Page 185 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 10.5 Green’s Function for Two Dimensional Problem

The theory of the Green function for the two dimensional Laplace equation may be developed
as follow s. It is w ell know n that if P(x, y) and Q(x, y) are functions defined inside and on the
boundary C of the closed area , then

Q P
dS = (Pdx Qdy ) ...(1)
x y
C
If we put

P , Q , in equation (1) we find that
y x

2
ds ds = dx dy
x x y y y x
C
= ds ...(2)
n
C

where denotes the derivative of in the direction of the outward normal to C and we have
n
used the relation

dy dx = ...(3)
x y n

If we interchange and in (2) and subtract the two equations, we find that

2 2
ds = ds ...(4)
n n
C
Figure 10.2

Suppose that P with co-ordinates (x, y) is a point in the interior of the region S in which the
function is assumed to be harmonic. Draw a small circle with center P and small radius (see

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