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Differential and Integral Equation
Notes 11.2 Boundary Conditions
We begin with a couple of definitions. The endpoint, x = a, of the interval (a, b) is a regular
endpoint if a is finite and the conditions (4) hold on the closed interval [a, c] for each c (a, b). The
endpoint x = a is a singular endpoint if a = or if a is finite but the conditions (4) do not hold on
the closed interval [a, c] for some c (a, b). Similar definitions hold for the other endpoint, x = b.
For example, Fourier’s equation has regular endpoints if a and b are finite. Legendre’s equation
has regular endpoints if 1 < a < b > 1, but singular endpoints if a = 1 or b =1, since p(x) = 1 x 2
= 0 when x = 1. Bessel’s equation has regular endpoints for 0 < a < b < , but singular endpoints
2
if a = 0 or b = , since q(x) = v /x is unbounded at x = 0.
We can now define the types of boundary conditions that can be applied to a Sturm-Liouville
equation.
(i) On a finite interval, [a, b], with regular endpoints, we prescribe unmixed, or separated,
boundary conditions, of the form
y(a, ) + y (a, ) = 0, y(b, ) + y (v, ) = 0. ...(6)
0 1 0 1
These boundary conditions are said to be real if the constants , , and are real,
0 1 0 1
with and .
(ii) On an interval with one or two singular endpoints, the boundary conditions that arise in
models of physical problems are usually boundedness conditions. In many problems,
these are equivalent to Friedrich’s boundary conditions, that for some c (a, b) there exists
+
A such that
|y(x, )| A for all x (a, c)
+
and similarly if the other endpoint, x = b, is singular there exists B such that y(x, )
B for all x (a, b)
We can now define the Sturm-Liouville boundary value problem to be the Sturm-Liouville
equation,
(p(x)y (x)) + q(x)y(x) = r(x)y(x) for x (a, b)
where the coefficient functions satisfy the conditions (4), to be solved subject to a separated
boundary condition at each regular endpoint and a Friedrich’s boundary condition at each
singular endpoint. Note that this boundary value problem is homogeneous and therefore always
has the trivial solution, y = 0. A non-trivial solution, y(x, ) 0, is an eigenfunction, and is the
corresponding eigenvalue.
Some Examples of Sturm-Liouville Boundary Value Problems.
Consider Fourier’s equation.
y (x, ) = x(x, ) for x 0, 1)
subject to the boundary conditions y(0, ) = y(1, ) = 0, which are appropriate since both endpoints
are regular. The eigenfunctions of this system are sin n for x = 1, 2,...., with corresponding
x
2
2
eigenvalues = = n .
Legendre’s equation is
2
{(1 x )y (x, )} = y(x, ) for x ( 1, 1).
Note that this is singular at both endpoints, since p( 1) = 0. We therefore apply Friedrich’s
boundary conditions, for example with c = 0, in the form
|y(x, )| A for x ( 1, 0), |y(x, )| B for x (0, 1),
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