Series Solutions of ODEs. Special Functions 상미분 방정식의 급수해법 특수함수

Engineering Mathematics I

Chapter 5 Series Solutions of ODEs Special Functions - Chapter 5. Series Solutions of ODEs. Special Functions - 5장. 상미분 방정식의 급수해법. 특수함수

민기복 민기복

Ki-Bok Min PhD Ki Bok Min, PhD

서울대학교 에너지자원공학과 조교수

Assistant Professor, Energy Resources Engineering, gy g g

Ch.4 Systems of ODEs. Phase Plane.

Q lit ti M th d Qualitative Methods

• Basics of Matrices and Vectors

• Systems of ODEs as Models

• Systems of ODEs as Models

• Basic Theory of Systems of ODEs

• Constant-Coefficient Systems. Phase Plane Method C it i f C iti l P i t St bilit

• Criteria for Critical Points. Stability

• Qualitative Methods for Nonlinear SystemsQ y

• Nonhomogeneous Linear Systems of ODEs

Series Solutions of ODEs. Special Functions 상미분 방정식의 급수해법 특수함수

상미분 방정식의 급수해법 . 특수함수

• 5.1 Power Series Method (거듭제곱 급수 해법)

• 5.2 Theory of the Power Series Method (거듭제곱 급수 해법의 이론)

• 5.3 Legendre’s Equation. Legendre Polynomials P_n(x) Legendre 방정식. Legendre 다항식 P_n(x)

5 4 Frobenius Method (Frebenius 해법)

• 5.4 Frobenius Method (Frebenius 해법)

• 5.5 Bessel’s Equation. Bessel functions J_v(x). Bessel의 방정식. Bessel 함수 J_v(x) 5 6 B l’ F ti f th S d Ki d Y ( ) 제2종 B l 함수 Y ( )

• 5.6 Bessel’s Functions of the Second Kind Y_v(x). 제2종 Bessel 함수 Y_v(x)

• 5.7 Sturm-Liouville Problems. Orthogonal Functions. Sturm-Liouville 문제. 직교함수

O f 직교 고유함수의 전개

• 5.8 Orthogonal Eigenfunction Expansions. 직교 고유함수의 전개

Series Solutions of ODEs. Special F ti

Functions.

• 변수계수를 갖는 선형미분 방정식을 풀이하는 표준적인 방법인 power series method

혹

(거듭제곱급수, 혹은 멱급수 해법)을 소개한다

• 거듭제곱급수 해법으로 얻을 수 있는 유명한거듭제곱급수 해법으로 얻을 수 있는 유명한 특수함수:

Bessel function (베셀 함수) – Bessel function (베셀 함수),

– Legendre function (르장드르 함수),

– Gauss의 hypergeometric function (초기하함수)

Power Series Method

– Linear ODEs with variable coefficient  power series method – Power series is an infinite series of the form*;

 Coefficients (계수) :

           

^



2 0 2

0 1

0 0

0

x x a x

x a a x

x

a ^m

m m

 a

a

 Coefficients (계수) : a

 Center (중심) :

 If x = 0;

, , , _{1 2}

0 a a

a x0







^{ a xm}  ^{a a x a x2}

 If x₀ = 0;

– Maclaurin series (맥클로린 급수)







 

 0 1 2

0

x a x a a x

a

m m

 









 ^



) 1 ( 1 1

1 ₂

0

x x

x x m x

m

  _











 

























1 1 cos

! 3

! 1 2

!

4 2 2

3 2

0

x x x x

x x x

m e x

m m m

m x

 

( ) 0

0 0

( ) : ( )

!

n n

n

f x

Taylor Series f x x x n





  

 

    ^{   }

 





















! 5

! 3

! 1 2 sin 1

! 4

! 1 2

! cos 2

5 3

0

1 2 0

x x x

m x x

x m

m

m m m

*통상 음의 거듭제곱이나 분수거듭제곱을

가지는 급수는 포함하지 않음.

Power Series Method

• Idea of power series

– For a given ODEFor a given ODE ^{y ''p} ^{x y'q} ^{x y 0}

– Represent and by power series in power of x A l ti i th f f i

 ^{y q} ^y

p y

 ^x

p ^q ^x

– Assume a solution in the form of power series











 ^



3 3 2 2 1

0 0

x a x a x a a x

a y

m

m m

– Differentiation of this series, and put into ODE

m0









 ^{ 1 2 3 2}

' ma x a a x a x

y   ^m    

 1 2 3

1

3 2a x a x a

x ma y

m m

 ^  ^{   }

 ^



 a a x

x a m

m y

m

m

m 2 3

2

1 2 3 2

1 ''

– Determine the unknown coefficients ^am

Power Series Method

• Ex 1. Solve the following ODE by power series

xy y 2' y y











 ^



3 3 2 2 1

0 0

x a x a x a a x

a y

m

m m  ^     

 1 2! 3!

!

3 2

0

x x x

m e x

m

m x









 ^



 2

3 2

1 1

1 2 3

' ma x a a x a x

y

m

m m

 ² 

2

m 0

 









2 2

2 3

2

2 3

2

3 2 2

1 0

2 3 2

1

2 2 1

0 2

3 2

1

x a x

a x a x

a x a a

x a x a a x x

a x a a





































! , 3 3

!, 2 2

,

, 2 6

, 2 5

, 2 4

, 2 3

, 2 2

, 0

0 4

6 0 2

4 0 2

4 6

3 5

2 4

1 3

0 2

1

a a a

a a a

a a

a a

a a

a a

a a

a a

a



























! 3 3

! 2 2

2

0 8

6 4

2

0 1 2! 3! 4!

x x x a e^x

x a

y  

 



     



 

Power Series Method

• Ex. 2 Solve the following ODE by power series.

0 '' y 

y  

^{ 1 mx2m x2 x4}

^





m 0

m mx a

y ^  



 



2

1 2

''

m

m mx a m m y

m a x a x

m  _m ^m  _m ^m 

 ^{ 1 2} ^{ 0}

  

   ^











 

 

















 



! 5

! 3

! 1 2 sin 1

! 4

! 1 2

! 2 cos 1

5 3

0

1 2 0

x x x

m x x

x x m

x x

m m m

 

s s a x a x  m s m s

s

s s s

s s

m m m

m















  











 





, 2

1

2

0 0

2 0

2 _m0 2m^1! 3! 5!

첫 번째 항은 두 번째 항은

   ^{s , , }

s s

a_s a^s 0 1

1

2 2 



 

 

! 3 2

3

!, 2 1

2

1 1

3 0

0 2

a a a

a

a a  

 





 





Recursion Formula (순환공식):

x x x

x a

a a

a _{2 3 4 5} ^{2 4 3 5} 







! 5 4 5

!, 4 3 4

! 3 2

3

! 2 1

2

1 3

5 0

2 4

a a a a

a a 

 



 





x a

x a

x x x

x a a x

a x a x

a x a x

x a a y

sin cos

! 5

! 3

! 4

! 1 2

! 5

! 4

! 3

! 2

1 0

1 0

1 5 0 4

1 3 0 2

1 0







 



   





 



   

















   

Theory of the Power Series Method

B i C t

Basic Concepts

• Basic Concepts

 0 0 1 0 2  0²  0 1 0 ¹ 2  0 ²

m n n n

m n n n

a x x a a x x a x x a x x a x x a x x

  

 

             

 ^{ 0 0 1 0 2  0   0 1 0 2  0 }

0

m n n n

m

 



n 

s x : n-th partial sum (부분합) R xn  : remainder (나머지)

– Series is convergent at x=x₁  – ^{s x( )1 } ^{ a xm( 1 x0)m}

 1 1

lim _n ( )

n s x s x

 

Value (수렴값) or sum (합)

– For every n,

1 1 0

0

( ) _m( )

m

1 1 1

( ) _n( ) _n( ) s x  s x R x

– If this sequence diverges at x=x₁, series (1) is called divergent at x=x₁

– In case of convergence, for any positive ε, there is an N such that

1 1 1

( ) ( ) ( )

n n

R x  s x s x  for all n N

Theory of the Power Series Method

C I t l (수렴구간)

Convergence Interval (수렴구간),

Radius of Convergence (수렴반지름)

– 수렴구간: 급수가 수렴하는 값들의 구간 ( 의 형태로 나타남)

R x

x  ₀ 

– 수렴반지름 (R ):

급수는 인 모든 x 에 대하여 수렴하고,

인 모든 x^{x }에 대하여 발산할 때^{x0  R} _R

인 모든 x 에 대하여 발산할 때 _{x  x  R}

0

R 1

 _{R } ¹

m m

m

lim

a





lim

^am 1^am

m





Theory of Power Series Method

• Case 1: (useless) The series always converges at the center.

• Case 2. (usual) If there are further values of x for which the series converges, these values form an interval, called the series converges, these values form an interval, called the convergence interval.

• Case 3. (best) The convergence interval may sometimes be infinite that is the series converges for all x

infinite, that is, the series converges for all x.

Theory of Power Series Method

• Example 1.

2 3

! ^m 1 2 6

m x x x x

     



^

m0

1 ?

m m

a a

  R = 0, converges only at the center x

• Example 2. _{2 3}

0

1 1

1

m m

x x x x

x





     

  ^

R = 1, converges when |x| < 1

• Example 3. ²

0

! 1 2!

m x

m

x x

e x

m





     ^

0 ! !

m

R = ∞, converges for all x

Theory of Power Series Method

• Example 4.

 

^



 

^1 ^m _{3  } x³ _ x⁶ _ x⁹ _{ }



 1 8 64 512 8

1 ₃

0

x x

x ^m x

m m

8

8 1 8

8

1

1  _   

 R

a a

m m m

3  8

 x

8 t

a_m 8

converges when |x| < 2 converges when |x| < 2

Theory of Power Series Method

O ti P S i

Operations on Power Series

– Termwise Differentiation:

 0^m ( 0 )

y  ^ a x x x x  R ^{y } ^{ ma x xm   0m1 ( x x 0  R)} – Termwise Addition:

 0 0

0

( )

m m

y a x x x x R



  ^{ 0 0}

1

( )

m m

y



 0  0   0

m m m

m m m m

a x x b x x a b x x

  

     

  

– Termwise Multiplication:

0 0 0

m m m

^{a b a b a b} ^{x x} ^m

   



If i h i i di f d

  

     

0 1 1 0 0

0

2

0 0 0 1 1 0 0 0 2 1 1 2 0 0

...

...

m m m

m

a b a b a b x x

a b a b a b x x a b a b a b x x

 

   

        



– If a power series has a positive radius of convergence and a sum that is identically zero throughout its interval of convergence, then each coefficient of the series must be zero.(Vanishing of All

Coefficients)

Theory of Power Series Method

E istence of Po er Series Sol tions of Existence of Power Series Solutions of ODEs. Real Analytic Functions.

• Definition: Real Analytic Function

– A real function f(x) is called analytic at a point x=xA real function f(x) is called analytic at a point x x₀₀ if it can be if it can be represented by a power series in powers of x-x₀ with radius of convergence R>0

• Theorem1. Existence of Power Series Solutions

     

''   '     (9)

'' ' (9)

y  p x y q x y r x 

  ^''   ^'     ⁽¹⁰⁾

h x y  p x y q x y r x    

Legendre’s Equation. Legendre P l i l P ( )

Polynomials P

_n

(x)

• Legendre’s Equation.



^{1 x2}



^{y ''2xy'nn 1y  0} n is a given real number

 

^{y y  y}

  



^{      }



2

2 1

1 0

1 '

, _m ^m _m ^m

m

mx y ma x y'' m m- a x

a

y ,

n is a given real number

¹  ^{ 1 2  1 0}

0 1

1 2

2

2    



   ^







 





m

m m m

m m m

m

mx x ma x n n a x

a m- m x





0 m 1 m 2

m

 ¹  ^{1 2}  ¹ ⁰

0 1

2 2

2     

    ^

















m

m m m

m m m

m m m

m

mx m m- a x ma x n n a x

a m- m

m 2 = s m = s

 ² ¹  ¹   ¹ ⁰

 ^{ s s a xs} ^{ s s a xs} ^{ sa xs} ^{ n n a xs}

m-2 = s m = s

 ² ¹  ¹  ¹ ⁰

0 1

2 0

2     





    







 

s

s s

s s

s s

s x s s- a x sa x n n a x

a s

s

Legendre’s Equation. Legendre P l i l P ( )

Polynomials P

_n

(x)

 

 

 ^{2 1}  ⁰

2 3

0 1

1 2

1 3

1

0 2

0





















a n

n a

x

a n n a x

 :

:

n sn s a s 

a 1 0 1

 



 





 s²s ¹a_s2 ss ¹²snn¹a_s ⁰



   a s , , 

s

a_s s _s 0 1

1

2 2 



 

 _

    

1 3

0

2 3!

2 1

! 2

1 n n a

a n a

a n  



 





               

1 3

5 0 2

4 5!

4 2

1 3

4 5

4 3

! 4

3 1

2 3

4

3

2 n n n n a

n a a n

n a n

n a n

n

a n    

 



 

 



 





 



     

            _

 



 ^{2 4}

1 4!

3 1

2

! 2

1 1 n n n n x

n x x n

y

 ^{x a y}  ^{x a y}  ^x

y  _{0 1}  _{1 2}

General solution:

              _

 

 

 ^{3 5}

2 5!

4 2

1 3

! 3

2 1

! 4

! 2

n x n

n x n

n x n

x y

Legendre’s Equation. Legendre P l i l P ( )

Polynomials P

_n

(x)

• Legendre Polynomials

 ^{2n ! 1 3 5  }^2n1 _{로 선택 P (1) 1 이 되도록 선택}

 

   

  

    

2

2

2 ! !

2 1

2 1

n n

s s

a n n

s s

a a s n

n s n s ^

 

 

   

  

로 선택 P_n(1)  이 되도록 선택1

  

 

    

   

      

2 2

1

1 1 2 !

2 2 1 2 2 1 2 !

1 2 2 1 2 2 ! 2 2 !

n n n

n s n s

n n n n n

a a

n n n



 

 

    

 

    

        

   

  

 

4 2

1 2 2 1 2 2 ! 2 2 !

2 2 1 2 1 ! 1 2 ! 2 1 ! 2 !

2 3 2

n n

n n

n n n n n n

n n n n n n n n

n n

a a

   

   

     

 

    

 ⁿ  ^{4 !} 



 

4 2

4 2 3

n n

 n     

   

   

2

2 2! 2 ! 4 !

2 2 !

1

2 ! ! 2 !

n

m

n m n

n n

n m

a ^ m n m n m

 

   

 

     

    ²

0

2 2 ! 1

1

2 ! ! 2 ! 2 2

M m n m

n n

m

n m n n

P x x M

m n m n m





   

       ^또는 

Legendre’s Equation. Legendre P l i l P ( )

Polynomials P

_n

(x)

• Legendre Polynomials

 ^{x P} ^{x x}

P₀ 1, ₁  ,

    ^{ }  

 ^x  ^{x x}  ^{P  x}  ^{x x x}

P

x x

x P x

x P

15 70

1 63 3

30 1 35

,

3 2 5

1

,

1 2 3

1

3 5

5 2

4 4

3 3

2 2





















 ^x  ^{x x}  ^{P  x}  ^{x x x}

P 63 70 15

8

3 30

8 35 ⁵

4  

Legendre’s Equation. Legendre P l i l P ( )

Polynomials P

_n

(x)

• Example. (problem set 5.3)



²





^{1 x y2}



^{'' 2 xy'  0  n 0}