Engineering Mathematics Icontents.kocw.or.kr/document/wcu/2012/JunNam/LeeBongKee... · 2012-08-20 · Engineering Mathematics I School of Mechanical Systems Engineering 5.1 Power

Bong-Kee Lee School of Mechanical Systems Engineering

Chonnam National University

Engineering Mathematics I

5. Series Solutions of ODEs. Special Functions

School of Mechanical Systems Engineering Engineering Mathematics I

5.1 Power Series Method

n계 선형상미분방정식

xryxpyxpyxpy n

n

n

01

1

1 '

xryayayay n

n

n

01

1

1 '

001

1

1

aaa n

n

n

characteristic equation

xey

xyxyxy ph

general solution of nonhomogeneous ODE

nn ycycxy 11

general solution of homogeneous ODE

constant coefficients



거듭제곱급수(멱급수, power series) 해법

– 변수계수를 가지는 선형상미분방정식을 풀이하는 표준해법 • 해의 수치적 값을 계산하거나 특성을 조사하는데 사용됨

• 해에 대한 다른 표현을 유도하는데 사용됨

거듭제곱급수

– x – x0의 거듭제곱을 가지는 무한급수의 형태

– x0 = 0 인 경우

2

02010

0

0 xxaxxaaxxam

m

m

coefficient center

2

210

0

xaxaaxam

m

m



거듭제곱급수

– 거듭제곱급수의 예: Maclaurin 급수

!5!3!12

1sin

!4!21

!2

1cos

!3!21

!

11

1

53

0

12

22

0

2

32

0

2

0

2

210

0

xxx

m

xx

xx

m

xx

xxx

m

xe

xxxx

xaxaaxa

m

mm

m

mm

m

mx

m

m

m

m

m



거듭제곱급수 해법의 개념

m

m

m

m

m

m

m

m

m

m

a

yxqyxpyyyy

xaaxammy

xaxaaxmay

xaxaxaaxay

xqxp

yxqyxpy

obtain

0''',',''

2321''&

32'&

solution

seriespower ~,

0'''

32

2

2

2

321

1

1

3

3

2

210

0




2

0

642

0

60402

00

3

3

2

210

026

02402

26

2402531

352413021

4

3

3

2

2

10

3

3

2

210

2

321

2

321

1

13

3

2

210

0

!3!21

!3!2

,!33

,!22

,

,3

,2

,&0

25,24,23,22,0

2222

232

32'

2'example

x

m

m

m

m

m

m

eaxx

xa

xa

xa

xaaxaxaxaay

aaa

aaaaa

aa

aaaaaaa

aaaaaaaaa

xaxaxaxa

xaxaxaaxxaxaa

xaxaaxmayxaxaxaaxay

xyy




– 첨수이동(shift of index)

,233

1,0

5

2,

22

1,0

3

2,

2

222,0

22

22

2''&

04635

0241302

221

0

1

0

1

21

0

1

2

1

1

01

1

1

1

0

aaaaa

aaaaaaa

as

aaasa

xaxasa

xaxmaaxaxxma

xyyxmayxay

ssss

s

s

s

s

s

s

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m – 2 = s m = s




xaxaxx

xaxx

a

xaxaxaxaxaaxay

aaa

aaa

aa

aa

sss

aaaass

xaxassxaxamm

xammyxmayxay

yy

m

m

m

ssss

s

s

s

s

s

s

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

sincos!5!3!4!2

1

,!545

,!434

,!3

,!2

,2,1,0 12

12

121

1'','

0''example

10

53

1

42

0

5

5

4

4

3

3

2

210

0

135

024

13

02

22

00

2

02

2

2

2

1

1

0

순환공식, 점화공식 (recursion formula)


5.2 Theory of the Power Series Method

거듭제곱급수 해법의 이론

– 기본 개념 • 거듭제곱급수: 무한급수의 형태

• n-번째까지의 부분합(n-th partial sum)

• 나머지(remainder)

• 수렴 및 발산: x = x1에서 수렴한다

2

02010

0

0 xxaxxaaxxam

m

m

nnn xxaxxaxxaaxs 0

2

02010

2

02

1

01

n

n

n

nn xxaxxaxR

11

0

0111lim xRxsxxaxsxs nn

m

m

mnn

수렴값




– 기본 개념 • 수렴

• 세 가지 대표적인 경우

– 경우 1. 무용: x = x0 에서만 유일하게 수렴하는 경우

– 경우 2. 보통: 수렴구간 내에서 수렴하는 경우

– 경우 3. 유용: 모든 구간에서 수렴하는 경우 (수렴구간이 무한대)

NnxsxsxR nn for 111

Rxx 0

중점(midpoint) 수렴반지름

m

m

m

mm

m

a

aR

aR

1lim

1or

lim

1




0

as 1!

!1!

621!1

1

32

0

R

mmm

m

a

ama

xxxxm

m

mm

m

m

1

as 111

1 11

12

1

32

0

R

ma

aa

xxxxxx

m

mm

m

m

R

mmm

m

a

a

ma

xx

m

xe

m

mm

m

mx

as 01

1

!1

!

!

1

!21

!3

1

2

0

280

8

8

1

8

8

8

1

5126481

8

14

33

1

1

963

0

3

xxRx

R

a

aa

xxxx

m

m

m

m

m

m

m

m

m

m

m




– 거듭제곱급수의 연산 • 항별 미분(termwise differentiation)

• 항별 덧셈(termwise addition)

• 항별 곱셈(termwise multiplication)

• 모든 계수가 0이 됨(vanishing of all coefficients)

– 만일 어떤 거듭제곱급수가 양의 수렴반지름을 갖고, 수렴구간 전체에서 합이 항등적으로 0이라면, 급수의 모든 계수는 0이다.

1

1

0

0

0 'm

m

m

m

m

m xxmaxyxxaxy

0

0

0

0

0

0

m

m

mm

m

m

m

m

m

m xxbaxxbxxaxgxf

0

00110

0

0

0

0

m

m

mmm

m

m

m

m

m

m xxbababaxxbxxaxgxf




– 실수 해석함수(real analytic function) • 실수함수 f(x)가 수렴반지름 R>0을 갖고, x – x0의 거듭제곱급수로

나타내어지면, f(x)는 x = x0에서 해석적이라 한다.

– 거듭제곱급수 해의 존재

.0~

and at analytic

are ~~'~''~

in ~ and ,~,~,~

if trueis same theHence .0 econvergenc

of radius with of powersin seriespower aby drepresente be can thus and at

analytic issolution every then ,at analytic are in and If

solution seriespower of existence

00

00

0

xhxx

xryxqyxpyxhrqphR

xxxx

xxxryxqy'xpy''rp, q,


5.3 Legendre’s Equation. Legendre Polynomials

Legendre의 방정식

Adrien-Marie Legendre

spherical coordinate system

01'2''1 2 ynnxyyx

Legendre function

매개변수(parameter), n

0

1

1'

1

2''

22

y

x

nny

x

xy

analytic at x = 0

2

2

1

1

0

1'','m

m

m

m

m

m

m

m

m

xammyxmay

xay

0211

1

01

1

2

22

m

m

m

m

m

m

m

m

m xakxmaxxammx

nnk

011 2

ynny

dx

dx

dx

dor




,2,1,0 12

1

012112

012212342:

012231:

01120:

02112

0211

0211

2

2

2224

2

113

1

02

0

0120

2

0122

2

01

1

2

22

sass

snsna

annsssass

annaaasx

annaasx

annasx

xkaxsaxassxass

xkaxmaxammxamm

xakxmaxxammx

ss

ss

s

s

s

s

s

s

s

s

s

s

s

s

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

순환공식, 점화공식 (recursion formula)




5

1

4

0

3

1

2

010

5

5

4

4

3

3

2

210

0

1

35

13

0

24

02

2

!5

4213

!4

312

!3

21

!2

1

2345

4213

45

4323

21

1234

312

34

3212

1

,2,1,0 12

1

xannnn

xannnn

xann

xann

xaa

xaxaxaxaxaaxaxy

annnn

ann

a

ann

a

annnn

ann

a

ann

a

sass

snsna

m

m

m

ss




53

2

42

1

2110

53

1

42

0

!5

4213

!3

21

!4

312

!2

11

!5

4213

!3

21

!4

312

!2

11

xnnnn

xnn

xxy

xnnnn

xnn

xy

xyaxyaxy

xnnnn

xnn

xa

xnnnn

xnn

axy



Legendre 다항식, Pn(x)

011 2

xPnnxP

dx

dx

dx

dnn

011 2

ynny

dx

dx

dx

dLegendre’s equation

Legendre polynomials

xyaxyaxy 2110

음이 아닌 정수, n = s ss a

ss

snsna

12

12

0m

m

mxaxy

polynomialorder th -n:

3

2

1

0

3

2

1

0

xP

nxP

nxP

nxP

nxP

n




1&integer positive:

!

12531

!2

!202

an

n

n

n

na

nn

!2!!2

!221

!2!12

!22

!21!12

!22122

122

1

!2

!2

122

1

122

1:2

2 1

12

12

1

22

22

22

mnmnm

mna

nn

na

nnnnn

nnn

n

nn

n

n

n

nna

n

nnans

nsasnsn

ssaa

ss

snsna

n

m

mnnn

nnnn

ssss

integer:

2

1or

2

!2!!2

!221

0

2

nnM

xmnmnm

mnxP

M

m

mn

n

m

n Legendre polynomials




xxxxP

xxxP

xxxP

xxP

xxP

xP

1570638

1

330358

1

352

1

132

1

1

35

5

24

4

3

3

2

2

1

0

xP0

xP1

xP2 xP3

xP4

xP5


5.4 Frobenius Method

Frobenius 해법

– 해석적이지 않은 계수를 가지는 특수한 2계 상미분방정식의 해법을 제공

– 정리 1. Frobenius method

Ferdinand Georg Frobenius

0'''

2 y

x

xcy

x

xby

0at analytic :, xxcxb

numbercomplex or realany :

0 0

2

210

0

r

axaxaaxxaxxy r

m

m

m

r



Frobenius 해법

– 결정방정식(indicial equation)

01

011,for

0

111

111''

1'

&

0'''0'''

00

00000000

2

210

2

210

10

2

21010

10

2

0

2

10

1

0

1

2

210

02

210

2

210

2

2

crbrr

acrbrracrabarrx

xaxaaxxcxcc

xarraxxbxbbxrararrx

xrararrxxarmrmxy

xarraxxarmxy

xaxaaxxaxxyxcxccxc

xbxbbxb

yxcyxxbyxyx

xcy

x

xby

r

r

rr

r

m

rm

m

r

m

rm

m

r

m

m

m

r



Frobenius 해법

– 정리 2. 해의 기저 • 경우 1. 두 근의 차가 정수가 아닌 서로 다른 근들

• 경우 2. 이중근

• 경우 3. 두 근의 차가 정수인 서로 다른 근들

2

2102

2

2101

2

1

xAxAAxxy

xaxaaxxyr

r

integer21 rr

21 rrr

integer21 rr

0 ln 2

2112

2

2101

xxAxAxxxyxy

xaxaaxxyr

r

0 bemay &0

ln

21

2

21012

2

2101

2

1

krr

xAxAAxxxkyxy

xaxaaxxyr

r



Frobenius 해법

2

210

2

2101

2

0

2

00

1

00

1

0

0

1

0

0

2

0

1

0

2

root double:00001

03

11

1'','

0at analytic :1

,1

13

01/

'1/13

''01

1'

1

13''

0'13''1example

xaxaaxaxaaxxy

rrarraarrx

xaxarmxarm

xarmrmxarmrm

xarmrmyxarmyxaxy

xx

xxc

x

xxb

yx

xxy

x

xxyy

xxy

xx

xy

yyxyxx

r

r

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

m

m

r



Frobenius 해법

x

xuyyxu

xuuxu

xy

xyyuxyuyuxx

yyxyxxuyuxyuyuxx

uyyuyuyuyyuyuyy

xx

xxxxaxaaxy

aaaaaaassasasass

xaxasxsaxsasxass

xaxmaxmaxammxamm

xaxaaxy

m

m

sssssss

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

1

lnln

1''''

1

1',

1

10'13''2''1

0'13''1'13''2''1

''''2'''','''orders ofreduction of method

1 1

11

101311

01311

0311&

12

211111

111111

111211212

0

22

2101

210111

01

1

01

1

0

00

1

00

1

0

2

2101



Frobenius 해법

ssssss

s

rs

s

s

rs

s

m

rm

m

m

rm

m

r

r

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

m

m

r

ass

saassasarsrsars

xarsrsxars

xarmrmxarm

xaxaaxxaxaaxxy

rrrrrrrarrx

xaxarmxarmrmxarmrm

xarmrmyxarmyxaxy

yxyyxx

1201211

011

011

0integer0,10,10101

011

1'','

0'''example

2

11

2

1

2

1

1

0

2

0

1

0

2

2

210

2

2101

21210

1

000

1

0

0

2

0

1

0

2



Frobenius 해법

1ln1

ln

1ln

111'

1ln1lnln2'ln

1

122

'

''0'2''0'2''

0''2'''''

'2'''',''orders ofreduction of method

1&012

12

222

2

222

2

222

2

2212

1021

2

1

xxxx

xuxuyy

xxu

xxx

xu

x

xxxu

xxxx

x

u

uuxuxxuxxxuxx

uxuxuxuxuxxyxyyxx

uxuyuxuyuxuyy

xyaaaass

sa ss


5.5 Bessel’s Equation. Bessel Functions

Bessel의 방정식

Friedrich Wilhelm Bessel

cylindrical coordinate system

0''' 222 yxxyyx

Bessel functions

매개변수(parameter), ν (음이 아닌 실수)

0'1

''2

22

yx

xy

xy

analytic at x = 0

0

2

0

1

0

1''

'

m

rm

m

m

rm

m

m

rm

m

xarmrmy

xarmy

xay

Frobenius method



Bessel의 방정식

21

0

22

0

2

00

2

2

1

2

11

0

2

00

0

2

2

2

00

0

2

0

2

00

0

22

0

1

0

22

222

,0

001

01,3,2

0111

010

01

01

01

0'''

rr

rrararaarr

aaarsarsrss

aarrars

araarrs

xaxaxarsxarsrs

xaxaxarmxarmrm

xaxxarmxxarmrmx

yxxyyx

ssss

s

rs

s

s

rs

s

s

rs

s

s

rs

s

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

rm

m

m

rm

m

결정방정식 (indicial equation)



Bessel의 방정식

022

04224022

22222222

753

22

22

2

2

2

2

111

2

1

2

11

1

21!2

1

12!22

1

222

1,

12

1

2

1022202

,3,2,1 with 2

0

021

01

001211011

amm

a

aaaaa

amm

aamamasas

mms

aaa

asasaasaasss

aaarsarsrs

aaaaarrar

rr

m

m

m

mmmmss

ssssss

ssss

계수 점화식(coefficient recursion formula)



Bessel 함수, Jn(x)

– ν = n (정수)인 경우

!!2

1

!2

1

21!2

1

!2

1&

21!2

1

21!2

1

2220

022022

mnmnmnnnma

na

amnnnm

aamm

a

nm

m

nm

m

mn

m

m

mm

m

m

0m

rm

mxaxy

02

2

!!2

1

mnm

mm

n

nmnm

xxxJ

nr 1

n차 제1종 Bessel 함수 (Bessel function of the first kind of order n)




– 제1종 Bessel 함수

022

2

0!2

1

mm

mm

m

xxJ

02

2

!!2

1

mnm

mm

n

nmnm

xxxJ

012

12

1!1!2

1

mm

mm

mm

xxJ

xJ0

xJ1




– 임의의 ν ≥ 0 에 대한 Bessel 함수 • 감마함수(gamma function), Γ(ν)

02

2

!!2

1

mnm

mm

n

nmnm

xxxJ

0 0

1

dtte t !1

1

nn

!2

10

na

n

!!2

122

mnma

nm

m

m

022

21!2

1a

mma

m

m

m

ν = n (양의 정수)

12

10

a

1!2

1

12

1

21!2

1

22

22

mma

mma

m

m

m

m

m

m

02

2

1!2

1

mm

mm

mm

xxxJ

ν차 제1종 Bessel 함수



정수가 아닌 임의의 ν에 대한 일반해

– 두 번째 1차 독립인 해: J-ν(x)

– Bessel 방정식의 일반해(정수가 아닌 ν≥0)

– 만일 ν가 정수일 경우(ν=n) • Bessel 함수 Jn(x)와 J-n(x)는 1차 종속

→ 제2종 Bessel 함수(Bessel function of the second kind)

02

2

02

2

1!2

1

1!2

1

mm

mm

mm

mm

mm

xxxJ

mm

xxxJ

xJcxJcxy 21 0''' 222 yxxyyx

xJxJ n

n

n 1



Bessel 함수의 특성

– 미분, 점화관계

– 반정수 차수 ν에 대한 초등(elementary) Bessel 함수 • ν = ±1/2, ±3/2, ±5/2, …

xJxJxJ

xJx

xJxJ

xJxxJx

xJxxJx

'2

2

'

'

11

11

1

1

xx

xJxx

xJ cos2

& sin2

2/12/1

xJxJx

xJxJx

xJxJ

xJxJx

xJxJx

xJxJ

xJx

xJxJ

2/12/12/32/12/12/3

2/12/12/32/12/32/1

11

112/1

112/1

2


5.6 Bessel Functions of the Second Kind

제2종 Bessel 함수(Bessel function of the second kind), Yn(x)

– ν = n (정수)인 경우 1차 독립인 두 번째 해를 결정

0

1'2

ln'''ln

ln'1'2

ln''

1'2

ln''''&ln''

ln&

root double :0&0'''0'''

11

10

1

2

2

00000

1

0

1

100

1

2

2

000

1

2

2

0002

1

1002

1

0201

222

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

xAxxmAx

J

xAmmx

J

x

JxxxJJxJxAxJx

xmAx

JxJxAmm

x

J

x

JxJx

xAmmx

J

x

JxJyxmA

x

JxJy

xAxxJxyxJxy

nrxyyxyyxxyyx




0012

0120:,,

01

0!1!2

1

!1!2

1',

!2

10'2

01'2

01'2

7531212

2

1

2

12

1

2

12

2

1

1

1

12242

1

0

1

20

1

1

1

12

122

12

112

12

0

022

2

0

1

1

1

12

0

1

1

1

1

1

1

0

11

10

1

2

2

00

AAAAAs

xAxAsxAxAmxxx

AxAx

xAxAmmm

x

mm

xxJ

m

xxJxAxAmJ

xAxmAxAmmJ

xAxxmAx

JxAmm

x

J

x

Jx

ss

s

s

s

s

s

s

m

m

m

m

m

m

s

m

m

m

m

m

m

mm

mm

mm

mm

mm

mm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m




642

0

1

2

22

1

02

22

1

21

22

1

2

424222

2

2

1

1231

22

2

0

1

1

1

1

12

122

12

13824

11

128

3

4

1ln

!2

1ln

!2

1,3,2

1

3

1

2

11,1

,3,2,1 1

3

1

2

11

!2

1

128

3016

8

1:1022

!!12

1:,,

4

102

!0!12

1

0!1!2

1

xxxxxJxm

hxxJxy

m

hAm

mhh

mmm

A

AAAsAAsss

xxx

AAx

xAxAmmm

x

m

m

m

m

m

m

m

m

mm

m

m

m

sss

s

s

m

m

m

m

m

m

mm

mm




1

2

22

1

00

1

002222012211

!2

1

2ln

2

ln1

lim05772156649.02ln,2

~

m

m

m

m

m

n

kn

xm

hxxJxY

nk

ba

xYbJyaxyxycxJcxycxycxy

Euler-Mascheroni constant

0차 제2종 Bessel 함수 또는 0차 Neumann 함수 (Bessel function of the second kind of order 0 or Neumann’s function of order 0)




– ν의 특성에 관계없는 통일된 형식의 함수 형태의 유도

xYxY

mhhhnx

xm

mnxx

nmm

hhxxxJxY

xYxY

xJxJxY

n

n

n

m

n

m

m

nm

n

m

m

nm

nmm

mn

nn

nn

1

1

3

1

2

11,1,0,,2,1,0,0

!2

!1

!!2

1

2ln

2

lim

cossin

1

10

1

0

2

20

2

2

1




xY0

xY1



Bessel 방정식의 일반해

– ν차 제3종 Bessel 함수, Hankel 함수 • 실수 x에 대하여 복소인 Bessel 방정식의 해

xYcxJcxy

yxxyyx

21

222 0'''

xiYxJxH

xiYxJxH

2

1


5.7 Sturm-Liouville Problems. Orthogonal Functions


5.8 Orthogonal Eigenfunction Expansions

Documents

Engineering Mathematics Icontents.kocw.or.kr/document/wcu/2012/JunNam/LeeBongKee... · 2012-08-20 · Engineering Mathematics I School of Mechanical Systems Engineering 5.1 Power