Page 182 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 10: Green’s Function Method

10.4 Green’s Function for Two Independent Variables Notes

Let us assume that a function z of x and y satisfies the differential equation
L(z) = f(x, y) ...(1)
Where L denotes the linear operator

2
a b c ...(2)
x y x y
Now let w be another function with continuous derivatives of the first order. We may write

2 z 2 w z w
w z w z
x y x y y x x y

z (aw )
wa z (awz )
x x x
z (aw )
wb z (bwz )
y y y
Defining the M operator by the relation

2 w (aw ) (bw )
Mw = cw ...(3)
x y x y
we find that

2 z z z
wLz z Mw = w a b cz
x y x y

2 w (aw ) (bw )
z cw
x y x y

w z
= (awz ) z (bwz ) w
x x y y y x

or
u v
wLz zMw = ...(4)
x y

w z
where u = awz z , v = bwz + w ...(5)
y x
The operator M defined by equation (3) is called the adjoint operator. If M = L, we say the operator
L is self-adjoint.
Now if is a closed curve enclosing an area , then it follows from equation (4) and a straight
forward use of Green’s theorem that

u
wLz zLw dxdy = dxdy
x y

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