MODEL PAPER FOR APPLICATIONS OF “PDE”
Marks:100 Time:3hours
PART—A (16marks)
ATTEMPT ANY 8 OF THE FOLLOWING:(8*2=16):
01.What are the possible solutions for the one dimensional wave equation?
02.State the fourier law of heat conduction. What is the constant a2 in the wave equation uu=a2uxx.
03.State any 2 laws under which Laplace transforms exists.
04.Classify the PDE:
(x+1)Zxx + √2 (x+y+1)Zxy + (y+1)Zyy + Yzy -Xzy + 2sinx = 0.
05.The ends A and B of a rod of length 10cm long have their temperature kept
at 200C and 700C.Find the steady state temperature distribution on the rod.
06.List out the possible solutions of the heat equation.
07.Distinguish between the solutions of one-dimensional wave and heat
equations.
08.State the one-dimensional wave and heat equations with the boundary
conditions.
09.Classify the following PDE: x2fxx + (1-y2)fyy = 0.
10.Solve by the method of separation of variables the following PDE:
2x Zx – 3y Zy = 0.
11.Find the solution using the method of separation of variables:
4 X'Y + XY' = 3XY.
12.Write the standard examples for the elliptic, parabolic and hyperbolic type of equations.
PART—B:(84 marks):
ANSWER ANY 7 CHOOSING ATLEAST 2 FROM EACH SET:(7*12=84)
SET—1:
13.(a)state and derive the possible solutions for the wave equation.
(b)Solve the following PDE:
Zxx – 2Zx + Zy = 0. (4+3+5)
(c)Using the method of Separation of variables, find the solution of:
ux = 2 ut + u where u(x,0)= 6e-3x
14.A string is tightly stretched and its ends are fastened at two points x=0 and
x=1. The mid-point of the string is displaced transversely through a small
distance 'b' and the string is released from rest in that position. Find an
expression for the transverse displacement of the string at any time during
the subsequent motion. (12)
15.(a)Find the displacement of any point of a string, if it is length 2l and
vibrating between fixed end points with initial velocity zero and initial
displacement given by
f(x) = kx/l in 0<x<l
= 2k – kx/l in l<x<2l. (7+5)
(b)A elastic string is stretched between 2 points at a distance ∏ apart. In its
equilibrium position the string is in the shape of the curve
f(x) = k(sinx – sin3x). Obtain y(x,t) the vertical displacement if it satisfies
the equation ytt = yxx.
16.A tightly stretched with fixed end points x=0 and x=l is initially at rest in its
equilibrium position. If it is set vibrating giving each point a velocity
λx(l-x), then find the value of y(x,t). (12)
17.Solve the following boundary for the vibration of string:
(i) y(0,t)=0 (iii)∂y(x,0)/∂t = x(x-l) 0<x<l
(ii)y(l,t)=0 (iv)y(x,0) = x in 0<x<l/2
= l-x in l/2 <x< l. (12)
SET—2
18.A rod of length l has its ends A and B kept at 0oC and 100oC until steady
state condition prevail. If the temperature at B is reduced to 0oC and kept
so while that of A is maintained, find the temperature u(x,t) at a
distance x from A and at time t. (12)
19.A bar, 10cm long, with insulated sides, has its ends A and B kept at 20oC and 40oC respectively until steady state conditions prevail. The temperature at A is then suddenly raised to 50oC and at the same instant that at B is lowered
to 10oC. Find the subsequent temperature at any point of the bar at any time.
(12)
20.Solve the following boundary value problem:
(i)∂u/∂t = (1/a2)(∂2u/∂x2) for 0<x<5;
(ii)∂u(0,t)/∂x = 0;
(iii)∂u(5,t)/∂x = 0;
(iv)u(x,0) = x; (12)
21.A rectangular plate is bounded by the lines x=0,y=0,x=a,y=b. It's surface
are insulated. The temperature along x=0 and y=0 are kept at 0oC and others
at 100oC. Find the steady state temperature at any point of the plate. (12)
SET—3
22.(a)An infinitely long-plane uniform plate is bounded by 2 parallel edges and
an end at right angle to them. The breadth of this edge x=0 is ∏, this end
is maintained at temperature as u = k(Πy-y2) at all points while the other
edges are at zero temperature. Determine the temperature u(x,y) at any
point of the plate in the steady state if u satisfies the Laplace equation.
(b)A rectangular plate with insulated surfaces is 8cm wide and so long
compared to its width. It may be considered as an infinite plate. If the
temperature along the short edge y=0 is u(x,0)= 100sinΠx/8, 0<x<8, while the 2 edges x=0 and x=8 as well as the other short edge are kept at 0oC. Find the steady state temperature at any point on the plate. (6+6)
23.An infinitely long-plane with insulated surface is 10cm wide. The 2 long
edges and one short edge are kept at zero temperature, while the other
short edge x=0 is kept at temp given by
u = 20y for 0≤ y ≤5
= 20(10-y) for 5≤ y ≤10
find the steady state temperature distribution in the plate. (12)
24.A rectangular plate is bounded by the lines x=0, x=a, y=0, y=b and the
temperatures at the edges are given by
(i) u(0,y) = y in 0<y<b/2
= b-y in b/2<y<b
(ii)u(a,y) = 0 = u(x,b)
(iii)u(x,0) = 5 sin(4Пx/a) + 3sin(3Пx/a)
find the steady state temperature distribution in the plate.(12)
25.A taut string of length l has its ends x=0, x=l fixed. The point where x=l/3 is
drawn aside a small distance h, the displacement y(x,t) satisfies the
following equation. Determine the value of y(x,t) at any time t.
∂2y/∂t2 = a2 ( ∂2y/∂x2 ). (12)
NOTE:
1.Draw diagrams for all the questions possible.
2.Apply the conditions in the correct order.
3.Write down all the corresponding formulas.
ALL THE BEST AND BEST OF EFFORTS!!!
set by,
RANGARAJAN T.R.:):)
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