z + o),
(b) (,
Or Solve the alifference eq,rntior, ,sing Z-t"^nsfo"m Jr,*s) 3J1,*r1+2J1,y=0 given that ta=4, h=0,
f2=8. (D Solve J1,*21 + 6 J1,*1; + 91,1 =2" given that la=h=0.
(8) (8)
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