Page 190 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 11: Sturm–Liouville’s Boundary Value Problems

solutions that can be used as the basis for series expansion of the solution of the physical Notes
problem in question, namely the Fourier-Legendre’s and Fourier-Bessel series. In this unit we
will see that Legendre’s, Bessel’s, Hermite and Laguerre’s equations are examples of Sturm-
Liouville’s equations which are also in self-adjoint form. Some of the properties of Sturm-
Liouville’s equations are examined in the previous unit also. In this unit we deduce some more
properties of such equations independent of the function form of the coefficients.
Sturm-Liouville equations are of the form
(p(x)y (x)) + q(x)y(x) = r(x)y(x) ...(1)
which can be written more concisely as
Sy(x, ) = r(x)y(x, ) ...(2)
where the differential operator S is defined as
d df
f
x
x
Sf p ( ) q ( ) . ...(3)
dx dx
This is a slightly more general equation. In (1) the number is the eigenvalue, whose possible
values, which may be complex, are critically dependent upon the given boundary conditions. It is
often more important to know the properties of than it is to construct the actual solutions of (1).
We seek to solve the Sturm-Liouville equation (1) on an open interval, (a, b) of the real line. We will
also make some assumptions about the behaviour of the coefficients of (1) for x (a, b), namely that
(i) p(x), q(x) and r(z) are real-valued and continuous
(ii) p(x) is differentiable, ...(4)
(iii) p(x) > 0 and r(z) > 0.
Some Example of Sturm-Liouville Equations
Perhaps the simplest example of a Sturm-Liouville equation is Fourier’s equations,
y (x, ) = y(x, ) ...(5)
)
which has solutions cos(x  and sin(x  ). We discussed a physical problem that leads naturally
to Fourier’s equation at the start of least unit.
We can write Legendre’s equation and Bessel’s equation as Sturm-Liouville problems. Recall
that Legendre’s equation is

2
d y 2x dy  y 0
dx 2 1 x 2 dx 1 x 2
and we are usually interested in solving this for 1 < x < 1. This can be written as
2
[(1 x )y ] = y.
If = n(n 1), we showed in unit 2 that this has solutions P (x) and Q (x). Similarly, Bessel’s
n n
equation, which is usually solved for 0 < x < a, is
2
2
2
x y + xy + ( x ) = 0.
This can be rearranged into the form
2
(xy ) y  xy .
x
)
Again, from the results of unit 1, we know that this has solutions of the form J v (x  and
)
Y v (x  .
Although the Sturm-Liouville forms of these equations may look more cumbersome than the
original forms, we will see that they are very convenient for the analysis that follows. This is
because of the self-adjoint nature of the differential operator.

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