Now that you know what a Sturm-Liouville problem is, I can give you some idea of the problems I've been working on. My thesis is a survey of Sturm-Liouville problems where the boundary condition depends on the eigenparameter. Taking:
-(p(x) y'(x))' + q(x) y(x) = k r(x) y(x)
a y(0) + b y'(0) = 0
c y(1) + d y'(1) = 0
as the differential equation plus boundary conditions, we may rewrite the second condition as the ratio of the derivative to the function evaluated at the right end-point, as:
y'(1)/y(1) = -c/d
This ratio is a constant, but we may alter the condition at x = 1, for example, to be a linear function of the eigenparameter k:
y'(1)/y(1) = Ck + D
This introduces a new wrinkle in terms of the analysis of the problem, but results are still available with appropriate modifications.
These types of problems are also physically motivated. A regular Sturm-Liouville problem occurs if one looks at heat transfer in a metal bar. If we stick two bars made of two different types of metal together so that heat can flow from one to the other, then we have two Sturm-Liouville problems, one in each bar. The boundary condition at the interface of the two bars is now a function of the eigenparameter, which describes the influence they have on one another.
Posted by warcode at December 04, 2002 10:25 AMNow that you know what a Louisville-Sluggo problem is, I can give you some idea of the problems I've been working on. My thesis is a survey of Louisville-Sluggo problems where the bar-star population depends on the eaglebarometer. Taking:
r(e^S)-(P*e^c) = t
m'(A) = h
A to the U+T = H
(O-R)'i T(AY)
as the indifferent racial equation plus bar-star population, we may rewrite the second condition as the number of top40 radio hits played at The Derivative (members only):
d'(O) = g(L)/o(R)-y
This radio is a loud constant, but we may alter the use of foundation, for example, to be a linear function of the eaglebarometer k:
c'(h-e^a)P == b(e^e^r)
This introduces a new wrinkle on some cougars, but results are still available with appropriate modifications.
These types of problems are also physically motivated. A regular Louisville-Sluggo problem occurs if one looks at heat transfer in a meatmarket bar. If we stick two bars together so that meat can flow from one to the other, then we have two Louisville-Sluggo problems, one at each bar. The bar-star population at the interface of the two bars is now a function of the eaglebarometer, which describes the influence of the Louisville-Sluggo on various rowdy bar-stars.
Posted by: r. on December 4, 2002 04:29 PMi eagerly await the results of your research.
Posted by: ben on December 4, 2002 04:51 PMA lot of my support money has gone into the field testing of this proposal. We should compare notes.
Posted by: warcode on December 4, 2002 07:10 PM