Page 193 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes where a superscript asterisk denotes the complex conjugate. This means that the inner product
has the properties
(i) , = ,
1 2 2 1
*
(ii) a , a = a a ,
1 1 2 2 1 2 1 2
(iii) , + = , + , , + , = , + ,
1 2 3 1 2 1 3 1 2 3 1 3 2 3
2
(iv) , = | | dx 0, with equality if and only if (x) 0 in I.
I
Note that this reduces to the definition of a real inner product if and are real. If , = 0
1 2 1 2
with 0 and 0, we say that and are orthogonal.
1 2 1 2
2
Let y (x), y (x) C [a, b] be twice-differentiable complex-valued functions. By integrating by
1 2
parts, it is straightforward to show that

x
y Sy Sy y p ( ){ ( )(y ( )) y ( ) ( )} ...(7)
y
x
x
y
x
x
2 1 2 1 1 2 1 2 
which is known as Green’s formula. The inner products are defined over a sub-interval [ , ]
+
(a, b), so that we can take the limits a and b when the endpoints are singular, and the
Sturm-Liouville operator, S, is given by (3). Now if x = a is a regular endpoint and the function
y and y satisfy a separated boundary condition at a, then
1 2
*
a
a
y
a
y
a
a
p ( ){ ( )(y * 2 ( )) y 1 ( ) ( )} 0. ...(8)
1
2
If a is a finite singular endpoint and the functions y and y satisfy the Friedrich’s boundary
1 2
condition at a,
*
*
y
x
y
x
x
y
x
x
p
lim [ ( ){ ( ) ( )) y 1 ( ) ( )}] 0
1
2
2
x a ...(9)
Similar results hold at x = b.
We can now derive several results concerning the eigenvalues and eigenfunctions of a Sturm-
Liouville boundary value problem.
Theorem 1: The eigenvalues of a Sturm-Liouville boundary value problem are real.
x
x
x
y

y * ( , )Sy ( , ) Sy * ( , ), ( , )

x



*
y


x
x

[ ( ){ ( , )( ( , )) y ( , ) ( , )}] b 0

y
y
x
x
x
p
a
*
Proof: If we substitute y (x) = y(x, ) and y (x) = y (x, ) into Green’s formula over the entire
1 2
x
x
y
x
interval, [a, b], we have y *( , ), Sy ( , ) Sy *( , ), ( , )
x
b
x
x
x
x
p ( ) y ( , )( *( , ) y ( , ) *( , ) 0
x
y
y
a
making use of (8) and (9). Now, using the fact that the function y(x, ) and y*(x, ) are solutions
of (1) and its complex conjugate, we find that
b * * b 2
x
y
x
y
y
x
r ( ) ( , ) ( , )(   ) dx (  ) r ( )[ ( , )] dx 0
x
x



a a
Since r(x) > 0 and y(x, ) is nontrivial, we must have = * and hence . i.e. the eigenvalues
are real.
Theorem 2: If y(x, ) and y(x,  ) are eigenfunctions of the Sturm-Liouville boundary value
p
problem, with  ), then these eigenfunctions are orthogonal over C [a, b] with respect to the
weighing function r(x), so that
b
y
y
x
x
r ( ) ( , ) ( , ) dx 0 ...(10)
x


a
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