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

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