Page 167 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
y (Ly ) y (Ly ) y (py ) qy y (py ) qy
1 2 2 1 1 2 2 2 1 1
= y 1 (py 2 ) y 2 (py 1 ) y 1 py 2 p y 2 y 2 py 1 p y 1

p
= p (y y y y ) p (y y y y ) [ (y y y y )]
1 2 1 2 1 2 1 2 1 2 1 2
Now recall that the space C[a, b] is a real inner product space with a standard inner product
defined by

b
, f g f ( ) ( )dx
x
g
x
a
If we now integrate (8) over [a, b] then
y 1 ,Ly 2 Ly 1 ,y 2 = [ (p y y y y )] b a ...(9)
1 2
1 2
This result can be used to motivate the following definitions. The adjoint operator of T, written
, T satisfies y 1 ,Ty 2 Ty 1 ,y 2 for all y and y . For example, let us see if we can construct the
1
2
adjoint to the operator
d 2 d
 2 ,
dx dx
with , R, on the interval [0, 1], when the functions on which  operates are zero at x = 0 and
x = 1. After integrating by parts and applying these boundary conditions, we find that

1 1 1 1 1 1
, ( )dx dx dx dx
1 2 1 2 2 2 1 2 0 1 2 1 2 0 1 2 1 2
0 0 0 0
1 1 1 1
= 1 2 1 2 dx 1 2 dx 1 2 ( 1 , 2 ),
0 0 0 0
where

d d
D 2
dx 2 dx
,
A linear operator is said to be Hermitian, or self-adjoint. If y 1 ,Ty 2 = Ty y 2 for all y and y .
1
1
2
It is clear from (9) that L is a Hermitian, or self-adjoint, operator if and only if
b
( p y y y y 0
1 2
1 2
a
and hence
a
y
a
y
a
a
a
y
b
b
y
y
b
y
b
p ( ){ ( ) ( ) y 1 ( ) ( )} p ( ){ ( ) ( ) y 1 ( ) ( )} 0 ...(10)
b
2
1
1
2
2
2
In other words, whether or not L is Hermitian depends only upon the boundary values of the
functions in the space upon which it operates.
There are three different ways in which (10) can occur.
(i) p(a) = p(b) = 0. Note that this doesn’t violate our definition of p as strictly non-zero on the
open interval (a, b). This is the case of singular boundary conditions.
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