This document contains a question paper for a civil engineering exam covering topics in transforms and partial differential equations. It has two parts with multiple choice and long answer questions. Part A contains 10 short answer questions worth 2 marks each, covering topics like Fourier series, Fourier transforms, partial differential equations, and Z-transforms. Part B contains 5 long answer questions worth 16 marks each, involving finding Fourier series expansions, solving partial differential equations, using Fourier transforms and Z-transforms to solve problems.

1. ., Question Paper Code:· 23768
B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2018.
Third/Fifth Semester
Civil Engineering
MA 2211- TRANSFORMS AND PA.R'l'IAL DiFFERENTIAL EQUATIONS
(Common to all branches)
·(Regulatione 2008)
Time.: Three hours -Maximum: 100 marks
·Answer ALL questions.
PART A- (10 x 2 = 20 marks)
1. If f(x) =x 2
in -2 < x < 2 and· f(x + 4) =f(x), then find the coefficient a0 ofits
·Fourier series.
2. Find the root mean squa're value of f(x) =cosx in (0, 2n) .. ,
3. State the co:p.volution theorem for Fourier: transform.
4. Show that the Fourier transform satisfies the linearity property.
5. For1Il the p~rtia! differential equation by eliminating the arbitrary function
from z=f x +y. ) • . · · .·
6. . Find the complete integral of p =2q X 0
7. Mathematically formulate the following vibratin,g string problem: "A string is _
stretched between two fixed pomts at a distance 2l apart and the points of the
string are given an. initial velocity f(:x), x · being the distance from an end
point".
8. . Classify the partial differential equation: uu + 2uxy + uYY =0.
9: Find the Z transform of u (n-1) .
10. State the initial valu~ theorem ofZ transform.
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BIBIN.C / ASSOCIATE PROFESSOR / MECHANICAL ENGINEERING / RMK COLLEGE OF ENGINEERING AND TECHNOLOGY 1

2. PART B - (5 >< 16 ~.. SO marks)
11. (a) (i) Find a Fourier series to represent f(x)"" x ~ x2
from.-~ ~o tt. (10)
12.
(ti) "-' Obtain the constant term a0 and the first two harmonics a1 a~d a2
. in the Fourier co&ine series representation of y ""'f(x) in· (0, 6) for
the given.table.oivaiues: (6)
X 0 ·1 2 3 4 . 5
y 4 8 15 7 6 2
Or
(b) (i) Find the half range Fourier sine· series expansion for
(a)
(b)
L''f(x)= 2k(L~x)
· {. 2kx O<x<.L/2 . ., . 1 .
. Hence evaluate L ··· ····- 4
· .
L12<x<L . · .. n=l (2n -1)
(8)
L . ,
(ii) Find the complex f<?rm of the Fourier series for /(x) = e-x,-l<x<l.
(8)
(i) Express the function f(:<) ~ {~ lxl ~ 1
as ~ Fourier integraL
lxl >1
· ·· · jsin..t ··
Hence e~alu~te
0
~ dA. . (10)
(ii) Find the Fourier sine transform of ! .
X
(6)
Or
=az.t:~. .. 0 ..(i) . Find the Fourier transform of e ,a.:> and henc~ .find the
Foudertransform of e~x212
~- (8)
(ii) Using Fourier · transform methods, evaluate
oO • dx .
J(2 .· 2R 2 b2)' a, b > 0.
0
X: +a X + .. ·
(8)
13. (a) (i). Solve the partial durerential equation~ z2
{p2
+ q1
}z.x2
+y 2
• (8)
(ii) Solve (D2
+2DD'+D'2
)ztisinh(x +y)+ e·t+2
Y.. - (8)
Or
(b) (i} Solve (3z-4y) p+(4x-2z)q c 2y~3x. (8)
(ii) Obtain the CQmplete . solution
z ""' p X + qy + p2 q2 .
and singular solution of
(8)
2 23768
BIBIN.C / ASSOCIATE PROFESSOR / MECHANICAL ENGINEERING / RMK COLLEGE OF ENGINEERING AND TECHNOLOGY 2

3. 14.
15.
(a) A rod; · 30 em long, has its ends A and B kept at 20°C ·and 80°C
_respectively until steady state conditions prevail. The temperature at
e~ch end is then suddenly reduced to O"C .and kept so. Find the resulting
temperature function u(x, t) taking ~ =0 at A (16)
Or
(b) An infinitely long plane uniform plate is bounded by two parallel edges
x = 0 and· x !!; l and an edge at right angles to them. The breadth of this
edge y == 0 is l and is ·maintabted at a temperatm:e f(x). All the other
three edges are· at· ,;ero temperatur~. Find.the steady state temperature
at any interior point of the plate. . .(16)
(a)
(b)
(r) Using Z-transf'orm method, solve the
x(n + 1)-2x(n.)e 1'given x(O) ""0.
difference equation:·
(10)
(ii) Find the Z transform of t" and hence deduce the result:
z(tk)==~T~ ddz {z v~e-i)}, where_ T is the sampling period_ with~
t == nT, n t:o;i.2,... . (6)
Or
(i) · Find the inverse Z trapefor'm of
(z~~)(z-~J
using partial
fractiofis method. (8)
{ii)
z.2
Find the inverse ~ t!'aneform -ttsing convolution theorem.
(z~af
(8)
.. ·..
3 23768
BIBIN.C / ASSOCIATE PROFESSOR / MECHANICAL ENGINEERING / RMK COLLEGE OF ENGINEERING AND TECHNOLOGY 3