-
1
(Generalized Vector) 3-Dimension (1) ),(),( uvvu (2) vuvu ),(),( kk k (3) ( , ) 0u 0 u u ( , ) 0u 0 u u (4) ),(),(),( wvwuwvu
(5) Norm 2),( uuu uuu
**
21, ff [ a, b ] 21, ff [ a, b ] [ Inner Product]
dxxfxfffb
a
)()(),( 2121
[ Orthogonal Function]
0)()(),( 2121 dxxfxfffb
a
21, ff
** ( Orthogonal ) ( Perpendicular )
Ex. 32
21 )(,)( xxfxxf [ 1, 1]
}),(),(),({ 210 xxx
( , ) ( ) ( ) 0,b
m n m na
x x dx m n
[a, b] Ex. 1sintcostcos2t ],0[ t
[78 ] sint
-
2
[Norm] )(xn Norm [ Generalized Length]
b
annnn dxxx )(),()(
2
b
ann dxxx )()(
22 n [ Square Norm]
** )(xn [a, b] n = 0, 1, 2, . 1)( xn
)(xn [ Orthonormal Set] Normalized )(xn n = 012. )(xn Norm
Ex. 21 0 2 0 1 3 0 1 2, ,g a g b b x g c c x c x 0 0 1 0 1 2, , , , ,a b b c c c
11 x [] [83 ] [85 ]
0 0 1 0 1 21 3 5 5, 0, , , 0, 3
2 8 82a b b c c c
Ex. {1, cos x, cos 2x, }[ , ] Ex. [Norm]
< Ans > 2 , , , ......... Ex. {1, cos x , cos 2x, }
1 cos cos2, , , .........2
x x
H.W.1 f (x) = 2x, g(x) = 3 + cx 10 x (1) c (2)
[83 ][84 ]
(1) 29
c (2) xx 32,3
H.W.2 cos2 , cos3 , cos7 , cos10x x x x bxa a b = ? [74 ]
ab b a = 246 .
-
3
H.W.3 1 2( ), ( ), ...g x g x ( , )a b 1 2( ), ( ), ...g ct k g ct k
0c ,a k b kc c
( ) ( ) 0,b
m nag x g x dx m n
,x ct k dx cdt
H.W. 4 (1) 2 2 31, sin , cos , sin , cos , sin , cos ,x x x x x xL L L L L L
,L L
(2) (1) [81 ] [83 ]
(1)
(2) 1 1 1 1 2 1 2 1 3 1, sin , cos , sin , cos , sin , cos ,2
x x x x x xL L L L L LL L L L L L L
Generalized Fourier Series
n x ba, xf ,a b , 0, 1, 2, ,nc n
xcxcxcxcxf nn 221100
2
2
, 0, 1, 2,
,
bna
n bna
n
n
f x x dxc n
x dx
f
x
xf
20
, nn
n n
ff x x
x
/ Orthogonal Set / Weight Function
, 0, 1, 2,n x n
0 ,b
m naw x x x dx m n
-
4
n x ba, xw
** xw ba, 0xw
** xn Weight Norm b
a nwndxxxwx 2
** xn ba, xw ba, xf
0n
nn xcxf
nc
2
bna
nwn
f x w x x dxc
x
2 2b
n nw ax w x x dx
Complete Set 1) xf xn xf
,2,1,0),( nxn xf xn 0, 0, 1, 2,nc n
2) , 0, 1, 2,n x n xn Complete
H.W.1 cos , 1, 2, 3, ...n x nL
(0, )L ?
[] [83 ]
( ), ( ) 0,m nx x m n
( ) 1f x 1, cos 0n xL
cos , 1, 2, 3, ...n x nL
H.W.2 ?
cos , 1, 2, 3, ...nx n [76 ] Complete Set
cos , 0, 1, 2, 3, ...nx n (0, ) sin , 1, 2, 3, ...nx n (0, )
H.W.3 ( )f t 02,4,6,... 1,3,5,...
( ) cos sinn mn m
f t a a nt b mt
-
5
1, cos2 , cos4 , ..., sin , sin3 , ...t t t t (Complete set) 2 ( )f t Gram-Schmidt
...,,, 321 xuxuxu bax ,
1. 1 1x u x ----------------- 1
2. xxux 122 ----------------- 2 x1 x2
0, 12 xx xxxxu 1112 ,,
21
12 ,x
xxu
------------------ 3
xx
xxuxux 12
1
1222
,
------------- 4
3. xxxux 221133 ------------- 5 x3 x1 x3 x2
0, 13 xx
xxxxxxu 12211113 ,,,
21
131
,
xxxu
------------------ 6
3 2, 0x x
xxxxxxu 22221123 ,,,
22
232
,x
xxu
------------------- 7
xx
xxux
xxxu
xux 222
2312
1
1333
,,
------------- 8
4.
= 0
= 0
-
6
xx
xxuxux n
k
n n
nkkk
1
12
, ------------------ 9
Ex. 21, ,x x 1,0x
61,
21,1 2321 xxxxxx
Ex. x,1
21,0x 81
1 212, 4 64
x x x
H.W.1 1,1x 22 1,1, xxx Ans
22
25
21,1, xxx 79 84
H.W.2
1 1,0,3,2,1 2321 xxvxvv
2 1,3,1,2,2,1,1,1,1 321 vvv 84 Ans Hint
1
213352,123,1 2 xxx
2 1 3 2 1 1 11, 1, 1 , , , , 0, 1, 12 3 3 33 2
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7
[Fourier Series]
[Def.] xf pp, xf
01
cos sin2 n nna n nf x a x b x
p p
p
pn
p
pn
p
p
xdxp
nxfp
b
xdxp
nxfp
a
dxxfp
a
sin1
cos1
10
xf [Fourier Coefficients of xf ]
Ex.
xx
xxf
0,0,0 xf
< Ans >
12 sin
1cos114 n
n
nxn
nxn
xf
-
8
** 0a na 0n
40
a 211
na
n
n
0n na 0a
[Conditions for Convergence]
[Th.] f f pp, [Piecewise Continuous] f xf
2
xfxf
xf xf f x Ex. xf 0x , x
xf 0x 22
02
00
ff
Ex. 4,f x x x xf [: P.V. ONeil ]
4025
a , 2 248 6 1 nna nn , 0nb
H.W. 1 ( )f t Fourier 0, 0
( )sin , 0
tf t
t t
[82 ]
01 cosa
, 2 2(1 cos cos )
( )nna
n
, 2 2( 1) sin
( )
n
nnb
n
H.W.2 (1) ( ) ( 2 )f x f x ( )f x Fourier 0, 0
( ), 0
xf x
x x
(2) x ( )f x = ? [83 ]
(1) 0 2a , 2
1 ( 1) 1nna n ,
11 ( 1)nnb n ; (2) ( ) / 2f
H.W.3 0, 0
( )sin , 0
xf x
x x
[81 ]
(1) ( )f x Fourier (2) 1 1 11 3 3 5 5 7
(1) 02a
, 2
0, 1, 3, 5, ...2 , 2, 4, 6, ...
(1 )n
na
nn
, 0, 1nb n , 112
b ; (2) 24
-
9
H.W.4 sin 2 , / 2
( ) 0, / 2 0sin 2 , 0
x xf x x
x x
( )f x Fourier
01a
, 2 0a ,
22 1 cos( / 2)
, 2( 4)n
na n
n
234
b , 22 sin( / 2) , 2
( 4)nnb n
n
[Refer to: Alan Jeffrey Advanced Engineering mathematics, 2002] [Periodic Expansion]
[Fourier Cosine and Sine Series]
[Even Function and Odd Function] 1) xf xfxf y 2) xf xfxf
1) 2) 3) 4) 5)
6) xf 0
2a a
af x dx f x dx
7) xf 0aa
f x dx
Fourier Cosine and Sine Series
1) xf pp, F.S. [Cosine Series]
xp
naaxfn
n
1
0 cos2
-
10
0 0
0
2
2 cos
p
p
n
a f x dxp
na f x xdxp p
2) xf pp, F.S. [Sine Series]
xp
nbxfn
n
1
sin
p
n xdxpnxf
pb
0sin2
Ex. xxf 22 x [81 ]
1
1
2sin14
n
n
xnn
xf
12
0
4 1sin
2
n
nnb x xdx
n
Ex.
x
xxf
0101 F.S. [82 ]
2
nb 1 1n
n
1
1 12 sinn
nf x nx
n
Gibbs Phenomenon
,15...,,3,2,1n n )(xf (d) ,0x Overshooting n Gibbs Phenomenon
x
y
1
1
-
11
H.W.1 xf 21
?nn
b
[82 ] 2
H.W.2 11,)( xxxxf f(x)
1
31
2( 1) 4( ) ( 1) 1 sin( )
nn
nf x n x
n n
[Half-Range Expansions]
)(xfy Lx 0 )(xfy 1) [ y ] F.S. 2) [] F.S. 3) i.e. )()( xfLxf =
** F.S. PL 2
Ex. 2)( xxf Lx 0 F.S.
[85 0 2x ] a) :
L
dxxfL
La0
20 )(
232
xdxL
nxfLn
LaLn
n
02
2
cos)(2)(
)1(4
-
12
b) :
]1)1[()(
4)1(2sin)(2 3212
0
nnL
n nL
nLxdx
Lnxf
Lb
c) :
2
00 32)(2 Ldxxf
La
L 22
2
0
2cos)(2
n
LxdxLnxf
La
L
n
nLxdx
Lnxf
Lb
L
n
2
0
2sin)(2
H.W.1 cos , 0xf x x LL
[84 ] [82 ]
21
4cos sin(1 4 )n
x nL n xf xL Ln
H.W.2 20),2()( xxxxf f(x)
(1) : 121
2 8( ) ( 1) 1 cos3 2( )
n
n
n xf xn
(2) : 31
16( ) 1 ( 1) sin2( )
n
n
n xf xn
-
13
F.S. Iterative Method [ Refer to: C.R. Wylie and L.C. Barrett Advanced Engineering Mathematics, 6th ed.]
1. f (t) T Dirichlet Conditionf (t) Ttt 00 , mttt 21 mJJJ ,, 21 [ )()(
KKK tftfJ ] f (t) F.S.
1
1
1 2sin2
1 2cos2
mk
n n kk
mk
n n kk
T n ta b Jn n T
T n tb a Jn n T
n 0 (A)
,n na b )(tf F.S.a0 Euler-Fourier Formula
2. a. f (t) Discontinuities b. f (t) c. d. (A) 0)()( tf n
Iterative Formula
1
21
14
3
13
2
1 12
2cos2/
2sin2/2cos2/2sin1
m
k
kk
m
k
kk
km
k
m
kk
kkn
TtnJ
nT
TtnJ
nT
TtnJ
nT
TtnJ
na
1
21
14
3
13
2
1 12
2sin2/
2cos2/2sin2/2cos1
m
k
kk
m
k
kk
km
k
m
kk
kkn
TtnJ
nT
TtnJ
nT
TtnJ
nT
TtnJ
nb
Ex. :
1, 2 10, 1 0
( )1, 0 10, 1 2
tt
f ttt
f 2 2t :
k 1 2 3 4 tk 2 1 0 1 Jk 1 1 1 1
k 1 2 tk Jk
-
14
2 1 ( 2) ( 1) (0) (1)( 1)sin (1)sin (1)sin ( 1)sin2 2 2 2n n
n n n na bn n
2 1 ( 2) ( 1) (0) (1)( 1)cos (1)cos (1)cos ( 1)cos2 2 2 2n n
n n n nb an n
1n ( ) 0f t 0n na b
( 1) / 2
0, even2 sin 22 ( 1) , oddn n
nna
n nn
10, even
1 ( 1) 1 2 , oddn
n
nb
n nn
2 1n N ( )f t :
1
1 1
2 ( 1) (2 1) 1 (2 1)( ) cos sin2 1 2 2 1 2
N
N N
N t N tf tN N
H.W.1 ( ) , 1 1f t t t
1
1
2 ( 1)( ) sinn
nf t n t
n
H.W.2 f(x)
433220
2932)1(
)(
2
xxx
xx
xxf
073
a , 28 3cos
2( )nna
n
, 3 2
8 8 31 ( 1) sin2( ) ( )
nn
nbn n
H.W.3 0, 0
( )sin , 0
tf t
t t
0 1 12 1, 0,
2a a b
, 2
1 cos , 0, 1(1 )n n
na b nn
H.W.4 2, 0( )2, 0
t tf tt
2 20 2 21 2 1 22, ( 1) , 2 1 ( 1) ( 1)3
n n nn na a b nn n
[ : Glyn James, Advanced Modern Engineering Mathematics, Ex.4.16, pp. 326-328]
-
15
[Periodic Driving Force]
F.S. I/P D.E.: D.E. :
)(22
tfkxdt
xdm
)(tf
1
sin)(n
n tpnbtf
1
sin)(n
nP tpnBtx
Ex. D.E.161
m 4k )(tf ?)( txP
ttf )( 10 t []
tnn
tfn
nsin)1(2)(
1
1
11
0
2 2( 1)sin1
n
nb t n td t n
D.E. 2 1
21
1 2( 1)4 sin16
n
n
d x x n tdt n
tnBtx nn
P sin)(1
tnsin 1
2 21 2( 1)416
n
nn B n
)64(
)1(3222
1
nnB
n
n
1
22
1
sin)64(
)1(32)(n
n
P tnnntx
** 1n N mkpN // [ mkw / ] nB
Pure Resonance H.W.1
)(1241
2
2
tfydt
yd
)()2(,20,2)( 2 tftfttttf
t
f (t)
1 2
-
16
2
2 21
116 cos18 ( 48)p n
y ntn n
H.W.2 Suppose a uniform beam of length L is simply supported at 0x and at x L . If the load
per unit length is given by 0( ) / , 0w x w x L x L , then the differential for the deflection ( )y x is
4
04
d y w xEILdx
where , ,E I and 0w are constants. (1) Expand ( )w x in a half-range sine series. (2) Use the method of above example to find a particular solution ( )y x of the differential
equation. [: D. G. Zill and M. R. Cullen, Advanced Engineering Mathematics, 2nd ed., Prob. 45, Sec. 12-3.]
(1) 101
2( ) ( 1) sinnn
w n xw xn L
(2) 4 1
05 5
1
2 ( 1)( ) sinn
pn
w L n xy xLEI n
H.W. 3 Proceed as in H.W. 2 to find a particular solution ( )y x when the load per unit length is as
given in following figure:
(1) 01
2 2( ) cos cos sin3 3n
w n n n xw xn L
(2) 24 cos cos
3 305 5
1
2( ) sinn n
pn
w L n xy xLEI n
[: D. G. Zill and M. R. Cullen, Advanced Engineering Mathematics, 2nd ed., Prob. 46, Sec. 12-3.]
x
w(x)
2L/3
w0
LL/3