ma1122 transforms and partial differential equations · 2009. 8. 1. · ma1122 transforms and...
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MA1122 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
QUESTION BANK
UNIT - I FOURIER SERIES
PART – A
1. If ( ) xxxf += 2 is expressed as Fourier series in the interval (-2, 2), to which value this series converges at x = a
Solution: The value of the Fourier series of f(x) at x =2 is
[ ] [ ] 42424
21)2()2(
21
=++−=+− ff
2. If the Fourier series of the function 2)( xxxf += in the interval ),( ππ− is
∑∞
⎟⎠⎞
⎜⎝⎛ −−+
12
2
sin2cos4)1(3
nxn
nxn
nπ , then find the value of the infinite series
......31
21
11
222 +++
Solution:
=)(xf ∑∞
⎟⎠⎞
⎜⎝⎛ −−+
12
2
sin2cos4)1(3
nxn
nxn
nπ
Put π=x , ∑∞
+=1
2
2 43
)(n
f ππ
[ ] [ ]
∑∑∞∞
=⇒=−∴
=+++−=+−=
1
2
21
2
22
222
321414
3
21)()(
21)(
πππ
ππππππππ
nn
fff
∴ ......31
21
11
222 +++ = 643
2 22 ππ=
×
3. State Dirichlet’s conditions for a given function to expand to Fourier series Solution: (1) )(xf is well defined, periodic and single valued. (2) )(xf has finite number of finite discontinuities and no infinite discontinuities. (3) )(xf has finite number of finite maxima and minima.
4. If the Fourier series for the function ⎢⎣
⎡≤≤≤≤
=ππ
π2,sin
0,0)(
xxx
xf is
⎥⎦⎤
⎢⎣⎡ ++++−= .....
7.56cos
5.34cos
3.12cos21
2sin)( xxxxxf
ππdeduce that
42......
7.51
5.31
3.11 −
=∞++−π
Solution:
Put x = 2π in the Fourier expansion of )(xf ,
⎥⎦⎤
⎢⎣⎡ +−+−+−=⎟
⎠⎞
⎜⎝⎛ .....
7.51
5.31
3.1121
21
2 πππf
ππ1
21.....
7.51
5.31
3.112
−=⎥⎦⎤
⎢⎣⎡ ++−⇒ since 0
2=⎟
⎠⎞
⎜⎝⎛πf
∴4
222
2.....7.5
15.3
13.1
1 −=×
−=⎥⎦
⎤⎢⎣⎡ ++−
πππ
π
5. What is the const6ant term a0 and the coefficient of cosnx an in the Fourier series of
3)( xxxf −= in ( )ππ ,− ?
Solution:
)()()()(
3
33
xfxxxxxfxxxf
−=−−=
+−=−⇒−=
)(xf∴ is an odd function of x in ( )ππ ,− .
∴The Fourier series of )(xf contains sine terms only.
∴ a0 = 0 and an = 0.
6. Find bn in the expansion of 2x as a Fourier series in ( )ππ ,− .
Solution:
)()()( 22 xfxxfxxf ==−⇒=
)(xf∴ is an even function of x in ( )ππ ,− .
∴ The coefficient bn of sinnx in the Fourier expansion is zero.
∴ bn = 0.
7. If )(xf is an odd function defined in ),( ll− , what are the values of a0 and an? Solution: Since )(xf is an odd function of x in ),( ll− , its Fourier expansion contains sine terms only. ∴ a0 = 0 and an = 0.
8. Find an in expanding xe− as Fourier series in ( )ππ ,− . Solution:
( )
[ ][ ]
)1(sinh)1(2
)1()1(
)1()1()1(
1
sincos1
1
cos1
22
12
2
nnee
een
nxnnxn
e
nxdxea
nn
nn
x
x
xn
+−
=+−−
=
−+−+
=
⎥⎦
⎤⎢⎣
⎡+−
+=
=
−
+−
−=
−
=
−∫
ππ
π
π
π
π
ππ
ππ
π
π
π
π
9. Find the Fourier constant bn for xx sin in ( )ππ ,− . Solution: Let xxxf sin)( = )(sin)sin()()( xfxxxxxf ==−−=∴ )(xf∴ is even function of x in ( )ππ ,− The Fourier series of )(xf contains cosine terms only ⇒bn = 0.
10. Determine bn in the Fourier series expansion of )(21)( xxf −= π in π20 ≤≤ x
With period 2π . Solution:
( )
nn
nnn
nnx
nnxx
nxdxxbn
1)11(2
0cos2cos21
sin)1(cos21
sin2
1
2
02
2
0
=+=
⎥⎦⎤
⎢⎣⎡ +=
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−−−⎟
⎠⎞
⎜⎝⎛−−=
⎟⎠⎞
⎜⎝⎛ −
= ∫
ππ
ππππ
ππ
ππ
π
π
11. If ⎩⎨⎧
≤≤≤≤
=ποπ
2,500,cos
)(xifxifx
xf with period 2π for all x, find the sum of the
Fourier series of )(xf at π=x ? Solution:
[ ] [ ]24950cos
21)()(
21)( =+=++−= ππππ fff
11. Find a sin series for the function .0,1)( π≤≤= xxf Solution:
[ ]nn nn
nxnxdxb )1(12cos2sin2
00
−−=⎥⎦⎤
⎢⎣⎡−== ∫ πππ
ππ
∴The Fourier sine series of )(xf =∑∞
1
.sin nxbn
⎥⎦⎤
⎢⎣⎡ +++= .....
55sin
33sinsin4)( xxxxf
π
12. State Parseval’s identity for full range expansion of )(xf as Fourier series in )2,0( l Solution:
[ ]∫ ∑∞
=
++=l
nnn ba
adxxf
l
2
0 1
222
02 )(21
4)(
21 where a0, an, bn are Fourier coefficients in
the expansion of )(xf as a Fourier series.
13. Define root mean square value of a function )(xf in .bxa ≤≤ Solution:
R.M.S. value [ ]∫−=
b
a
dxxfab
y 2)(1
14. Find the constant term in the Fourier expansion of xxf 2cos)( = in ),( ππ− . Solution:
x2cos is an even function of x in ),( ππ− .
∑
∑∞
∞
+=+
⇒
+=∴
1
0
1
02
cos22
2cos1
cos2
cos
nxaax
nxaa
x
n
n
∴The constant term is a0 = 1 15. Find the constant term a0 and the coefficient an of cosnx in the Fourier series expansion of 3)( xxxf −= in ),( ππ− . Solution:
)()()(
)(33
3
xfxxxxxfxxxf
−=+−=+−=−⇒
−=
)(xf∴ is an odd function ∴ a0 = 0 and an = 0.
PART – B
1. Obtain the Fourier series for 21)( xxxf ++= in ),( ππ− . Deduce that
6....
31
21
11 2
222
π=+++ .
2. Expand the function xxxf sin)( = as a Fourier series in the interval .ππ ≤≤− x 3. Determine the Fourier series of xxf =)( in the interval .ππ ≤≤− x 4. Find the half range cosine series for xsinx in (0,π ).
5. Find the Fourier series of period π2 for the function ( )( )⎩
⎨⎧
=ππ
π2,2
,01)(
inin
xf and
hence find the sum of the series ∞+++ ....51
31
11
222
6. Find the Fourier series for xxf cos)( = in the interval ),( ππ− .
7. Find the Fourier series of 2)( xxf = in ),( ππ− . Hence find ∞+++ ....31
21
11
444
8. Obtain the half range cosine series for xxf =)( in (0,π ).
9. Expand xxxf −= 2)( as Fourier series in ),( ππ− .
10. Expand ⎢⎣
⎡≤≤≤≤
=πππ2,0
0,sin)(
xxx
xf as a Fourier series of periodicity π2 and
hence evaluate ∞+++ ......7.5
15.3
13.1
1
11. Obtain the Fourier series of ⎢⎣
⎡−
=)2,(2
),0()(
ππππ
inxinx
xf .
12. Find the half range Fourier sine series of 2)( xxf = in (0,π ). 13. Determine the Fourier series for the function 2)( xxf = of period 2π in
π20 ≤≤ x .
14. Determine the Fourier series of the function ⎩⎨⎧
≤≤+≤≤−+−
=π
πxx
xxxf
0,10,1
)( . Hence
deduce that 4
......71
51
311 π
=+−+− .
15. Find the half range cosine series for the function 2)( xxxf −= π in π≤≤ x0 .
Deduce that 90
....31
21
11 4
444
π=+++
16. By finding the Fourier cosine series for xxf =)( in π≤≤ x0 , show that
∑∞
= −=
14
4
)12(1
90 n nπ
17. Find the half range sine series expansion of xxf −=2
)( π in (0,π ) and deduce
the sum of the series ∑∞
=12
1n n
.
18. Obtain Fourier series for )(xf of period 2l and defined as follows
⎢⎣
⎡≤≤≤≤−
=lxlinlxinxl
xf20
0)( . Hence deduce that
4......
71
51
311 π
=+−+− and
8
.....51
31
11 2
222
π=++
19. Find the Fourier series for the function⎩⎨⎧
≤≤−≤≤
=211
10)(
xinxxinx
xf . Deduce that
8
.....51
31
11 2
222
π=++
20. Obtain the Fourier series for the function⎩⎨⎧
≤≤−≤≤
=21),2(
10,)(
xxxx
xfππ
.
21. Find the Fourier series expansion of 2)( xxf = in (0, l2 ).
22. Find the Fourier series of ⎢⎣
⎡≤≤≤≤−
=10,101,0
)(xx
xf .
23. Find the Fourier series expansion of period ‘l’ for the function ( )( )⎪⎩
⎪⎨⎧
−=
llinx
linxxf
,212,0
)( . Hence deduce the sum of the series ∑∞
= −14)12(
1n n
.
24. Obtain the half range cosine series of 2)2()( −= xxf in the interval 20 ≤≤ x .
Deduce that ∑∞
=
=−1
2
2 8)12(1
n nπ
25. Obtain sine series for ( )( )⎪⎩
⎪⎨⎧
−=
llinx
linxxf
,212,0
)( .
26. Find half range cosine series, given ⎩⎨⎧
≤≤−≤≤
=21210
)(xinxxinx
xf .
27. Find the half range sine series of axf =)( in (0, l). Deduce the sum of
∞++ ....51
31
11
222 .
28. Find the half range cosine seires of 2)()( xxf −= π in the interval (0, π ). Hence
find the sum of the series ∞+++ ....31
21
11
444 .
29. Obtain a Fourier series for xcos1− in ππ ≤≤− x . 30. Find a0, a1, a2, a3, b1, b2, b3 given
x 0 3
π 32π
π3
4π 35π
π2
y 1.0 1.4 1.9 1.7 1.5 1.2 1.0
Hence find )(xf . 31. Find the Fourier series upto second harmonic for the following data.
x 0 1 2 3 4 5
y 9 18 24 28 26 20
32. Find upto the first two harmonics in the Fourier series of )(xfy = in (0, 360 )
given in the following tabular value. x 0 60 120 180 240 300 360
y 2 2.1 3 3.2 2.5 2.2 2
UNIT –II FOURIER TRANSFORMS
1. State the Fourier Integral theorem. Statement: If f(x) is the piecewise continuously differentiable and absolutely integrable on ( )∞∞− , , then
dtdsetfxf stxi )()(
21)( −
∞
∞−
∞
∞−∫ ∫=
π
2. Define the Fourier Transform pair. Statement: If f(x) is defined in ( )∞∞− , , then its Fourier Transform is defined by
dxexfsF isx∫∞
∞−
= )(21)(π
and the inversion formulas given by
dsesFxf isx∫∞
∞−
−= )(21)(π
3. Define the Fourier Cosine Transform pair. Fourier cosine Transform of f(x) is defined by
dxsxxfsFc ∫∞
=0
cos)(2)(π
and the inversion formula is given by
dssxsFxf c∫∞
=0
cos)(2)(π
4. Write the Parseval’s Identity for Fourier Transforms. Sol.
If F(s) is the Fourier Transforms of f(x), then dxxfdssF ∫∫∞
∞−
∞
∞−
= 22 )()(
5. State the Convolution theorem on Fourier Transforms.
Sol. The Fourier transforms of Convolution of f(x) and g(x) is the product of their
Fourier Transforms { } )()( sGsFgfF =∗ where dttxgtfgf )()(21
−=∗ ∫∞
∞−π
6. If the Fourier Transform of f(x) is F(s), what is the Fourier Transform of f(x-a)? Sol.
dxeaxfaxfF isx∫∞
∞−
−=− )(21))((π
)(
)(21 )(
sFe
axtwheredtetf
isa
atis
=
−== ∫∞
∞−
+
π
7. Find { })(xfxF n . \ Sol.
dxexfsF isx∫∞
∞−
= )(21)(π
diff on both sides w.r.to s, n times
)()())((
)(21)()(
))((21)(
sFdsdixfxF
dxexxfsFdsdi
dxeixxfsFdsd
n
nnn
isxnn
nn
isxnn
n
−=
=−
=
∫
∫∞
∞−
∞
∞−
π
π
8. Prove that { } ⎟⎠⎞
⎜⎝⎛=
asF
aaxfF 1)( , a>0
Sol.
dxexfsF isx∫∞
∞−
= )(21)(π
.0,1
)(211
)(21)]([
>⎥⎦⎤
⎢⎣⎡=
=
==
∫
∫∞
∞−
∞
∞−
aasF
a
dtetfa
axtwheredxeaxfaxfF
aist
isx
π
π
9. What is the Fourier transform of axxf cos)( ?
Sol. dxexfsF isx∫∞
∞−
= )(21)(π
[ ])()(21)cos)((
))((21
21
2)(
21
cos)(21)cos)((
asFasFaxxfF
dxeeexf
dxeeexf
dxeaxxfaxxfF
isxiaxiax
isxiaxiax
isx
−++=
⎥⎦
⎤⎢⎣
⎡+=
⎟⎟⎠
⎞⎜⎜⎝
⎛ +=
=
∫
∫
∫
∞
∞−
−
∞
∞−
−
∞
∞−
π
π
π
10. What is the Fourier sine transforms of .)(axf Sol.
⎥⎦⎤
⎢⎣⎡=
⎟⎠⎞
⎜⎝⎛=
==
=
∫
∫
∫
∞
∞
∞
asF
aaxfF
duuasuf
a
axuwheredxsxaxfaxfF
dxsxxfsF
ss
s
s
1))((
sin)(21
sin)(2))((
sin)(2)(
0
0
0
π
π
π
11. Find the Fourier sine transform of xe− .
Sol.
⎥⎦⎤
⎢⎣⎡+
=
⎥⎦
⎤⎢⎣
⎡−−
+=
=
=
∞−
∞−−
∞
∫
∫
2
02
0
0
12
)cossin(1
2
sin2)(
sin)(2)(
ss
sxssxs
e
dxsxeeF
dxsxxfsF
x
xxs
s
π
π
π
π
Part – B
1. Find the Fourier transform of )(xf given by ⎪⎩
⎪⎨⎧
>>
<=
.00
1)(
axfor
axforxf
and hence evaluate .sinsin
0
2
0∫∫∞∞
⎟⎠⎞
⎜⎝⎛ dx
xxanddx
xx
2. Find the Fourier transform of )(xf given by ⎪⎩
⎪⎨⎧
>
<−=
.10
11)(
2
xfor
xforxxf
and hence evaluate .cossin
03∫
∞ − dss
ss
3. Find the Fourier transform of )(xf given by ⎪⎩
⎪⎨⎧
>
<−=
.10
11)(
xfor
xforxxf
and hence evaluate .sinsin
0
4
0∫∫∞∞
⎟⎠⎞
⎜⎝⎛ dt
ttanddt
tt
4. Show that 2
2x
e−
is self reciprocal under Fourier transforms.
5. Find the Fourier transform of 0>− aife xadeduce that
(1) .04 3
022 >=
+∫∞
aifaax
dx π
(2) 222 )(2}{
saasixeF xa
+=−
π
6. Evaluate ∫∞
++02222 ))(( bxax
dx using Fourier transforms.
7. Solve for from the integral equation aedxaxxf −∞
=∫ cos)(0
8. Derive the Parseval’s Identity for Fourier Transform.
9. Find the Fourier cosine transform of2xe−
.
10. Find the Fourier cosine transform ⎩⎨⎧ <<−
=otherwise
xxxf
:010:1
)(2
Hence prove that .163
2coscossin
3
π=⎟
⎠⎞
⎜⎝⎛−
∫ dsss
sss
11. Find the Fourier transform of 211x+
.
12. Find the Fourier cosine transform of 22xae− and hence find the Fourier sine transform
of22xaxe− .
13. Prove that 2
2x
e−
is self reciprocal under Fourier cosine transform.
14. Find the Fourier sine transform of ( )x
exfax−
= .
15. Find the Fourier sine transform of 2
2x
xe−
. 16. Find the Fourier sine transform and Fourier cosine transform of xe− and hence find
the Fourier sine transform of 21 xx+
& Fourier cosine transform of 211x+
.
17. Find the Fourier sine transform of axe− and hence find the Fourier cosine transform
of axxe− 18. Find the Fourier sine transform and Fourier cosine transform of axe− , .0>a Hence
evaluate ( ) ( )( )∫∫
∞∞
+++ 02222
0222
2
bxaxdxanddx
axx
UNIT III PARTIAL DIFFERENTIAL EQUATIONS
PART-A
1.Form the Partial differential equation equation by eliminating a and b from z=(x2+a)(y2+b) Solution: z=(x2+a)(y2+b) -------------------1
Differentiating (1)partially w.r.to x and y,
xz∂∂ = p = 2x(y2 +b)
yaxqxz 2)( 2 +==∂∂
Pq = 4xy(x2+a) (y2 +b)
Pq = 4xyz is the required P.D.E
2.Find the PDE of all plane having equal intercepts on the x and y axes Solution: Equation of the plane 1=++
cz
by
ax (intercept form)
x intercept=y intercept ⇒ a=b
1===cz
by
ax
Differentiating (1)partially w.r.to x and y, 0)(11&0)(11
=+=+ qca
pca
acqp
ac −
==− & The required PDE is p=q
3.Form the PDE of all spheres whose centres lie on the z –axis Solution: Equation of the sphere with centre on z axis is x2+y2+(z-c)2=r2
Differentiating (1)partially w.r.to x and y, 2x+2(z-c)p=0 and 2y+2(z-c)q=0 z-c =
qycz
px
=−− &
qy
px=
Py –qx = 0 is the required PDE 4 . Form a partial differential equation by eliminating ‘f’ from z=f(x2+y2) Solution: Given z=f(x2+y2)------------------1 Differentiating (1)partially w.r.to x and y, P=f΄(x2+y2)2x----------------------2 q= f΄(x2+y2)2y---------------------3 (2)+(3) gives
yx
qp=
Py=qx is the required PDE 6. Eliminate the arbitrary function ‘f’ from z = f
zxy and form the partial differential
equation Solution: Given z = f
zxy
Differentiating (1)partially w.r.to x and y, P= f΄ ( )zxy / [ 2z
xypzy− ] ------------(2)
q= f΄(xy/z)[zx-xyq/z2] ----------(3) (2)÷(3) gives
xyqzxxypzy
qp
−−
=
P(zx-xyq)=q(zy-xyp) Pzx-pqxy=qzy-pqxy Pzx=qzyPx-qy=0 is the required PDE 7.Form the PDE by eliminating the arbitrary function from φ ⎥⎦
⎤⎢⎣⎡ −
zxxyz ,2
Solution: Given φ ⎥⎦
⎤⎢⎣⎡ −
zxxyz ,2 = 0
)( 2 xyzf
zx
−= -----------------(1) Differentiating (1)partially w.r.to x and y, )2)((1 2
2 yzpxyzfpzx
z−−′=− ------------------------(2)
And )2)(( 2
2 xzqxyzfqz
x−−′=
− ------------------------------(3) z-px = z2 f ′ (z2- xy)(2zp –y) -qx = z2 f ′ (z2 – xy)(2zq –x) ⇒
xzqyzp
qxpxz
−−
=−−
22
∴(z-px)(2zp-x) = qxy -2zpqx ∴ 2z2q – 2zpqx –xz +px2 = qxy -2zpqx ⇒px2 + q(2z2 –xy) = xz is the required PDE. 8.Find the solution of ptanx+qtany=tanz Solution : Subsidiary equations are
zdz
ydy
xdx
tantantan==
Cotxdx = cotydy =cotzdz We know that ∫ = xxdx sinlogcot Cotxdx = cotydy
Integrating logsinx = logsiny +c Log c
yx log
sinsin
=⎟⎟⎠
⎞⎜⎜⎝
⎛
∴ c
yx=
sinsin
Similarly cotydy =cotzdz
1sinsin c
zy=
The general solution is Φ 0
sinsin,
sinsin
=⎟⎟⎠
⎞⎜⎜⎝
⎛zy
yx
9.Solve: p + q =1 Solution: The equation is of the form f(p,q) = 0 The complete integral is z = ax+by+c Where f(a,b) = 0 f(a,b) = 0 ⇒ ( )211 abba −=⇒=+ The complete integral is z = ax + ( 1 - a )2 y +c 10. Find the complete integral of p+q=pq Solution: The equation can be written p +q –pq =0 The complete integral is z = ax+by+c Where a+b =ab ⇒ b =
1−aa
Therefore z = ax + cy
aa
+−1
11. Find the complete integral of q=2px Solution: q=2px = a ⇒ q=a and 2px= a ⇒p=
xa2
Consider dz = pdx+qdy dz= ∫ ∫ ++ bdya
xdxa
2
z = .log
2solutionrequiredtheisbayxa
++
12. Solve:p+q=x-y Solution: P+q = x-y ⇒ p-x = -(y+q) x-p = y+q = a x-p = a ⇒ p = x-a and y+q=a ⇒ q = a-y dz = pdx +qdy ⇒dz = (x-a)dx +(a-y)dy Integrating z = byayaxx
+−+−22
22
13. . Find the complete integral of pq
py
qx
pqz
+= Solution: Z = px+qy+(pq)3/2 This is Clairaut’s equation The Complete integral is z = ax+by+(ab)3/2 14. Solve : y
xz sin2
2
=∂∂
Solution: Integrating w.r.t ‘x’ )(sin yfyx
xz
+=∂∂ --------------------------(1)
Integrating (1) w.r.t ‘x’ Z = )()(sin
2
2
ygyxfyx++
Where f(y) and g(y) are functions of y alone. 15. Solve : (D2 -3D D′ 2 + 2 3D′ )z = 0 Solution: The auxiliary equation is m3 -3m +2 =0 m=1 is a root (m-1)(m2+m-2) =0 m=1,1,-2 The complete solution is z = )2()()( 321 xyfxyxfxyf −++++
PART –B
1.Find the differential equation of all planes which are at a constant distance ‘k’ from the origin . 2.Form the partial differential equation by eliminating the arbitarary functions f and g in=x2f(y)+y2g(x) 3.Form the partial differential equation by eliminating f and Φ from z=f(y)+ Φ(x+y+z) 4.Form the PDE from z=yf(x)+xΦ(y) by eliminating the arbitrary functions 5.Obtain the PDE by eliminating f and g from z=f(y+2x)+g(y-3x) 6.solve:(y-z)p-(2x+y)q=2x+z 7.Find the general solution of (3z-4y)p+(4x-2z)q=2y-3x. 8. solve:x(y-z)p+y(z-x)q=z(x-y). 9.solve:y2p-xyq=x(z-2y). 10.(x2+y2+yz)p+( x2+y2-xz)q=z(x+y). 11.(x-2z)p+(2z-y)q=y-x. 12.(y+z)p+(z+x)q=x+y 13.(y2+z2)p-xyq+xz=0 14. Find the general solution of z(x-y)=px2-qy2. 15.solve:x(y2+z2)p+y(z2+x2)q=z(y2-x2). 16.solve:x(y2+z)p+y(x2+z)q=z(x2-y2). 17.solve:z=1+p2+q2
18.solve:p(1-q2)=q(1-z) 19.solve:9(p2z+q2)=4 20.Find the complete integral of p2+q2=x+y 21.solve:x2p2+y2q2=z2 22.solve:z2(p2+q2)=x2+y2
23.solve:p2+q2=z2(x2+y2) 24.Find the singular solution of z=px+qy+c√1+p2+q2
25. solve : (D2-DD΄-20 D΄2)Z=e5x+y+sin(4x+y)
26.solve:(D2-2DD΄)=e2x+x3y
27.solve:(D3+D2 D΄-D D΄2- D΄3)Z=COS(2X+Y)
28. solve 2
22
2
2
2y
zyxz
xz
∂∂
−∂∂
∂+
∂∂
=sin h(x+y)+xy
29. solve: (D2+2DD΄+ D΄2 -2D-2 D΄)Z=sin(x+2y)
30. solve: (D2-D΄2 -3D+3 D΄)Z=xy+7
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UNIT-4 APPLICATION OF PARTIAL DEFFERENTIAL EQUATION
PART-A
1. Classify the p.d.e 2 24 4 12 7 .xx xy yy x yu u u u u u x y+ + − + + = +
2. Classify the p.d.e 2 2 2
2 32 2
x yu u u ex x y y
+∂ ∂ ∂+ + =
∂ ∂ ∂ ∂
3. Classify the p.d.e 2 2 2
2 23 4 6 2 0.u u u u u ux x y y x y
∂ ∂ ∂ ∂ ∂+ + − + − =
∂ ∂ ∂ ∂ ∂ ∂
4. Classify the p.d.e 2 2 2(1 )(4 ) (5 2 ) 0.xx xy yyx x u x u u+ + + + + =
5. In wave equation
2 22
2 2 ,y yct x
∂ ∂=
∂ ∂ what is the physical meaning for
2c
and what is the agent for vibration. 6. What is the Fourier law of heat conduction?
7. Write all possible solution for O.D.H.E.
8. State the law assumed to derive the O.D.H.E.
9. Write three possible solutions of Laplace equations in two – dimensions?
10. What are the possible solutions of one dimensional wave equation?
11. In one dimensional heat equation 2 .t xxu uα= what does
2α stands for?
12. In steady state conditions derive the solution of one dimensional heat flow
equation.
13. Write the Initial conditions of the wave equation if the string has an initial
displacement.
(Or)
Write the initial conditions of the wave equation if the string has an initial
displacement but no initial velocity.
14. A tightly stretched string of length 2L is fixed at both ends. The mid point of
the string is displaced to distance ' 'b and released from rest in this position
write the initial conditions.
15. State the two solutions of the Laplace equation by method of variable separable.
16. State one dimensional heat equation with the initial and boundary conditions.
17. Write the boundary conditions and initial conditions for solving the vibration of
string equation, if the string is subjected to initial displacement ( )f x and initial
velocity ( )g x .
18. Solve the equation 2 0 ,u ux y
∂ ∂+ =
∂ ∂given that ( ,0) 4 xu x e−= by the
method of separation of variables. 19. A rod of 50cm long with insulated sides has its ends A and B kept at
20 cο and 70 cο respectively. Find the steady state temperature distribution of
the rod.
20. An infinitely long uniform plate is bounded by the edges
x l= 0,x x l= = and an end right angle to them. The breadth of the edge
0y = is l and is maintained at
2 22
2 2y ya
t x∂ ∂
=∂ ∂
All the other edges are
kept at .o cο Write down the boundary conditions in mathematical form
PART – B
1. A string is stretched and fastened to two points l apart. Motion is started by
displacing the string into the form 250( )y lx x= − (Or)
2( )y k lx x= −
from which it is released at time 0.t = Find the displacement of any point on
the string at a distance x from one end at time .t
2. A string of length 2l is fastened at both ends. The mid point of a string is taken
to a height b and then released from rest in that position. Find the displacement
of any point of the string at any subsequent time.
3. A tightly stretched string with fixed end points x o= and 2x l= is initially in
a position given by
, 0( ,0)
(2 ), 2 .
kx x lly x
k l x l x ll
⎧⎪⎪ ≤ ≤⎪⎪⎪=⎨⎪⎪ − ≤ ≤⎪⎪⎪⎩
If it is released from rest from this position, find the displacement
function ( , )y x t at any point of the string.
4. A string of length l has its ends 0x and x l= = fixed. The point where
3lx = is drawn aside a small distance h the displacement
( , )y x t satisfies2 2
22 2 .y ya
t x∂ ∂
=∂ ∂
Find at any time .t
5. A tightly string with fixed end points 0x and x l= = is initially at rest in its
equilibrium position. If it is set vibrating by giving each point a velocity
( )x l xλ − (or) ( ) 0 .kx l x for x l− < < Find the displacement of the
string at any distance from one end at any time t (or) Find the displacement
function ( , ).y x t
6. A string is stretched between two fixed points at a distance 2l apart and the points on the string are given initial velocities v where
: 0
( 2 ) : 2 .
c x x llvc l x l x ll
⎧⎪⎪ < <⎪⎪⎪= ⎨⎪⎪ − < <⎪⎪⎪⎩
7. If a string of length l is initially at rest in its equi9librium position and each of its
points is given a velocity v such that
0
2
( ) .2
lv c x fo r x
lc l x fo r x l
= < <
= − < <
Show that the displacement at any time is given by
2
3 34 1 3 3( , ) sin sin sin sin ...... .
3l c x at x aty x t
l l l laπ π π π
π
⎡ ⎤⎢ ⎥= − +⎢ ⎥⎣ ⎦
8. The ends A and B of a rod 30cm long have their temperatures kept at
20 c° and 80 c° until steady prevails. The temperatures at the end B is then
suddenly reduced to 60 c° and that of A is raised to 40 c° and maintained so.
Find the temperature distribution.
9. A metal bar 10cm long with insulated sides has its ends A and B kept at 20 c°
and 40 c° respectively until steady state conditions prevail. The temperatures at
A is then suddenly raised to 50 c° and B is lowered to 10 c° . Find the
subsequent temperature at any point at the bar at any time.
10. A rod 30cm long has its ends A and B kept at 20 c° and 80 c° until steady
conditions prevail. The temperature at both ends reduced to 0 c° and kept so.
Find the temperature distribution.
11. A rod of length l has its ends A and B kept at 0 c° and 100 c° until steady
state conditions prevail. The temperature at A raised to 25 c° and B is
reduced to 75 c° . Find the temperature distribution.
12. A rectangular plate with insulated surfaces is 10cmwide and so long compared
to its width that it may be considered infinite in length. The temperature along
the short edge 0y = is given by
( , 0 ) 2 0 : 0 52 0 (1 0 ) : 5 1 0 .
u x x xx x
= < <= − < <
Other edges are kept at 0 c° . Find the steady state temperature distribution.
13. A square plate is bounded by the lines 0, 0, 20, 20x y x y= = = = . It faces are
insulated. The temperature along the upper horizontal edge is given by
( , 20) (20 )u x x x= − When 0 20x< < while the other three
edges are kept at 0 c° . Find the steady state temperature in the plate.
14. A rectangular plate is bounded by the lines 0, 0, ,x y x a y b= = = = . Its
surfaces are insulated and temperature along two adjacent edges is kept at
100 c° and the temperature along other tow edges is kept at 0 c° . Find the
steady state temperature distribution.
15. An infinitely long rectangular plate with insulated surfaces is 10cmwide. The
two long edges and one short edge is kept at 0 c° while the other short edge
0x = is kept at
2 0 , 0 52 0 ( 2 0 ) , 5 1 0 .
x y yu y y
= < <= − < <
Find the steady state temperature distribution in the plate.
16. Find the steady state temperature at any point of a square plate whose two
edges are kept at 0 c° and other two edges are kept at the constant
temperature100 c° .Find the steady state temperature distribution at any point
of a square plate.
17. A rectangular plate with insulated surface is 10cmwide and so long compared
to its width that it may be considered as an infinite plate. If the temperature at
short edge 0y = is given by 3 (10 )x x− and all the other three edged
are kept at 0 c° . Find the steady state temperature at any point of the plate.
UNIT - 5 Topics: Z-Transform
Part – A
1. Find the Z-Transform of cosnπ/2. Ans: Z [cosn π/2] = (1+1/z2)-1=z2/z2+1 if |Z|>1
2. State Convolution theorem on Z-Transform.
Ans: If f (z) and G (z) are the Z-transform of f (n) and g (n) respectively, then Z[f(n)*g(n)]= F(z). G(z) Where, f(n) .g(n) is defined as the convolution of f(n) and g(n) given by f(n)*g(n)=∑k=0f(k) g(n-k).
3. Find the Z-Transform of 1/ (n+1)!
Ans: Z[e1/z-1]
4. Find the Z-Transform of 1/n!
Ans: e1/z
5. Express Z {f (n+1)} in terms of f⎯ (Z).
Ans: Z[f(n)=Z F(z)-z F(0)
6. Find the valve of Z{f(n)} When f(n)=n.an
Ans: z [nan] = az/ (z-a) 2
7. Find z [an/n!] in Z-Transforms.
Ans: z [an/n!] =ea/z
8. Find z [e-iat] using Z-Transforms.
Ans: zeiat/zeiat-1
9. State and Prove Initial valve theorem in Z-Transform.
Ans: Lt F (z) = Lt f (n) Z-∞ n-0
10. Find the Z-Transform of (n+1). (n+2)
Ans: z (z+1)/ (z+1)3 +3z/ (z-1)2+2z/ (z-1)
Part – B
11. Solve the difference equation y (n+3)-3y(n+1)+2y(n)=0 . Given that y(0)=4, Y(1)=0 and y(2)=8.
12. Find Z-1[Z2/(Z+2)(Z2+4)], by the Method of Partial fraction.
13. To find Z-1[(Z2-4Z)/(Z-2)2]
14. Prove that Z[1/n+1]=Z log (Z/Z-1)
15. Find a) F[xnf(x)]=(-i)n dnF/dsn
b) F [dnf(x)/dxn] = (-is)n F(s)
16. State and Prove the second shifting theorem in Z-Transform.
17. Find F[dnf(x)/dxn]=(-is)n F(s)
18. a) Find the Z-transform of n (n+1). b) Find the Inverse Z-transform of Z2/ (Z-a)2 using Convolution
theorem.
19. a) Find the Z-transform of 2n(1-n), n≥0 b) Solve, Using Z-transform Yn+2- 4yn=0, given y0=2, y1=1 20. a) Find the Z-transform of CosnӨ and SinnӨ. Hence find Z [cos n π/2]
b) Find the Inverse Z-transform of Z2-3Z/ (Z+2) (Z-5)