Page 186 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 10: Green’s Function Method

Notes
1
figure) and apply the result (4) to the region k bounded by the curves C and with log   .
r r 1
Since both and are harmonic, it follows that if S is measured in the direction shown in the
fig.,

1 1
x ,y log   log   = 0 ...(5)
n r r 1 r r 1 n
C
we can show that
1
y
x
log   ds 2 ( , ) 0( )
n r r 1
and that

1
log   ds 2 M log ,
r r 1 n

where M is an upper bound of . Inserting these results into equation (5), we find that
r

1 1 x ,y 1
(x, y) = log   x ,y log   ds ...(6)
2 r r 1 n n r r 1
C
we now introduce a Green’s function G(x, y, x , y ) defined by the equations

1
G(x, y, x , y ) = W ( , , , ) log   ...(7)
y
x
x
y
r r 1
where the function W(x, y, x , y ) satisfies the relations
2 2
W ( , , , ) = 0 ...(8)
x
y
y
x
2
2
x y
 
W(x, y, x , y ) = log r r 1 onC ...(9)
then for satisfying equations
2 = 0 within ,
and = f(x, y) on C ...(10)
is given by the expression

1 G
x
y
(x, y) = x ,y G x , , ,y ds ...(11)
2 n

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