Unit-2

Section-A

Q1: Write the following Equations

a. One Dimensional Wave Equation

b. One Dimensional Heat Equation
c. One Dimensional Laplace Equation
d. Two Dimensional Laplace Equation
e. Specify with suitable example, the classification of partial differential equation for
elliptic, parabolic and hyperbolic differential equation.

Q2: Classify the following equations

 2z 2z 2z
a. (1  x 2 )  2xy  (1  y 2 ) 2  2z  0 Ans: Hyperbolic in the region x 2  y2  1 ,
x 2 xy y
parabolic in the region x 2  y 2  1 , and the elliptic in the region x 2  y 2  1.
 2z  2z
b.  0 Ans: Elliptic
x 2
y 2
 2z  2z  2z z z
c. 2 3  3   0 Ans: Hyperbolic
x 2 xy y 2 x y

 2z  2z 1 u
d.   Ans: Parabolic
x 2
y 2
c 2 t
e. Show that the equation Z xx  2xZ xy  (1  y 2 )Z yy  0 is elliptic for the values of x and y in

the region x 2  y 2  1 , parabolic on the boundary and hyperbolic outsides the region.

Section-B and C

Q3: Solve by the method of separation of variables

a. u  u  2u; u(x, 0)  10e x  6e4x Ans: u(x, y)  10e ( x3t )  6e 2(2x3t )
t x

b.  u u ;
2
given that u = 0 when t → ∞ as well as u = 0 at x = 0 and x = l.

x 2 t
 n 2 2 
  2 t
 l  nx
Ans: u   b n e   sin .
n 1 l
1
 u u  Ans: u ( x , y)  2 x  e 3y sin x
2
c.   2u, u (0, y )  1  e3 y sinh
x 2 y x 2
d. 4 u  u  3u; u(0, y)  4e y  e5 y Ans: u(x, y)  4e ( x y)  e (2x 5y)
x y

e. 2
u u
 3  5u  0; u (0, y)  2e y Ans: u(x, y)  2e ( xy)
x y

nct nx
Q4: Solve one dimensional wave equation  z  c 2  z . Ans: u   a n cos
2 2
sin .
t 2
x 2
n 1 l l
Q5: A string is stretched and fastened to two point’s l apart. Motion is started by displacing the
x
string in the form y  A sin from which it is released at time t = 0. Show that the
l
displacement of any point at a distance x from one end at time t is given by
x ct
y( x, t )  A sin cos .
l l

Q6: Show how the wave equation  z  c 2  z can be solved by the method of separation of
2 2

2 2
t x
variables. If the initial displacement and velocity of a string stretched between x = 0 and x = l
z
are given by z = f(x) and  g (x) determine the constants in the series solution.
t


 nct nct  nx 2 l nx 2 l nx
Ans: z    an cos  bn sin  sin . where a n   f ( x) sin dx , bn   g ( x) sin dx
n1  l l  l l 0 l nc 0 l

Q7: If a string of length l is initially at rest in equilibrium position and each of its points is
 y  x
given by the velocity    b sin 3 , find the displacement y.
 t  t 0 l

bl  x ct 3x 3ct 

Ans: y   9 sin sin  sin sin .
12c  l l l l 

Q8: A tightly stretched string with fixed end points x= 0 and x = l is initially in a position given
x
by y  y0 sin 3 . if it is released from rest from its position; find the displacement y(x, t).
l

 3y x ct y 0 3x 3ct 

Ans: y   0 sin cos  sin cos .
 4 l l 4 l l 

Q9: Find the deflection u(x, t) of a tightly stretched vibrating string of unit length that is
1 1
initially at rest and whose initial position is given by sin x  sin 3x  sin 5x, 0  x  1.
3 5
1 1
sin x cos ct  sin 3x cos 3ct  sin 5x cos 5ct
Ans: 3 5

Q10: Solve one dimensional Heat equation  z  c 2  z . Ans: u  c1 cos px  c2 sin pxc3e c p t .
2 2 2 2

2 t x

Q11: A rod of length l with insulated sides is initially at a uniform temperature u0. Its ends are
suddenly cooled 0°c and are kept at that temperature. Find the temperature function u(x, t).

c 2 ( 2 n 1) 2 2t

4u 1 (2n  1)x 
Ans: u  0

 2n  1 sin l
e l2 ..
n 1
Q12: Find the temperature of a bar of length 2 whose ends are kept at zero and lateral surface
x 5x
insulated if the initial temperature is sin  3 sin .
2 2

x 2c 2t / 4 5x 252c 2t / 4

Ans: sin e  3 sin e .
2 2

Q13: A bar of 10cm length with insulated sides A and B are kept and 20°C and 40°C
respectively until steady state conditions prevail. The temperature at A is then suddenly varied
to 50°C and the same instant at B, lowered at 10°C. Find the subsequent temperature at any
point of the bar at any time.

120   1n
n 22t
nx  100
Ans: u ( x, t )  50  4 x  
 n1 n
sin
10
e .

Q14: Solve two dimensional Laplace equations  z   z  0 .

2 2

2 2
x y


Ans: z  c1 cos px  c2 sin px c3e py  c4 e  py . 
Q15: Use separation of variables method to solve the equation  z   z  0 subject to boundary
2 2

2 2
x y
nx
conditions z(0, y)  z(l , y)  z( x,0)  0. and z( x, a)  sin .
l

ny
sinh
z ( x, y )  l sin nx .
Ans: n a l
sinh
l

Q16: A rectangular plate with insulated surfaces is 8 cm wide and so long compared to its
width that it may be considered infinite in length without introducing an appreciable error. If
nx
the temperature along one short edge y = 0 is given by u ( x,0)  100 sin ,0  x  8
8

While the two long edges x= 0 and x = 8 as well as are kept at 0°C, show that the steady state
y
 nx
temperature at any point of the plate is given by u( x, y)  100e 8 sin .
8

 2u  2u
Q17: Solve the Laplace equation   0 in a rectangle in the xy-plane with
x 2 y 2
u(0, y)  u( x,0)  u( x, b)  0, and u(a, y)  f ( y) parallel to y-axis.


nx ny b
ny
Ans: u( x, y)   bn sinh . where bn  2
na 
sin f ( y ) sin dy.
n1 b b b
b sinh 0
b
Q18: A square plate is bounded by the lines x = 0, y = 0, x = 20 and y = 20. Its faces are
insulated. The temperature along the upper horizontal edge is given by u(x, 20) = x(20-x) when
0< x < 20 while other three edges are kept at 0°C. Find the steady state temperature in the
plate.

(2n  1)x (2n  1)y

 sin sin
3200
Ans: u ( x, y ) 
3
 20 20
(2n  1) 3 sinh( 2n  1)
.
n 1

Q19: A rectangular plate with insulated surface is 10 cm wide and so long compared to its
width that it may be considered infinite in length without introducing an appreciable error. If
the temperature along the short edge y = 0 is given by:

20x, 0x5
u(x,0)   
20(10  x) 5  x  10

While the two edges x=0 and x=10 as well as the other short edge are kept at 0 oC. Find the
steady state temperature at any point (x, y) of the plate.
n
sin ny
800  nx  10
Ans: u ( x , y )  
 2 n 1 n 2
2 sin
10
e .

Q20:

An insulated rod of length 𝑙 its ends A and B maintained at 0oC and 100oC respectively until
the steady state condition prevails. If B is suddenly reduced to 0oC and maintained at 0oC, Find
the temperature at a distance x from A at time t.

n 2 2 c 2 t
200  (1) n nx 
Ans: u ( x, y)   
 n 1 n
sin
l
e l2 .

x a x
Q21: Find the Fourier Transform of F ( x)  e . Hence Evaluate F ( x)  e .

2 2a
Ans: and 2
1 p 2
a  p2

1, x a  sin ap cos px

Q22: Find the Fourier Transform of F ( x)  
0, x a
Hence Evaluate 
 p
dp

 sin p
and 
0 p
dp

2 sin ap  ,
 x a 
Ans: , p  0 , F ( x)   and
p 
0, x a 2
 1  x 2 , x 1
Q23: Find the Fourier Transform of F ( x)   Hence Evaluate
0, x 1
 x cos x  sin x 
 x


 x 3 

cos dx .
2

4(sin p  p cos p) 3
Ans: 3
, 
p 16

e  ax
Q24: Find the Fourier Sine Transform of F ( x)  , a  0. Hence find Fourier sine transform
x
1
of .
x

a 
Ans: ,
a p22
2

1
Q25: Find the Fourier cosine Transform of F ( x)  Hence find Fourier sine transform of
1 x2
x
.
1 x2

 
Ans: e p , e p .
2 2

u  2 u
Q26: Solve the equation  , x  0, t  0 subject to the conditions (i) u  0 at x  0, t  0
t x 2

1, 0  x 1
(ii) u   at t = 0.
0, x 1

2   1  cos p   p 2t

Ans: u ( x, t )   e sin pxdp
0
 p 

Q27: The temperature u in the semi-infinite rod 0  x   is determined by the differential

u  u 2
equation k 2 subject to the conditions u  0 when t  0, x  0 and
t x
u
  when x  0, t  0.
x

Ans: u ( x, t ) 
2
 0


p

2
1  e cos pxdp
 kp 2t