Page 180 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 10: Green’s Function Method

Generalized Green’s Function Notes

Let us consider the inhomogeneous equation
L y = (x)
x
whose solution y(x) satisfies the boundary conditions. Let us assume that there exists a non-
trivial solution y (x) 0 of the equation L y(x) = 0. We can show that the function (x) must
0 x
satisfy
b
x
x
( )y 0 ( )dx 0 ...(9)
a
where y (x) also satisfying the boundary conditions. To see this we have
0
b b
x
y
x
( )y ( )dx = y ( )L ( ) y ( )L (y ) dx
x
x
0 0 x x 0
a a
b
x
= p ( )(y 0 ( ) ( ) y 0 ( ) ( ) a 0
x
x
y
x
y
x
On the other hand the solution y(x) may be written in the form
y(x) = z(x) + c y (x)
0
where z(x) is a solution of L (z) = (x), satisfying the boundary conditions. Since y (x) 0 we can
x 0
choose the constant C so that
b
y ( )y ( )dx 0 ...(10)
x
x
0
a
Now it can be proved that such a function y(x) of the boundary value problem satisfying (10) can
be written as
b
x
y(x) = G ( , ) ( )d ...(11)
a
by means of the generalized Green’s function G(x, ).
By a generalized Green’s function, we mean a such G(x, ) satisfying the following five conditions:
1. Continuity of G(x, ) at any point (x, ) in the domain a x < b. As a function of x,
G(x, ) satisfies the given boundary conditions.
2. If x , G(x, ) satisfies the equation
G(x, ) = y (x) y ( )
0 0
as a function of x. G (x, ) is bounded in the region x .
x
3. If a < x < b then as x x , x, keeping the relation x > and as x x , x keeping
0 0 0 0
the relation x < , G (x, ) tends to finite values G (x + 0, x ) and G (x 0, x ), respectively,
x x 0 0 x 0 0
and
1
G (x + 0, x ) G (x 0, x ) =
x 0 0 x 0 0 ( p x )
0
4. G(x, ) = G( , x)
b
x
y
x
5. G ( , ) ( )dx 0
0
a
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