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(1)

2008_Fourier Series(2)

Naval Architecture & Ocean Enginee ring

Engineering Mathematics 2

Prof. Kyu-Yeul Lee

Department of Naval Architecture and Ocean Engineering, Seoul National University of College of Engineering

[2008][10-1]

November, 2008

(2)

2008_Fourier Series(2)

Naval Architecture & Ocean Enginee ring

Fourier Series(2)

: Sturm-Liouville Problem

(3)

2008_Fourier Series(2)

Sturm-Liouville Problem

Review

 0

General solutions Linear Equations

 0

  y y

2

 0

  y y

2

 0

  y

y    0

e

x

c y

1

x c

x c

y

1

cos  

2

sin 

 

) sinh

cosh

,

2 1

2 1

x c

x c

y

or e

c e

c

y

x x

General solutions Cauchy-Euler Equation

2

0

2

y   x y   y

x

 

0 ,

ln

0 ,

2 1

2 1

x c

c y

x c x

c 0 y

 

 0 x

Linear Equations

When is a finite interval

x

When is an infinite or half finite interval

x

(4)

2008_Fourier Series(2)

Sturm-Liouville Problem

Review

General solutions Parametric Bessel equation

) ( )

(

2 0

0

1

J x c Y x

c

y    

Particular solutions are polynomials

Legendre‟s equation

0 )

1 (

2 )

1

(  x

2

y   x y   n ny

0 1

2 2

( ) 1, ( ) ,

( ) 1 (3 1), 2

y P x

y P x x

y P x x

 

 

  

 0 x

 0

2 2 2

0 x y    y x y

,...

2 , 1 ,

 0

n

(5)

2008_Fourier Series(2)

Sturm-Liouville Problem

Eigenvalues and Eigenfunctions

0 )

( ,

0 )

0 ( ,

0

"  yyy Ly

When (Case III )   0 0

2 , 

  

Write

x c

x c

y1 cos   2 sin 

or

1  0 c

2 sin

n

y c n x

L

 

(nontrivial solution)

m i i

m

1

 ,

2

 

Then roots of auxiliary equation is

2  0 c 0

) 0 (  y

2 2

) (

, L

n n

L   nn

   

0 )

( Ly

Recall example 2 of section 3.9

Eigenvalues

Eigenfunctions

(6)

2008_Fourier Series(2)

Sturm-Liouville Problem

Eigenvalues and Eigenfunctions

0, (0) 0, ( ) 0 y   yyy L

2 sin

n

y c n x

L

 

2 2

) ( L

n

n n

 

  

Eigenvalues

Eigenfunctions

It is important to recognize the set of functions generated by this B.V.P the orthogonal set of functions on the interval used as the basis for the Fourier sine series

) , 0 ( L

0, (0) 0, ( ) 0

y    yy y L the Fourier cosine series

(7)

2008_Fourier Series(2)

Sturm-Liouville Problem

 Example 1

Eigenvalues and Eigenfunctions

0 ) ( , 0 ) 0 ( ,

0    

  y y y L

y

It is left as an exercise to show, by considering the three possible

cases for the parameter (zero, negative, or positive; that is,

that the eigenvalues and )

eigenfunctions for the boundary- value problem

0 ,

0 ,

0 ,

0 ,

0  2    2  

      

and

are, respectively,

is an

eigenvalue for this BVP and is the corresponding eigenfunction.

The latter comes from solving subject to the same boundary

conditions . Note also that can be incorporated into the family by

permitting . The set

is orthogonal on the interval [0,L].

, , 2 , 1 , 0 ,

/

2

2 2 2

 

n

n L n

n

 

1

cos( / ) ,

1

0.

0

0

yc n x Lc   

1 y

 0 y 

0 ) ( , 0 ) 0

(   

y L

y

 1 y

) / cos( n x L y  

 0

n {cos( nx / L )}, ,

, 3 , 2 , 1 ,

0 

n

(8)

2008_Fourier Series(2)

Sturm-Liouville Problem

Regular Sturm-Liouville Problem p , q , r , r

real-valued functions

continuous on an interval

are not both zero

] , [ b a

1 1

, B A Solve : [ r ( x ) y  ]  [ q ( x )  p ( x )] y  0

dx

d

Subject to:

0 )

( )

(

0 )

( )

(

2 2

1 1

 

 

b y B b

y A

a y B a

y A

0 ) ( , 0 )

( xp xr

for every in the interval x [ b a , ]

are not both zero

2 2

, B A

0, (0) 0, ( ) 0 y   yyy L

L b

a B

A B

A

1

 1 ,

1

 0 ,

2

 1 ,

2

 0 ,  0 , 

L b

a B

A B

A

1

 0 ,

1

 1 ,

2

 0 ,

2

 1 ,  0 , 

0, (0) 0, ( ) 0 y    yy y L

1 )

( , 0 )

( , 1 )

( xq xr xp

Special case

B.V.P

Sturm-Liouville Problem :

Homogeneous Boundary Value problem -> Trivial solution y=0

-> goal : find nontrivial solution y

(Eigenvalues, Eigenfunctions)

(9)

2008_Fourier Series(2)

Sturm-Liouville Problem

“Homogeneous”

Homogeneous D.E.+

Homogeneous B/C

nonzero C

C a

y B a

y

A Nonhomogeneous B/C

1

( ) 

1

 ( ) 

2

,

2

:

Regular Sturm-Liouville Problem p , q , r , r

real-valued functions

continuous on an interval

are not both zero

] , [ b a

1 1

, B A Solve : [ r ( x ) y  ]  [ q ( x )  p ( x )] y  0

dx

d

Subject to:

0 )

( )

(

0 )

( )

(

2 2

1 1

 

 

b y B b

y A

a y B a

y A

0 ) ( , 0 )

( xp xr

for every in the interval x [ b a , ]

are not both zero

2 2

, B

A

B.V.P

(10)

2008_Fourier Series(2)

12.5 Sturm-Liouville Problem

“trivial solution is not our interest”

Homogeneous B.V.P always

possesses the trivial solution y  0

Regular Sturm-Liouville Problem p , q , r , r

real-valued functions

continuous on an interval

are not both zero

] , [ b a

1 1

, B A Solve : [ r ( x ) y  ]  [ q ( x )  p ( x )] y  0

dx

d

Subject to:

0 )

( )

(

0 )

( )

(

2 2

1 1

 

 

b y B b

y A

a y B a

y A

0 ) ( , 0 )

( xp xr

for every in the interval x [ b a , ]

are not both zero

2 2

, B

A

B.V.P

(11)

2008_Fourier Series(2)

Sturm-Liouville Problem

 By utilizing the inner product

Properties of the Regular Sturm-Liouville Problem

(a) There exist an infinite number of real eigenvalues that can be arranged in increasing

order such that as (b) For each eigenvalues there is only one eigenfunction (except for nonzero constant

multiples)

(c) Eigenfunctions corresponding to different eigenvalues are linearly independent

(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function on interval

Theorem 12.3

) (x p

  

   

n

1 2 3

n

  n

] , [ b a

Solve : [ r ( x ) y  ]  [ q ( x )  p ( x )] y  0 dx

d

Subject to:

0 )

( )

(

0 )

( )

(

2 2

1 1

 

 

b y B b

y A

a y B a

y

A

(12)

2008_Fourier Series(2)

Sturm-Liouville Problem

n b

a

p x y

m

x y

n

x dx

m

( ) ( ) ( ) 0 ,

Proof of (d)

) 1 ( 0 )]

( )

( [ ] )

(

[ r x y

m

q x

m

p x y

m

  dx

d

) 2 ( 0 )]

( )

( [ ] ) (

[ r x y

n

q x

n

p x y

n

  dx

d

: )

2 ( )

1

(  y

n

  y

m n

d [ ( )

m

]

m

d [ ( ) ] (

n m n

) ( )

n m

0

y r x y y r x y p x y y

dx dx     

(

n m

) ( )

n m n

d [ ( )

m

]

m

d [ ( ) ]

n

p x y y y r x y y r x y

dx dx

   

[ ( ) ] [ ( ) ] [ ( ) ] [ ( ) ]

n m m n m n n m

d d d d

y r x y r x y y y r x y r x y y

dx   dx dx   dx

   

n

[ ( )

m

]  

m

[ ( ) ]

n

d d

y r x y y r x y

dxdx

 

( )

n m

r x y y    r x y y ( )

n

 

m

(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]

(13)

2008_Fourier Series(2)

Sturm-Liouville Problem

n b

a

p x y

m

x y

n

x dx

m

( ) ( ) ( ) 0 ,

Integrating

   

b

a n m m n

b

a n m

m

n

y r x y dx

dx y d

x r dx y

dx d y y x

p ( ) [ ( ) ] [ ( ) ]

) (  

 

( )[ ( ) ( )] ( )[ ( ) ( )]

( )[ ( ) ( )] ( )[ ( ) ( )]

n m n m

m n m n

y b r b y b y a r a y a y b r b y b y a r a y a

 

 

 

 

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y a

y a y a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

0 )

( )

(

0 )

( )

(

2 2

1 1

 

 

b y B b

y A

a y B a

y A

Boundary Condition

) 4 ( 0 )

( )

(

) 3 ( 0 )

( )

(

1 1

1 1

 

 

a y B a

y A

a y

B a

y A

n n

m m

) 6 ( 0 )

( )

(

) 5 ( 0

) ( )

(

2 2

2 2

 

 

b y B b

y A

b y B b

y A

m n

m m

Proof of (d)

   

(

n m

) ( )

n m

d

n

[ ( )

m

] d

m

[ ( ) ]

n

p x y y y r x y y r x y

dx dx

      

(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]

(14)

2008_Fourier Series(2)

Sturm-Liouville Problem

n b

a

p x y

m

x y

n

x dx

m

( ) ( ) ( ) 0 ,

dx y

y x

b

p

a n m

m

n

)( )

(  

Boundary Condition

As are not both zero A 1 , B 1

0 )

( ) ( )

( )

( a yaya y a

y

m n m n

) 4 ( 0

) ( )

(

) 3 ( 0

) ( )

(

1 1

1 1

 

 

a y B a

y A

a y

B a

y A

n n

m

m

 

 

 

 

 

 

0 0 )

( )

(

) ( )

(

1 1

B A a

y a

y

a y

a y

n n

m m

 

 

) ( )

(

) ( )

det (

a y a

y

a y

a y

n n

m m

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y

a y a y

a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

Proof of (d)

(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]

(15)

2008_Fourier Series(2)

Sturm-Liouville Problem

n b

a

p x y

m

x y

n

x dx

m

( ) ( ) ( ) 0 ,

dx y

y x

b

p

a n m

m

n

)( )

(  

Boundary Condition

As are not both zero A 2 , B 2

0 )

( ) ( )

( )

( b ybyb y b

y

m n m n

 

 

 

 

 

 

 

0 0 )

( )

(

) ( )

(

2 2

B A b

y b

y

b y b

y

n n

m m

 

 

 ) ( )

(

) ( )

det (

b y b

y

b y b

y

n n

m m

) 6 ( 0

) ( )

(

) 5 ( 0

) ( )

(

2 2

2 2

 

 

b y B b

y A

b y B b

y A

n n

m m

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y

a y a y

a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

Proof of (d)

(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]

(16)

2008_Fourier Series(2)

Sturm-Liouville Problem

n b

a

p x y

m

x y

n

x dx

m

( ) ( ) ( ) 0 ,

dx y

y x

b

p

a n m

m

n

)( )

(  

From Boundary Condition:

0 )

( ) ( )

( )

( b ybyb y b

y

m n m n

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y

a y a y

a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

0 )

( ) ( )

( )

( a yaya y a

y

m n m n

zero zero

0 )

( )

(  

 

n

m

ab

p x y

n

y

m

dx

m n

b

a

p x y

n

y

m

dx    

( ) 0 ,

Proof of (d)

(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]

(17)

2008_Fourier Series(2)

Sturm-Liouville Problem

n b

a

p x y

m

x y

n

x dx

m

( ) ( ) ( ) 0 ,

dx y

y x

b

p

a n m

m

n

)( )

(  

From Boundary Condition:

0 )

( ) ( )

( )

( b ybyb y b

y

m n m n

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y

a y a y

a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

0 )

( ) ( )

( )

( a yaya y a

y

m n m n

zero zero

0 )

( )

(  

 

n

m

ab

p x y

n

y

m

dx

m n

b

a

p x y

n

y

m

dx    

( ) 0 ,

Orthogonal relation Proof of (d)

(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on interval[ ba, ]

(18)

2008_Fourier Series(2)

Sturm-Liouville Problem

 Example 2

A Regular Sturm-Liouville Problem

Slove the boundary-value problem

. 0 ) 1 ( )

1 ( , 0 ) 0 ( ,

0    

  y y y y

y

, 0 ,

0

0  

2

 

   

and where

 0 y solution trivial

the

, 0 ,

2

 0 

  

x c

x c

y is

y y

of solution general

the

 sin

cos

0

2 1

2

 

2

sin

2

cos

2

(sin cos ) 0 c   c    c     

2

 0 c

The second boundary condition is satisfied if

0 ) 1 ( )

1

(  y   y

Choosing , we see that the

last equation is equivalent to tan    

The eigenvalues of problem are

then , where , , are the consecutive positive roots of

2 n

n

  

n

n  1 , 2 , 3 , 

,  ,

,

2 3

1

 

 

   tan

1 2

(0) 0 sin

y     c y cx

y

tan 

y   

(19)

2008_Fourier Series(2)

Sturm-Liouville Problem

 Example 2

A Regular Sturm-Liouville Problem

Slove the boundary-value problem

. 0 ) 1 ( )

1 ( , 0 ) 0 ( ,

0    

  y y y y

y

, 0 ,

0

0  

2

 

   

and where

 0 y solution trivial

the

, 0 ,

2

 0 

  

x c

x c

y is

y y

of solution general

the

 sin

cos

0

2 1

2

 

2

sin

2

cos

2

(sin cos ) 0 c   c    c     

2

 0 c

The second boundary condition is satisfied if

0 ) 1 ( )

1

(  y   y

Choosing , we see that the

last equation is equivalent to tan    

and the corresponding solutions are

1

2 . 0288 ,

2

4 . 9132 ,

3

7 . 9787 ,

4

11 . 0855 ,

, 9787 .

7 sin ,

9132 .

4 sin ,

0288 .

2

sin

2 3

1

x y x y x

y   

x y

4

 sin 11 . 0855

In general, the eigenfunctions of the problem are {sin

n

}, n 1 , 2 , 3 ,

1 2

(0) 0 sin

y     c y cx

y

tan 

y   

(20)

2008_Fourier Series(2)

Sturm-Liouville Problem

 Example 2

A Regular Sturm-Liouville Problem

Slove the boundary-value problem

. 0 ) 1 ( )

1 ( , 0 ) 0 ( ,

0    

  y y y y

y

1 1 2 2

( ) 1, ( ) 0, ( ) 1

1, 0, 1, 1

r x q x p x

A B A B

  

   

Regular Sturm-Liouville Problem

Solve :

[ r ( x ) y  ]  [ q ( x )  p ( x )] y  0 dx

d

Subject to:

0 ) ( )

(

0 ) ( )

(

2 2

1 1

 

 

b y B b y A

a y B a y A

B.V.P

m n

b

a

p x y

n

y

m

dx    

( ) 0 ,

Orthogonal relation Regular Sturm-Liouville Problem

In general, the eigenfunctions of the problem are {sin 

n

}, n  1 , 2 , 3 ,

,  3 , 2 , 1 },

{sin 

n

n

is an orthogonal set with respect to the weight function

on the interval [0,1].

( ) 1

p x

(21)

2008_Fourier Series(2)

Sturm-Liouville Problem

dx y

y x

b

p

a n m

m

n

)( )

(  

should be zero for Orthogonal relation

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y

a y a y

a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

m n

b

a

p x y

n

y

m

dx    

( ) 0 ,

Orthogonal relation

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d( ( ) ) ( ( ) ) 0 0

2 2

1 1

 

 

b y B b

y A

a y B a

y A

Boundary Condition:

 Regular Sturm-Liouville Problem

In some circumstances, we can prove the orthogonality of the solutions of without the necessity of specifying a

boundary condition at x=a and at x=b

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d

 Singular Sturm-Liouville Problem

]

,

[ b a

(22)

2008_Fourier Series(2)

Sturm-Liouville Problem

dx y

y x

b

p

a n m

m

n

)( )

(  

should be zero for Orthogonal relation

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y

a y a y

a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

m n

b

a

p x y

n

y

m

dx    

( ) 0 ,

Orthogonal relation

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d

If then may be a singular and the equation

may become unbounded as

however

0 )

( ar

0 )

( )

(

0 )

( )

(

2 2

1 1

 

 

b y B b

y A

a y B a

y A

)]

( ) ( )

( ) ( )[

( )]

( ) ( )

( ) ( )[

( b y b y b y b y b r a y a y a y a y a

r

m

n

m n

 

m

n

m n

dropped from the problem : no boundary condition at

Singular Sturm-Liouville Problem

Orthogonal relation hold on

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d

a x

a x

] ,

[ b a xa

] , (

, a b

Boundary Condition:

 Regular Sturm-Liouville Problem

zero

]

,

[ b a

(23)

2008_Fourier Series(2)

Sturm-Liouville Problem

If then may be a singular and the equation

may become unbounded as

however

0 )

( br

Orthogonal relation hold on

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d

b x

b x

] , [ b a

)]

( ) ( )

( ) ( )[

( )]

( ) ( )

( ) ( )[

( b y b y b y b y b r a y a y a y a y a

r

zerom

n

m n

 

m

n

m n

dropped from the problem

: no boundary condition at

xb

) , [

, a b

Boundary Condition:

dx y

y x

b

p

a n m

m

n

)( )

(  

should be zero for Orthogonal relation

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y

a y a y

a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

m n

b

a

p x y

n

y

m

dx    

( ) 0 ,

Orthogonal relation

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d( ( ) ) ( ( ) ) 0 0

2 2

1 1

 

 

b y B b

y A

a y B a

y A

Singular Sturm-Liouville Problem

 Regular Sturm-Liouville Problem

]

,

[ b a

(24)

2008_Fourier Series(2)

Sturm-Liouville Problem

m n

b

a

p x y

n

y

m

dx    

( ) 0 ,

Orthogonal relation

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d

example*)

Singular Sturm-Liouville Problem

Legendre‟s equation is a Sturm-Liouville equation

( 1 x

2

) y y 0 ( 1 x

2

) y  2 x y y 0

) (x r 0

) 1 (   r

Since need no boundary conditions, but have a singular Sturm-Liouville problem on the interval . We know that , the Legendre polynomials are solutions of the problem for

Hence these are the eigenfunctions. They are orthogonal on the interval

1 1  

x

) 1 ( 

n n

) (x P

n

,...)

3 2 , 2 1 , 0 ,...(

2 , 1 ,

0   

 

n

0 )

1 ( 2

) 1

(  x

2

y   x y   n nyLegendre‟s equation

) (

, 0 )

( )

1

(

1

p

m

x p

n

x dxmn

]

,

[ b a

(25)

2008_Fourier Series(2)

Sturm-Liouville Problem

If then

Periodic r ( a )  Sturm-Liouville Problem r ( b )

∴Orthogonal relation hold on with

[ b a , ]

Boundary Condition:

( ( ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( ))

) (

)]

( ) ( )

( ) ( )[

( )]

( ) ( )

( ) ( )[

(

b y b y a

y a y

a y a y

b y b y a

r

a y a y

a y a y

a r b

y b y b

y b y b r

n m

n m

n m

n m

n m

n m

n m

n m

 

 

 

 

 

 

 

) ( )

( ),

( )

( a y b y a y b

y    

dx y

y x

b

p

a n m

m

n

)( )

(  

should be zero for Orthogonal relation

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y

a y a y

a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

m n

b

a

p x y

n

y

m

dx    

( ) 0 ,

Orthogonal relation

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d( ( ) ) ( ( ) ) 0 0

2 2

1 1

 

 

b y B b

y A

a y B a

y

 Regular Sturm-Liouville Problem A

]

,

[ b a

(26)

2008_Fourier Series(2)

Sturm-Liouville Problem

with

with boundary Condition:

( ) ( ), ( ) ( ) y ay b y a   y b

dx y y x

b

p

a n m

m

n

)( )

(  

)]

( ) ( )

( ) ( )[

(

)]

( ) ( )

( ) ( )[

(

a y a y a

y a y a r

b y b y b

y b y b r

n m

n m

n m

n m

 

 

 

 

m n

b

a

p x y

n

y

m

dx    

( ) 0 ,

Orthogonal relation hold on.. [a,b]

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d

0 ) ( )

(

0 ) ( )

(

2 2

1 1

 

 

b y B b

y A

a y B a

y A

Sturm-Liouville Problem

• Regular

• Singular

• Periodic

0 )

( ar

0 )

( br

] , [ b a

) ( )

( a r b r

( ) 0 r x

] , [ b a

without B/C at x=a without B/C at x=b

2

( )

2

( ) 0

A y bB y b  

1

( )

1

( ) 0

A y aB y a  

By assuming the solution (y) are bounded on the closed interval [a,b] , then

(27)

2008_Fourier Series(2)

Sturm-Liouville Problem

Self-Adjoint Form

If the coefficient are continuous and for all in some interval, then any second-order differential equation

can be recast into the so-called „self-adjoint form‟.

0 )

( xa

[ ( ) ] [ ( ) ( ) ] 0 d r x y q x p x y

dx     

0 ))

( )

( ( )

( )

( x y   b x y   c xd x y

a

x

Recall, ch. 2.3 integrating factor

1

( )

0

( ) 0

a x y  a x y  [ y ]  0 dx

d

) (

) ) (

( ,

1 ) 0

(

x a

x x a

p

e

p x dx

(28)

2008_Fourier Series(2)

Sturm-Liouville Problem

Self-Adjoint Form

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d

0 ))

( )

( ( )

( )

( x y   b x y   c xd x y

a

dx

x a

x b

e

( )

) (

) 0 (

) ( )

( ) ( )

( )

( 

 

  

 

  y

x a

x d x

a x y c

x a

x

y b

) 0 (

) ( )

( ) ( )

( )

(

(( )) (( )) (( ))

) (

) (

 

 

 

 

 

y

x a

x e d

x a

x e c

y x e

a x y b

e

x dx a

x dx b

x a

x dx b

x a

x dx b

x a

x b

) 0 (

) ( )

( )

(

(( )) (( ))

) (

) (

 

 

 

 

 

 

e

y

x a

x e d

x a

x y c

dt e

d

ab xx dx ab xx dx

ba xx dx

multiply

) (x

r q (x ) p x ( )

( ) a x

divided by

(29)

2008_Fourier Series(2)

Sturm-Liouville Problem

Self-Adjoint Form

0 )]

( )

( [ ] ) (

[ r x y   q xp x ydx

d

0 ))

( )

( ( )

( )

( x y   b x y   c xd x y

a

a x dx x b

x e a

x x d

p

( )

) (

) (

) ) (

(

a x dx x b

x e a

x x c

q

( )

) (

) (

) ) (

(

a x dx x b

e x

r

( )

) (

)

Example)

3 y   6 y    y  0 (

x dx dx

x a

x b

e e

e

( ) 2 2

) (

3 0 3

6   

  y y

y

3 0

2

2 2

2

y   e y   e y

e

x x

x

2

2

3 0

x

d

x

e y e

dx     y

(30)

2008_Fourier Series(2)

Sturm-Liouville Problem

1

ln

0

dx x

e

x

e x

x

 

General solution

yc

1

J

n

(  x )  c

2

Y

n

(  x )

Ex.)Parametric Bessel Series*

,...

2 , 1 , 0 ,

0 )

(

2 2 2

2

y   x y   xn yn

x

converges on when converges on

) (x

J

n

[ 0 ,  ) )

(x

Y

n

( 0 ,  )

 0 n

0 )

1 (

2 2

2

 

 

  y

x y n

y x

multiply

0 )

(

2

2

 

 

  y

x x n

y y

x

)  (x

r q (x ) p (x )

  (

2

2

x ) y 0

x y n

dx x

d

Self-Adjoint Form

x

2

divided by

m n b

a

p x y

n

y

m

dx    

( ) 0 ,

0 )]

( ) ( [ ] ) (

[r x y  q xp x ydx

d

]

,

[ b a

참조

관련 문서

Department of Naval Architecture and Ocean Engineering, Seoul National University of College of Engineering.. 학부 4학년 교과목“창의적

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Department of Naval Architecture and Ocean Engineering, Seoul National University... 2009

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School of Mechanical and Aerospace Engineering Seoul National University..