5233-l12 Special Solutions of Laplace's Equation

8/18/2019 5233-l12 Special Solutions of Laplace's Equation

1/6

5 0


2/6

L E C T U R E 1 2

S p e c i a l S o l u t i o n s o f L a p l a c e ' s E q u a t i o n

1 . S e p a r a t i o n o f V a r i a b l e s w i t h R e s p e c t t o C a r t e s i a n C o o r d i n a t e s

S u p p o s e

x y = X x Y y 1 2 . 1

i s a s o l u t i o n o f

@

2

@ x

2

+

@

2

@ y

2

= 0 1 2 . 2

T h e n w e m u s t h a v e

Y

d

2

X

d x

2

+ X

d

2

Y

d y

2

= 0 1 2 . 3

A p p l y i n g t h e u s u a l s e p a r a t i o n o f v a r i a b l e s a r g u m e n t , w e n d

d

2

X

d x

2

+

2

X = 0 X x = A s i n + B c o s

d

2

Y

d y

2

,

2

Y = 0 Y y = C e

y

+ D e

y

1 2 . 4

w h e n t h e s e p a r a t i o n c o n s t a n t K i s a p o s i t i v e r e a l n u m b e r

2

0 . T h e p o s s i b i l i t y t h a t t h e K = 0 s h o u l d

n o t b e e x c l u d e d , h o w e v e r . F o r t h i s s p e c i a l c a s e w e w o u l d h a v e

d

2

X

d x

2

= 0 X x = a x + b

d

2

Y

d y

2

= 0 Y y = c y + d

1 2 . 5

N o r s h o u l d w e e x c l u d e t h e c a s e w h e n K = ,

2

0

d

2

X

d x

2

,

2

X = 0 X x = A e

x

+ B e

x

d

2

Y

d y

2

+

2

Y = 0 Y y = C s i n y + D c o s y

1 2 . 6

T h u s s e p a r a t i o n o f v a r i a b l e s y i e l d s t h r e e 4 - p a r a m e t e r f a m i l i e s o f s o l u t i o n s .

0

x y = a x y + b x + c y + d

x y = A e

y

s i n x + B e

y

s i n x + C e

y

c o s x + D e

y

c o s x

i

x y = A e

x

s i n y + B e

x

s i n y + C e

x

c o s y + D e

x

c o s y

1 2 . 7

2 . S e p a r a t i o n o f V a r i a b l e s w i t h R e s p e c t t o P o l a r C o o r d i n a t e s

I f w e m a k e a c h a n g e o f v a r i a b l e s t o p o l a r c o o r d i n a t e s

x = r c o s

y = r s i n

r =

p

x

2

+ y

2

= t a n

1

,

y

x

1 2 . 8

5 1


3/6

5 2 1 2 . S P E C I A L S O L U T I O N S O F L A P L A C E ' S E Q U A T I O N

t h e n

@

@ x

=

@ r

@ x

@

@ r

+

@

@ x

@

@

= c o s

@

@ r

,

s i n

r

@

@

@

@ y

=

@ r

@ y

@

@ r

+

@

@ y

@

@

= s i n

@

@ r

+

c o s

r

@

@

@

2

@ x

2

=

c o s

@

@ r

,

s i n

r

@

@

c o s

@

@ r

,

s i n

r

@

@

= c o s

2

@

2

@ r

2

, 2

s i n c o s

r

@

@ r

@

@

+

s n

2

r

2

@

2

@

2

+

c o s s i n

r

2

@

@

+

s n

2

r

@

@ r

+

s i n c o s

r

@

@

@

2

@ y

2

=

s i n

@

@ r

+

c o s

r

@

@

s i n

@

@ r

+

c o s

r

@

@

= s i n

2

@

2

@ r

2

+

2 c o s s i n

r

@

@ r

@

@

+

c o s

2

r

2

@

2

@

2

,

s i n c o s

r

2

@

@

+

c o s

2

r

@

@ r

,

c o s s i n

r

2

@

@

a n d

@

2

@ x

2

+

@

2

@ y

2

=

@

2

@ r

2

+

1

r

@

@ r

+

1

r

2

@

2

@

2

1 2 . 9

T h u s , i n p o l a r c o o r d i n a t e s L a p l a c e ' s e q u a t i o n t a k e s t h e f o r m

@

2

@ r

2

+

1

r

@

@ r

+

1

r

2

@

2

@

2

= 0 1 2 . 1 0

I f w e s e t

r = R r 1 2 . 1 1

a n d p l u g 1 2 . 1 1 i n t o 1 2 . 1 0 a n d t h e n d i v i d e b y R r t h e n w e o b t a i n

R

R

+

R

r R

+

r

2

= 0

o r

r

2

R

R

+ r

R

R

= ,

A p p l y i n g t h e s e p a r a t i o n o f v a r i a b l e s a r g u m e n t w e n o w l o o k f o r s o l u t i o n s o f

r

2

R + r R ,

2

R = 0

+

2

= 0

1 2 . 1 2

I f

2

6= 0 , t h e n t h e s e c o n d e q u a t i o n h a s a s s o l u t i o n s

= A c o s + B s i n 1 2 . 1 3

I n o r d e r f o r s u c h s o l u t i o n s t o b e c o n t i n u o u s a c r o s s t h e r a y = 0 i . e . s o t h a t 2 = 0 w e m u s t

d e m a n d t h a t = n 2 N = f 1 2 3 ; : : : g . F o r s u c h t h e r s t e q u a t i o n i n 1 2 . 1 2 i s a n E u l e r - t y p e e q u a t i o n

w h i c h h a s a s s o l u t i o n s

R r = A r

n

+ B r

n

1 2 . 1 4

I f

2

= 0 , t h e n 1 2 . 1 2 r e d u c e s t o

r R + R = 0

= 0

1 2 . 1 5

T h e g e n e r a l s o l u t i o n o f t h e s e c o n d e q u a t i o n i s o b v i o u s l y

= a + b 1 2 . 1 6

T o s o l v e t h e r s t , w e s e t W = R s o t h a t w e c a n r e d u c e i t t o t h e f o l l o w i n g r s t o r d e r O D E

,

1

r

=

W

W

=

d

d r

l n W 1 2 . 1 7


4/6

3 P O L Y N O M I A L S O L U T I O N S 5 3

I n t e g r a t i n g b o t h s i d e s o f t h i s e q u a t i o n y i e l d s

, l n r = l n W + C 1 2 . 1 8

o r

W =

C

r

1 2 . 1 9

R e p l a c i n g t h e l e f t h a n d s i d e o f 1 2 . 1 9 R a n d t h e n i n t e g r a t i n g b o t h s i d e s o f y i e l d s

R r = C l n r + D 1 2 . 2 0

W e t h u s a r r i v e a t t h e f o l l o w i n g t w o f a m i l i e s o f s o l u t i o n s

n

r = A r

n

c o s n + B r

n

s i n n + C r

n

c o s n + D r

n

s i n n 1 2 . 2 1

0

r = a l n r + b l n r + c + d 1 2 . 2 2

F r o m t h e s o l u t i o n

0

r w i t h 0 = a = c = d , w e c a n i n f e r t h e e x i s t e n c e o f a s o l u t i o n o f t h e f o r m

x y = l n x , x

o

2

+ y , y

o

2

I t c a n b e e a s i l y c h e c k e d t h a t t h i s i s t h e o n l y s o l u t i o n u p t o a m u l t i p l i c a t i v e a n d o r a d d i t i v e c o n s t a n t

o f L a p l a c e ' s e q u a t i o n t h a t d e p e n d s o n l y t h e d i s t a n c e f r o m t h e p o i n t x

o

y

o

. I t i s c a l l e d t h e f u n d a m e n t a l

s o l u t i o n o f L a p l a c e ' s e q u a t i o n .

3 . P o l y n o m i a l S o l u t i o n s

W e h a v e a l r e a d y s e e n t h a t

x y = a x y + b x + c y + d

i s a s o l u t i o n o f L a p l a c e ' s e q u a t i o n . L e t u s n o w l o o k t o s e e i f t h e r e a r e o t h e r p o l y n o m i a l s o l u t i o n s . S u p p o s e

x y = A x

2

+ B y

2

t h e n

0 =

@

2

@ x

2

+

@

2

@ y

2

= A + B B = , A

T h u s ,

x y = A

,

x

2

, y

2

i s a s o l u t i o n o f L a p l a c e ' s e q u a t i o n .

S i m i l a r l y , i f w e s e t

x y = A x

3

+ B x

2

y + C x y

2

+ D y

3

t h e n

0 =

@

2

@ x

2

+

@

2

@ y

2

= 6 A x + 2 B y + 2 C x + 6 D y

C = , 3 A

B = , 3 D

a n d s o

x y = A

,

x

3

, 2 x y

2

+ B

,

y

3

, 3 x

2

y

w i l l b e a s o l u t i o n o f L a p l a c e ' s e q u a t i o n .

N o w l e t x y b e a n a r b i t r a r y h o m o g e n e o u s p o l y n o m i a l o f d e g r e e n

x y =

n

X

i = 0

a

i

x

n i

y

i

1 2 . 2 3


5/6

5 4 1 2 . S P E C I A L S O L U T I O N S O F L A P L A C E ' S E Q U A T I O N

T h e n i f t h i s i s t o b e a s o l u t i o n o f L a p l a c e ' s e q u a t i o n w e m u s t h a v e

0 =

@

2

@ x

2

+

@

2

@ y

2

1 2 . 2 4

=

n

X

i = 0

n , i n , i , 1 a

i

x

n i 2

y

i

+

n

X

i = 0

i i , 1 a

i

x

n i

y

i 2

1 2 . 2 5

=

n 2

X

i = 0

n , i n , i , 1 a

i

x

n i 2

y

i

+

n 2

X

i = 0

i + 2 i + 1 a

i + 2

x

n 2 i

y

i

1 2 . 2 6

=

n 2

X

i = 0

n , i n , i , 1 a

i

+ i + 2 i + 1 a

i + 2

x

n 2 i

y

i

1 2 . 2 7

a n d s o 1 2 . 2 3 w i l l b e a s o l u t i o n o f L a p l a c e ' s e q u a t i o n i f t h e c o e c i e n t s a

i

s a t i s f y t h e f o l l o w i n g r e c u r s i o n

r e l a t i o n

a

i + 2

=

n , i n , i , 1

i + 1 i + 2

a

i

1 2 . 2 8

N o t e t h a t t h e s e r e c u r s i o n r e l a t i o n s i m p l y t h a t f o r e a c h n 2 N t h e r e a r e p r e c i s e l y t w o l i n e a r l y i n d e p e n d e n t

h o m o g e n e o u s p o l y n o m i a l s o l u t i o n s o f L a p l a c e ' s e q u a t i o n s i n c e 1 2 . 2 8 t e l l s t h a t a l l t h e c o e c i e n t s a

i

a r e

c o m p l e t e l y d e t e r m i n e d b y a

0

a n d a

1

4 . S e r i e s S o l u t i o n s

C o n s i d e r t h e f o l l o w i n g P D E B V P

r r

+

1

r

r

+

1

r

2

= 0

R = f

1 2 . 2 9

w h i c h i s j u s t L a p l a c e ' s e q u a t i o n i n p o l a r c o o r d i n a t e s w i t h D i r i c h l e t b o u n d a r y c o n d i t i o n s i m p o s e d o n t h e

b o u n d a r y o f t h e c i r c l e r = R

T o c o n s t r u c t a s o l u t i o n o f 1 2 . 2 9 w e s h a l l r s t e x p a n d t h e s o l u t i o n r i n a s e r i e s o f - d e p e n d e n t

e i g e n f u c t i o n s :

r =

1

2

a

0

r +

1

X

n = 1

a

n

r c o s n + b

n

r s i n n 1 2 . 3 0

W e n o t e t h a t t h e t r i g o n o m e t r i c f u n c t i o n s c o s n a n d s i n n a r e e i g e n f u n c t i o n s o f t h e S t u r m - L i o u v i l l e

p r o b l e m w i t h d i e r e n t i a l e q u a t i o n

+

2

= 0

c o m p a r e w i t h 1 2 . 1 2 a n d b o u n d a r y c o n d i t i o n s

0 = 2

I n s e r t i n g 1 2 . 3 0 i n t o 1 2 . 2 9 w e o b t a i n

0 = a

0

+

1

r

a

0

+

P

1

n = 1

, ,

a

n

+

1

r

a

n

c o s n +

,

b

n

+

1

r

b

n

s i n n

+

P

1

n = 1

,

, n

2

a

n

c o s n , n

2

b

n

s i n n

1 2 . 3 1


6/6

4 S E R I E S S O L U T I O N S 5 5

M u l t i p l y i n g b o t h s i d e s b y c o s n o r s i n n , i n t e g r a t i n g w i t h r e s p e c t t o f r o m 0 t o 2 , a n d e m p l o y i n g t h e

i d e n t i t i e s

R

2

0

c o s n d =

2 n = 0

0 n 6= 0

R

2

0

s i n n c o s m d = 0 8 n m 2 Z

R

2

0

c o s n c o s m d =

n m

R

2

0

s i n n s i n m d =

n m

1 2 . 3 2

w e o b t a i n

a

n

+

1

r

a

n

,

n

2

r

2

a

n

= 0 n = 0 1 2 3 ; : : :

b

n

+

1

r

b

n

,

n

2

r

2

b

n

= 0 n = 1 2 3 ; : : :

1 2 . 3 3

T h e s e a r e a l l E u l e r t y p e O D E s w h i c h h a v e a s t h e i r g e n e r a l s o l u t i o n

R r = A r

n

+ B r

n

n 6= 0

R r = A + B l n r n = 0

1 2 . 3 4

I n o r d e r f o r o u r s o l u t i o n t o b e r e g u l a r a t t h e o r i g i n w e m u s t e x c l u d e s o l u t i o n s p r o p o r t i o n a l t o r

n

a n d l n r

W e t h e r e f o r e t a k e

a

n

r = A

n

r

n

n = 0 1 2

b

n

r = B

n

r

n

n = 1 2 3

1 2 . 3 5

t o b e t h e a p p r o p r i a t e s o l u t i o n s o f 1 2 . 3 3 . H e n c e , 1 2 . 2 5 b e c o m e s

r =

1

2

A

0

+

1

X

n = 1

A

n

r

n

c o s n + B

n

r

n

s i n n 1 2 . 3 6

T o x t h e c o n s t a n t s A

n

B

n

w e n o w i m p o s e t h e b o u n d a r y c o n d i t i o n a t r = R

f =

1

2

A

0

+

1

X

n = 1

A

n

R

n

c o s n + B

n

R

n

s i n n 1 2 . 3 7

M u l t i p l y i n g b o t h s i d e s o f 1 2 . 3 7 b y c o s m o r s i n m a n d i n t e g r a t i n g f r o m 0 t o 2 t h e n y i e l d s

A

n

=

1

R

n

R

2

0

f c o s n d ; n = 0 1 2 3 ; : : :

B

n

=

1

R

n

R

2

0

f s i n n d ; n = 1 2 3 ; : : :

1 2 . 3 8

H o m e w o r k : P r o b l e m 4 . 5 . 4

Documents

5233-l12 Special Solutions of Laplace's Equation