Page 192 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 11: Sturm–Liouville’s Boundary Value Problems

+
for some A, B  . In unit 2 we used the method of Frobenius to construct the solutions of Notes
Legendre’s equation, and we know that the only eigenfunctions bounded at both the endpoints
are the Legendre polynomials, P (x) for n = 0, 1, 2,..., with corresponding eigenvalues = =
n n
n(n + 1).
Let’s now consider Bessel’s equation with v = 1, over the interval (0, 1),

y
(xy )  xy .
x
Because of the form of q(x), x = 0 is a singular endpoint, whilst x = 1 is a regular endpoint. Suitable
boundary conditions are therefore
1
|y(x, )| A for x 0, , y(1, ) = 0
2
+
for some A  . In unit 1 we constructed the solutions of this equation using the method of
Frobenius. The solution that is bounded at x = 0 is J x , . The eigenvalues are solutions of
1

J 1 n 0,

2
which we write as =   2 2 , ...., where J ( ) = 0.
,
1
1
n
Finally, let’s examine Bessel’s equation with v = 1, but now for x (0, ). Since both endpoints
are now singular, appropriate boundary conditions are
1 1
|y(x, )| A for x 0, , |y(x, )| B for x , ,
2 2
+
for some A, B  . The eigenfunctions are again J x ,  , but now the eigenvalues lie on the
1
half-line [0, ). In other words, the eigenfunctions exist for all real, positive . The set of eigenvalues
for a Sturm-Liouville system is often called the spectrum. In the first of the Bessel function examples
above, we have a discrete spectrum, whereas for the second there is a continuous spectrum. We
will focus our attention on problems that have a discrete spectrum only.

Self Assessment

1. Put the equation
2
2 2
x y + xy + ( x 4) y = 0
in Sturm-Liouville’s form
2. Put the equation
2
d y dy
2x 2 y 0
dx 2 dx
into Sturm-Liouville’s form

11.3 Properties of the Eigenvalues and Eigenfunctions

In order to study further the properties of the eigenfunctions and eigenvalues, we begin by
defining the inner product of two complex-valued functions over an interval I to be

x
f ( ), f ( ) f * ( ) ( ) ,
f
dx
x
x
x
1 2 1 2
I
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