{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/66450"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/66450","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Difference Methods for Stiff Delay Differential Equations","abstract":"Delay differential equations of the form y'(t) = f(y(t), z(t)), where z(t) = {y(,1)((alpha)(,1)(y(t))),..., y(,n)((alpha)(,n)(y(t)))}('T) and (alpha)(,i)(y(t)) (LESSTHEQ) t arise in many scientific and engineering fields when transport lags and propagation times are physically significant in a dynamic process. Difference methods for approximating the solutions of stiff delay systems require special stability properties that are generalizations of those employed for stiff ordinary differential equations. Using the model equation (y'(t) = py(t) + qy(t-1), with complex p and q, the definitions of A-stability, (A((alpha))-stability, and stiff stability have been generalized to delay equations. For linear multistep difference formulas, these properties extend directly from ordinary to delay equations. This is not true for implicit Runge-Kutta methods, as illustrated by the mid-point formula, which is A-stable for ordinary equations, but not for delay equations.","abstract_html":"Delay differential equations of the form y'(t) = f(y(t), z(t)), where z(t) = {y(,1)((alpha)(,1)(y(t))),..., y(,n)((alpha)(,n)(y(t)))}('T) and (alpha)(,i)(y(t)) (LESSTHEQ) t arise in many scientific and engineering fields when transport lags and propagation times are physically significant in a dynamic process. Difference methods for approximating the solutions of stiff delay systems require special stability properties that are generalizations of those employed for stiff ordinary differential equations. Using the model equation (y'(t) = py(t) + qy(t-1), with complex p and q, the definitions of A-stability, (A((alpha))-stability, and stiff stability have been generalized to delay equations. For linear multistep difference formulas, these properties extend directly from ordinary to delay equations. This is not true for implicit Runge-Kutta methods, as illustrated by the mid-point formula, which is A-stable for ordinary equations, but not for delay equations.","abstract_has_math":false,"creators":["Roth, Mitchell Godfrey"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-13T18:02:21Z","date_published":"2014-12-13T18:02:21Z","updated_at":"2026-07-22T22:25:56Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8114471"],"render_values":[{"text":"(UMI)AAI8114471","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/66450","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Roth, Mitchell Godfrey"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-13T18:02:21Z","10000-01-01","1981"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/66450","(UMI)AAI8114471"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Delay differential equations of the form y'(t) = f(y(t), z(t)), where z(t) = {y(,1)((alpha)(,1)(y(t))),..., y(,n)((alpha)(,n)(y(t)))}('T) and (alpha)(,i)(y(t)) (LESSTHEQ) t arise in many scientific and engineering fields when transport lags and propagation times are physically significant in a dynamic process. Difference methods for approximating the solutions of stiff delay systems require special stability properties that are generalizations of those employed for stiff ordinary differential equations. Using the model equation (y'(t) = py(t) + qy(t-1), with complex p and q, the definitions of A-stability, (A((alpha))-stability, and stiff stability have been generalized to delay equations. For linear multistep difference formulas, these properties extend directly from ordinary to delay equations. This is not true for implicit Runge-Kutta methods, as illustrated by the mid-point formula, which is A-stable for ordinary equations, but not for delay equations.","A computer code for stiff delay equations was developed using the stiffly stable backward differentiation formulae (BDF). In a three-way comparison with non-stiff Adam's and Runge-Kutta methods, the BDF method required significantly less computation for stiff problems and only slightly more computation for non-stiff problems.","Made available in DSpace on 2014-12-13T18:02:21Z (GMT). No. of bitstreams: 1 8114471.pdf: 3838652 bytes, checksum: 9da590d77361d271f219672807648195 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 66628 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","142 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."]},{"key":"dc:title","label":"Title","values":["Difference Methods for Stiff Delay Differential Equations"]}]}],"canonical_facts":{"dc:creator":["Roth, Mitchell Godfrey"],"dc:date":["2014-12-13T18:02:21Z","10000-01-01","1981"],"dc:description":["Delay differential equations of the form y'(t) = f(y(t), z(t)), where z(t) = {y(,1)((alpha)(,1)(y(t))),..., y(,n)((alpha)(,n)(y(t)))}('T) and (alpha)(,i)(y(t)) (LESSTHEQ) t arise in many scientific and engineering fields when transport lags and propagation times are physically significant in a dynamic process. Difference methods for approximating the solutions of stiff delay systems require special stability properties that are generalizations of those employed for stiff ordinary differential equations. Using the model equation (y'(t) = py(t) + qy(t-1), with complex p and q, the definitions of A-stability, (A((alpha))-stability, and stiff stability have been generalized to delay equations. For linear multistep difference formulas, these properties extend directly from ordinary to delay equations. This is not true for implicit Runge-Kutta methods, as illustrated by the mid-point formula, which is A-stable for ordinary equations, but not for delay equations.","A computer code for stiff delay equations was developed using the stiffly stable backward differentiation formulae (BDF). In a three-way comparison with non-stiff Adam's and Runge-Kutta methods, the BDF method required significantly less computation for stiff problems and only slightly more computation for non-stiff problems.","Made available in DSpace on 2014-12-13T18:02:21Z (GMT). No. of bitstreams: 1 8114471.pdf: 3838652 bytes, checksum: 9da590d77361d271f219672807648195 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 66628 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","142 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."],"dc:identifier":["http://hdl.handle.net/2142/66450","(UMI)AAI8114471"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Difference Methods for Stiff Delay Differential Equations"],"dc:type":["text"],"thesis:degree_discipline":["Computer Science"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:56Z"}