Boris P. Belinskiy, Tanner A. Smith
Abstract:
We find an optimal design of a structure described by a Sturm-Liouville (S-L)
problem with a spectral parameter in the boundary conditions.
Using an approach from calculus of variations, we determine a set of critical points
of a corresponding mass functional. However, these critical points - which we call
predesigns - do not necessarily themselves represent meaningful solutions:
it is of course natural to expect a mass to be real and positive. This represents
a generalization of previous work on the topic in several ways.
First, previous work considered only boundary conditions and S-L coefficients
under certain simplifying assumptions. Principally, we do not assume that one
of the coefficients vanishes as in the previous work.
Finally, we introduce a set of solvability conditions on the S-L problem data,
confirming that the corresponding critical points represent meaningful solutions
we refer to as designs. Additionally, we present a natural schematic for testing
these conditions, as well as suggesting a code and several numerical examples.
Submitted August 6, 2023. Published January 24, 2024.
Math Subject Classifications: 34L15, 74P05, 49K15, 49S05, 49R05.
Key Words: Sturm-Liouville problem; vibrating rod; calculus of variations;
optimal design; boundary conditions with spectral parameter.
DOI: 10.58997/ejde.2023.08
Show me the PDF file (440 KB), TEX file for this article.
 |
Boris P. Belinskiy University of Tennessee at Chattanooga Department of Mathematics Dept 6956, 615 McCallie Ave. Chattanooga, TN 37403-2598, USA email: boris-belinskiy@utc.edu |
|---|---|
 |
Tanner A. Smith University of Alabama at Birmingham Department of Mathematics Room 4005, 10th Ave S Birmingham, AL 35294, USA email: tsmith46@uab.edu |
Return to the EJDE web page