i partial differential equations - · pdf filepart – b 1) form the partial differential...
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PART – B
1) Form the partial differential equation by eliminating the arbitrary functions
f and g in ).2()2( 33 yxgyxfz
2) Form the partial differential equation by eliminating the arbitrary functions
f and g in ).()( 22 xgyyfxz
3) Form the partial differential equation by eliminating the arbitrary functions
f and from ).()( zyxyfz
4) Find the singular solution of .1622 qpqypxz
5) Solve .1 22 qpqypxz
6) Find the singular integral of the partial differential equation
.22 qpqypxz
7) Solve .1 22 qpz
8) Solve ).1()1( 2 zqqp
9) Solve .4)(9 22 qzp
10) Solve .)1( qzqp
11) Solve ).)(()()( yxyxqxyzpxzy
12) Solve .2)2()( zxqyxpzy
13) Find the general solution of .32)24()43( xyqzxpyz
14) Solve )()()( yxzqxzypzyx
15) Solve ).2(2 yzxxyqpy
16) Solve .)()( 222 xyzqzxypyzx
17) Solve .)()( 22 yxzqyxzpyx
18) Solve .)2()2( xyqyzpzx
19) Find the general solution of .)( 22 qypxyxz
20) Solve ).()()( 222222 xyzqxzypzyx
21) Solve ).4sin()20( 522 yxezDDDD yx
22) Solve ).2sin(3)54( 222 yxezDDDD yx
23) Solve ).cos()( 23223 yxezDDDDDD yx
24) Solve .)30( 622 yxexyzDDDD
25) Solve .)6( 3222 yxeyxzDDDD
26) Solve .)sinh(22
22
2
2
xyyxy
z
yx
z
x
z
27) Solve .cos62
22
2
2
xyy
z
yx
z
x
z
28) Solve .sin)65( 22 xyzDDDD
29) Solve .)1222( 222 yxezDDDDDD
30) Solve .7)33( 22 xyzDDDD
UNIT – I PARTIAL DIFFERENTIAL EQUATIONS
PART-B
1) Find the Fourier series of period 2 for the function )2,(;2
),0(;1)(xf and
hence find the sum of the series ........5
1
3
1
1
1222
.
2) Obtain the Fourier series for )2,(;2
),0(;)(
x
xxf .
3) Expand 2;0
0;sin)(
x
xxxf as a Fourier series of periodicity 2 and
hence evaluate ..........7.5
1
5.3
1
3.1
1.
4) Determine the Fourier series for the function 2)( xxf of period 2 in
20 x .
5) Obtain the Fourier series for 21)( xxxf in ),( . Deduce that
6.........
3
1
2
1
1
1 2
222.
6) Expand the function xxxf sin)( as a Fourier series in the interval
x .
7) Determine the Fourier expansion of xxf )( in the interval x .
8) Find the Fourier series for xxf cos)( in the interval ),( .
9) Expand xxxf 2)( as Fourier series in ),( .
10) Determine the Fourier series for the function xx
xxxf
0,1
0,1)( .
Hence deduce that 4
.........5
1
3
11 .
11) Find the half range sine series of xxxf cos)( in ),0( .
12) Find the half range cosine series of xxxf sin)( in ),0( .
13) Obtain the half range cosine series for xxf )( in ),0( .
14) Find the half range sine series for )()( xxxf in the interval ),0( .
15) Find the half range sine series of 2)( xxf in ),0( .
Hence find )(xf .
UNIT – II FOURIER SERIES
18. Write down the appropriate solutions of the two dimensional heat
Equations.
19. In two dimensional heat flow, what is the temperature along the
Normal to the xy- plane?
20. If a square plate has its faces and the edge y = 0 insulated, its edges
x = 0 and x = n are kept at zero temperature and its fourth edge is
kept at temperature u, then what are the boundary conditions for this
problem?
PART –B
21. A tightly stretched string with fixed end points x = 0 and x = l is
initially in a position given by y = y0 sin3( π x / l ). If it is released from
rest from this position, find the displacement y( x, t).
22. A tightly stretched string of length l has its ends fastened at x = 0 ,
x = l. The mid-point of the string is then taken to height h and then
released from rest in that position. Find the lateral displacement of a
point of the string at time t from the instant of release.
23. A tightly stretched string with fixed end points x = 0 and x = l. At time
t = 0, the string is given a shape defined by F(x) = μ x ( l - x ), where μ
is constant, and then released . Find the displacement of any point x of
the string at any time t >0.
24. The points of trisection of a string are pulled aside through the same
distance on opposite sides of the position of equilibrium and the string
is released from rest. Derive an expression for the displacement of the
string at subsequent time and show that the mid-point of the string
always remains at rest.
25. A tightly stretched string of length l with fixed ends is initially in
equilibrium position. It is set vibrating by giving each point a velocity
v0 sin3( π x / l ). Find the displacement y(x,t).
26. A tightly stretched string with fixed end points x = 0 and x = l is
initially at rest in its equilibrium position. . It is set vibrating by giving
each point a velocity λ x ( l - x ), find the displacement of the string at
any distance x from one end at any time t.
27. A taut string of length 20cms.fastened at both ends is displaced from its
position of equilibrium , by imparting to each of its points an initial velocity
given by: v = x in 0 < x < 10 and, x being the
20 – x in 10 < x <20 distance from one end.
Determine the displacement at any subsequent time.
28. An insulated rod of length l has its ends A and B maintained at 00C and
1000c respectively until steady state conditions prevail. If B is suddenly
reduced to 00C and maintained at 00C, find the temperature at a distance x
from A at time t.
29. A homogeneous rod of conducting material of length 100cm has its ends
kept at zero temperature and the temperature initially is
u(x,0) = x in 0 < x < 50
100 – x in 50 < x <100 Find the temperature u(x,t) at any time.
30. An insulated rod of length l has its ends A and B maintained at 00C and 1000c
respectively until steady state conditions prevail. If the change consists of
raising the temperature of A to 200c and reducing that of B to 800c ,
find the temperature at a distance x from A at time t.
31. The ends A and B of a rod 20cm.long have the temperature at 300c and 800c
until steady state prevails. The temperature of the ends are changed to 400c
and 600c respectively. Find the temperature distribution in the rod at time t.
32. The ends A and B of a rod 10cm.long have the temperature at 500c and 1000c
until steady state prevails. The temperature of the ends are changed to 900c
and 600c respectively. Find the temperature distribution in the rod at time t.
33. 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 the other three edges
are kept at 00c. Find the steady state temperature in the plate.
34. Find the steady state temperature at any point of a square plate whose two
adjacent edges are kept at 00c and the other two edges are kept at the constant
temperature 1000c.
35. Find the steady temperature distribution at points in a rectangular plate
With insulated faces the edges of the plate being the lines x = 0 , x = a ,
y = 0 and y = b. When three of the edges are kept at temperature zero
and the fourth at fixed temperature a0c.
36. Solve the BVP Uxx + Uyy = 0 , 0 < x, y < π with u( 0 ,y ) = u(π , y) =
u( x , π) = 0 and u(x,0) = sin3x.
37. A rectangular plate is bounded by the lines x = 0 , y = 0 , x = a , y = b . Its
Surfaces are insulated. The temperature along x = 0 and y = 0 are kept
at 00 c and the others at 1000 c . Find the steady state temperature at any
point of the plate.
38. A long rectangular plate has its surfaces insulated and the two long sides
as well as one of the short sides are maintained at 00 c . Find an
expression for the steady state temperature u(x,y) if the short side y = 0
is π cm long and is kept at uo0c.
39. An infinitely long rectangular plate with insulated surface is 10cm wide.
The two long edges and one short edge are kept at zero temperature
while the other short edge x = 0 is kept at temperature given by
U = 20y for 0 < y < 5
20 ( 10 – y ) for 5 < y < 10 .Find the steady state temperature
distribution in the plate.
40. An infinitely long – plane uniform plate is bounded by two parallel edges
and an end at right angle to them. The breadth of this edge x =0 is π, 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 Laplace equation.
PART –B
1. A function f(x) is defined as f(x) = 1 if |x| < 1
0 , otherwise . Using Fourier
integral representation of f(x). Hence evaluate [sin s/ s] cos sx ds in 0 < x< .
2. A function f(x) is defined as f(x) = 1 if |x| < 1
0 , otherwise . Using Fourier
cosine integral representation of f(x). Hence evaluate [sin s/ s] cos sx ds in
0 < x < .
3. Find the F.T of f(x) is defined as f(x) = 1 if |x| < 1
0 , otherwise . Hence evaluate
(i) [sin / ] d (ii) [sin
2 /
2]
d in (0 , ).
4. Find the F.T of f(x) is defined as f(x) = a -|x| if |x| < a
0 , otherwise . Hence evaluate
(i) [sin / ] 2
d in (0 , ). (ii) [sin / ] 4 d in (0 , ).
5. Find the F.T of f(x) is defined as f(x) = 1 –x2 if |x| < 1
0 , otherwise . Hence evaluate
(i) [sin t – t cos t/ t 3] dt in (0 , ). (ii) [x cos x - sin x / x
3] cos (x /2)dx in (0 , ).
6. Find the F.T of f(x) is defined as f(x) = e-a2x2
,a >0. Hence S.T e-x2 / 2
is self reciprocal
under F.T.
7. Find the F.T of e-|x|
and hence find the F.T of e-|x|
cos 2x.
8. Obtain the F.S.T of f(x) = x if 0 <x < 1
2 – x if 1<x<2
0 , otherwise
9. Find the F.C.T of f(x) is defined as f(x) = cos x if 0 <x < a
0 , otherwise .
10. State and Prove Parseval’s Identity.
11. Find the F.S.T and F.C.T of x n-1
, where 0 < n< 1, x >0 . Deduce that 1/ x is self-
Unit – IV
FOURIER TRANSFORMS
reciprocal under both F.S.T and F.C.T.
12. Find the F.S.T of e-ax
/ x . Hence find F.S.T of 1 / x.
13. Evaluate [dx / (a2 + x
2 ) (b
2 + x
2) ]
dx in (0 , ).
14. Find F c {f ’(x)}.
15. Solve the integral equation [f(x) cos x] dx in (0 , ) and also [cos x / ( 1 +
2)]
d
in (0 , ).
Z – TRANSFORM PART –B
z2/ ( z -a ) ( z - b )
1. Find Z [ an cos nθ ] and Z [an sin nθ ]
2. Find Z [an n2 ].
3. Find Z [ cos nπ/2 ] and Z [ sin nπ/2 ]
4. Find the Z – transforms of the following (i) ean (ii) n ean
5. Find the Z – transform of (i) cosh nθ (ii) an cosh nθ
6. Find Z [ cos ( nπ/2 + π/4 ) ]
7. Find the Z – transform of (i) ncp (ii) n+p cp
8. Find the Z – transform of unit impulse sequence and unit step sequence.
9. Find the Z – transform of (i) sinh nθ (ii) an sinh nθ
10. Find Z [ et sin2t ] and Z [ e-2t sin3t ].
11. Find the inverse Z – transform of z / ( z + 1 )2 by division method.
12. Find the inverse Z – transform of { 2 z2 + 3z } / ( z + 2) ( z – 4 ) by partial
fractions method.
13. Find the inverse Z – transform of ( z3 – 20 z ) / ( z – 2 ) 3 ( z – 4 ) by partial
fraction method.
14. Find the inverse Z – transform of 10 z / ( z-1) ( z-2) by inversion integral
method.
15. Find the inverse Z – transform of 2z / ( z -1 ) ( z - i ) ( z + I )
by inversi on intergral method
16. Using convolution theorem , evaluate the inverse Z – transform of
17. Using convolution theorem , evaluate the inverse Z – transform of
z2/ ( z –a) 2
18. Show that ( 1/ n! ) * (1/ n! ) = 2n / n!
19. Solve yn+2 + 6 y n+1 + 9yn = 2n with y0 = y1 = 0, using Z – transform.
20. Solve yn+2 - 2 y n+1 + yn = 3n + 5.