MATSLISE: (Regular) Sturm-Liouville equation | ![]() ![]() |
The (regular) Sturm-Liouville equation
The regular Sturm-Liouville problems (SLP) read
( -p(x)y')'+ q(x)y = Ew(x)y,
where a
and b
are finite, functions
p
, q
, and w
are defined on the closed interval
[a,b]
with p
and w
strictly positive.
It is also assumed that q
is continuous and that p
and w
can be twice differentiated on the specified interval. The boundary
conditions are:
A0*y(a) + B0*p(a)y'(a) = 0
A1*y(b) + B1*p(b)y'(b) = 0
where the constants A0
and B0
are not both zero and similarly
for A1
and B1
.
Sturm-Liouville problems model phenomena such as the earth's seismic behaviour, the propagation of sonar in water stratified by varying density, the stability and velocity of large-scale waves in the atmosphere,...
Solving the Sturm-Liouville equation with MATSLISE
Figure 3 shows the problem-specification window for a Sturm-Liouville problem (paine_slp.mat
).
This window contains the following inputfields:
p(x)
: a stricly positive function that can be twice differentiated
on the closed interval [a,b]
. q(x)
: a continuous function defined on the closed interval [a,b]
.w(x)
: a stricly positive function that can be twice differentiated
on the closed interval [a,b]
. a,b
: the integration interval (real values)p
, q
or w
or in the boundary conditions ( in a, b, A0, A1, B0
or B1
).
He must enter the name of this parameter in the "parameter name(s)"- field and a value for the parameter in the
"parameter value(s)"-field. The parameter can then be used in the expressions entered in the other fields.
When the SLP has to be solved for several different values for the parameter, a vector has to be entered in
the "parameter value"-field e.g. [1,2,3,4] or 1:4).
It is possible to define more than one parameter, but it is not allowed to use parameters
in the specification of the values of the parameters. When multiple parameters are used,
then the names of the parameters are entered in the "parameter name(s)"-field and separated by a
comma. The corresponding values for the parameters are entered in the parameter value(s)-field,
also separated by a comma.tol
: a positive double precision constant representing the accuracy requested by the
user.The user is allowed to enter -inf
or inf
in the integration-interval-fields: a
and b
.
The integration interval is then chosen by the program and will be wide enough to provide
the eigenvalues within the desired tolerance.
Solving SLP means calculating the eigenvalues E
and the associated eigenfunctions y
. This can be done
by clicking the construct-button.
![]() | Regular Schrödinger: The File Menu | The Construct-button |
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