{"id":{"repo_id":"dcu","oai_identifier":"oai:doras.dcu.ie:16640"},"canonical_url":"https://search.dev.ndltd.org/etd/dcu/oai:doras.dcu.ie:16640","repository":{"repo_id":"dcu","name":"Dublin City University","base_url":"http://doras.dcu.ie/cgi/oai2"},"display":{"title":"Explosions and unbounded growth in nonlinear delay differential equations: Numerical and asymptotic analysis","abstract":"This thesis investigates the asymptotic behaviour of a scalar, nonlinear dierential equation with a fixed delay, and examines whether the properties of this equation can be replicated by an appropriate discretisation. We begin by considering equations for which the solution explodes in finite time. Existing work on such explosive equations has dealt with devising numerical schemes for equations with polynomially growing instantaneous feedback, and methods to deal with delayed feedback have not been fully explored. We therefore set out a discretised scheme which replicates all the qualitative features of the continuous-time solution for a more general class of equations. Next, for non-explosive equations which exhibit extremely rapid growth, the rate of growth of the solution depends on the comparative asymptotic nonlinearities of the coefficients of the equation and the magnitude of the delay. Thus we set out conditions on these parameters which characterise the growth rate of the solution, and investigate numerical methods for recovering this rate. Using constructive comparison principles and nonlinear asymptotic analysis, we extend the numerical methods devised for explosive equations for this purpose.","abstract_html":"This thesis investigates the asymptotic behaviour of a scalar, nonlinear dierential equation with a fixed delay, and examines whether the properties of this equation can be replicated by an appropriate discretisation. We begin by considering equations for which the solution explodes in finite time. Existing work on such explosive equations has dealt with devising numerical schemes for equations with polynomially growing instantaneous feedback, and methods to deal with delayed feedback have not been fully explored. We therefore set out a discretised scheme which replicates all the qualitative features of the continuous-time solution for a more general class of equations. Next, for non-explosive equations which exhibit extremely rapid growth, the rate of growth of the solution depends on the comparative asymptotic nonlinearities of the coefficients of the equation and the magnitude of the delay. Thus we set out conditions on these parameters which characterise the growth rate of the solution, and investigate numerical methods for recovering this rate. Using constructive comparison principles and nonlinear asymptotic analysis, we extend the numerical methods devised for explosive equations for this purpose.","abstract_has_math":false,"creators":["McCarthy, Michael J."],"institution":"Dublin City University","degree_name":"phd","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-11","date_published":"2011-11","updated_at":"2026-07-24T02:00:48Z","subjects":["Differential equations","Numerical analysis"],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.sponsor","label":"Sponsor","values":["Irish Research Council for Science Engineering and Technology"]},{"key":"dc:creator","label":"Author","values":["McCarthy, Michael J."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-11"]},{"key":"dc:date.issued","label":"Date","values":["2011-11"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["Dublin City University"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://doras.dcu.ie/16640/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["phd"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Differential equations","Numerical analysis"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://doras.dcu.ie/16640/1/MMcC-ThesisFinal.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis investigates the asymptotic behaviour of a scalar, nonlinear dierential equation with a fixed delay, and examines whether the properties of this equation can be replicated by an appropriate discretisation. We begin by considering equations for which the solution explodes in finite time. Existing work on such explosive equations has dealt with devising numerical schemes for equations with polynomially growing instantaneous feedback, and methods to deal with delayed feedback have not been fully explored. We therefore set out a discretised scheme which replicates all the qualitative features of the continuous-time solution for a more general class of equations. Next, for non-explosive equations which exhibit extremely rapid growth, the rate of growth of the solution depends on the comparative asymptotic nonlinearities of the coefficients of the equation and the magnitude of the delay. Thus we set out conditions on these parameters which characterise the growth rate of the solution, and investigate numerical methods for recovering this rate. Using constructive comparison principles and nonlinear asymptotic analysis, we extend the numerical methods devised for explosive equations for this purpose."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Explosions and unbounded growth in nonlinear delay differential equations: Numerical and asymptotic analysis"]}]}],"canonical_facts":{"dc:contributor.sponsor":["Irish Research Council for Science Engineering and Technology"],"dc:creator":["McCarthy, Michael J."],"dc:date":["2011-11"],"dc:date.issued":["2011-11"],"dc:description.abstract":["This thesis investigates the asymptotic behaviour of a scalar, nonlinear dierential equation with a fixed delay, and examines whether the properties of this equation can be replicated by an appropriate discretisation. We begin by considering equations for which the solution explodes in finite time. Existing work on such explosive equations has dealt with devising numerical schemes for equations with polynomially growing instantaneous feedback, and methods to deal with delayed feedback have not been fully explored. We therefore set out a discretised scheme which replicates all the qualitative features of the continuous-time solution for a more general class of equations. Next, for non-explosive equations which exhibit extremely rapid growth, the rate of growth of the solution depends on the comparative asymptotic nonlinearities of the coefficients of the equation and the magnitude of the delay. Thus we set out conditions on these parameters which characterise the growth rate of the solution, and investigate numerical methods for recovering this rate. Using constructive comparison principles and nonlinear asymptotic analysis, we extend the numerical methods devised for explosive equations for this purpose."],"dc:format":["application/pdf"],"dc:identifier.uri":["https://doras.dcu.ie/16640/1/MMcC-ThesisFinal.pdf"],"dc:language":["en"],"dc:publisher.institution":["Dublin City University"],"dc:relation.isreferencedby":["https://doras.dcu.ie/16640/"],"dc:subject":["Differential equations","Numerical analysis"],"dc:title":["Explosions and unbounded growth in nonlinear delay differential equations: Numerical and asymptotic analysis"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["phd"]},"updated_at":"2026-07-24T02:00:48Z"}