Page 178 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 10: Green’s Function Method

Self Assessment Notes

1. Find the Green function for the equation

d 2
L y = 2 y = 0
x dx
with the conditions
y(0) = 0, y (1) = 0

10.3 Periodic Solutions Generalized Green’s Function

A system of important boundary conditions not included earlier is
y(a) = y(b), y (a) = y (b) ...(1)
If the coefficients p(x), g(x), r(x) are periodic functions with period b a, that is
p(x + b a) = p(x), q (x + b a) = q(x), r(x + b a) = r(x)
Then the conditions (1) are just the conditions that the solution y(x) of the equation
(p(x)y ) q(x)y + r(x)y = 0 ...(A)
is periodic with the same period b a, that is
y(x + b a) = y(x)
For in each case, y(x), y (x + b a) both satisfy the equation (A) together with the same initial
a, b
conditions
y(a) = y (a), y (a) = y (a)
a,b a,b
Hence by the uniqueness of the solutions, we must have
y(x) = y (x)
a,b
In the following we shall be concerned with more general conditions, which include the
conditions (1), of the form

p ( )
b
y(a) = y(b), p(a) y (a) = y (b) ...(2)
1
or y(a) = p(b), y (b), p(a) y (a) = y ( ) ...(3)
b
where is a non-zero constant. It is easily seen that if y(x) and z(x) both satisfy either (2) or (3),
then the relation
b
x
z
x
z
x
x
x
p ( ) ( ( ) ( ) y ( ) ( )) 0 ...(4)
y

a
holds.
10.3.1 Construction of Green’s Function
Suppose that a Green’s function exists. Then since L (G(x, )) = 0 for x , y(x, ) must be
x
represented by means of a fundamental system y (x), y (x) of the solution of L (y) = 0 as follows:
1 2 x
c y ( ) c y ( ) (a x )
x
x
2 2
1 1
G(x, ) = ...(5)
x
x
c y ( ) c y ( ) ( x ) b
3 1
4 2
LOVELY PROFESSIONAL UNIVERSITY 171