{"id":{"repo_id":"utc","oai_identifier":"oai:scholar.utc.edu:theses-1951"},"canonical_url":"https://search.dev.ndltd.org/etd/utc/oai:scholar.utc.edu:theses-1951","repository":{"repo_id":"utc","name":"University of Tennessee - Chattanooga","base_url":"https://scholar.utc.edu/do/oai/"},"display":{"title":"Optimization for a Sturm--Liouville problem with the spectral parameter in the boundary condition","abstract":"We find an optimal mass of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. While previous work on the subject focused on a somewhat simplified model, we consider a more general S-L problem. We use the calculus of variations approach to determine a set of critical points of a corresponding mass functional, yet these critical points - which we call \\textit{predesigns} - do not necessarily themselves represent meaningful solutions. It is natural to expect a mass to be real and positive. To this end, we additionally introduce a set of solvability conditions on the S-L problem data, confirming that these critical points represent meaningful solutions we refer to as \\textit{designs}. We further present the analytic continuation of these predesigns in regards to the spectral parameter as well as a discussion of the stability of these (pre)designs. We present a code that allows us to for the given data of the S-L problem check conditions of solvability, plot the design, and calculate the value of the functional that represents the optimal mass.","abstract_html":"We find an optimal mass of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. While previous work on the subject focused on a somewhat simplified model, we consider a more general S-L problem. We use the calculus of variations approach to determine a set of critical points of a corresponding mass functional, yet these critical points - which we call \\textit{predesigns} - do not necessarily themselves represent meaningful solutions. It is natural to expect a mass to be real and positive. To this end, we additionally introduce a set of solvability conditions on the S-L problem data, confirming that these critical points represent meaningful solutions we refer to as \\textit{designs}. We further present the analytic continuation of these predesigns in regards to the spectral parameter as well as a discussion of the stability of these (pre)designs. We present a code that allows us to for the given data of the S-L problem check conditions of solvability, plot the design, and calculate the value of the functional that represents the optimal mass.","abstract_has_math":false,"creators":["Smith, Tanner"],"institution":"University of Tennessee at Chattanooga","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Belinskiy, Boris P.","Kong, Lingju; Cox, Christopher; Wang, Jin; Nichols, Roger","College of Engineering and Computer Science"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":null,"date_issued":"","date_published":null,"updated_at":"2026-07-24T05:47:06Z","subjects":["Calculus of variations","Differential equations"],"languages":["English","eng"],"rights":[],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://scholar.utc.edu/theses/778","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Belinskiy, Boris P.","Kong, Lingju; Cox, Christopher; Wang, Jin; Nichols, Roger","College of Engineering and Computer Science"]},{"key":"dc:creator","label":"Author","values":["Smith, Tanner"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-12-01T08:00:00Z"]},{"key":"dc:publisher","label":"Institution","values":["University of Tennessee at Chattanooga","Chattanooga (Tenn.)"]},{"key":"dc:relation","label":"Dc Relation","values":["Masters Theses and Doctoral Dissertations"]},{"key":"dc:type","label":"Dc Type","values":["Doctoral dissertations","Text"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Calculus of variations","Differential equations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholar.utc.edu/theses/778"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Dept. of Computational Science","Ph. D.; A dissertation submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Doctor of Philosophy."]},{"key":"dc:description.abstract","label":"Abstract","values":["We find an optimal mass of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. While previous work on the subject focused on a somewhat simplified model, we consider a more general S-L problem. We use the calculus of variations approach to determine a set of critical points of a corresponding mass functional, yet these critical points - which we call \\textit{predesigns} - do not necessarily themselves represent meaningful solutions. It is natural to expect a mass to be real and positive. To this end, we additionally introduce a set of solvability conditions on the S-L problem data, confirming that these critical points represent meaningful solutions we refer to as \\textit{designs}. We further present the analytic continuation of these predesigns in regards to the spectral parameter as well as a discussion of the stability of these (pre)designs. We present a code that allows us to for the given data of the S-L problem check conditions of solvability, plot the design, and calculate the value of the functional that represents the optimal mass."]},{"key":"dc:title","label":"Title","values":["Optimization for a Sturm--Liouville problem with the spectral parameter in the boundary condition"]}]}],"canonical_facts":{"dc:contributor":["Belinskiy, Boris P.","Kong, Lingju; Cox, Christopher; Wang, Jin; Nichols, Roger","College of Engineering and Computer Science"],"dc:creator":["Smith, Tanner"],"dc:date":["2022-12-01T08:00:00Z"],"dc:description":["Dept. of Computational Science","Ph. D.; A dissertation submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Doctor of Philosophy."],"dc:description.abstract":["We find an optimal mass of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. While previous work on the subject focused on a somewhat simplified model, we consider a more general S-L problem. We use the calculus of variations approach to determine a set of critical points of a corresponding mass functional, yet these critical points - which we call \\textit{predesigns} - do not necessarily themselves represent meaningful solutions. It is natural to expect a mass to be real and positive. To this end, we additionally introduce a set of solvability conditions on the S-L problem data, confirming that these critical points represent meaningful solutions we refer to as \\textit{designs}. We further present the analytic continuation of these predesigns in regards to the spectral parameter as well as a discussion of the stability of these (pre)designs. We present a code that allows us to for the given data of the S-L problem check conditions of solvability, plot the design, and calculate the value of the functional that represents the optimal mass."],"dc:identifier":["https://scholar.utc.edu/theses/778"],"dc:language":["English","eng"],"dc:publisher":["University of Tennessee at Chattanooga","Chattanooga (Tenn.)"],"dc:relation":["Masters Theses and Doctoral Dissertations"],"dc:rights":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Calculus of variations","Differential equations"],"dc:title":["Optimization for a Sturm--Liouville problem with the spectral parameter in the boundary condition"],"dc:type":["Doctoral dissertations","Text"]},"updated_at":"2026-07-24T05:47:06Z"}