{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/11153"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/11153","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Volterra Systems with Realizable Kernels","abstract":"We compare an internal state method and a direct Runge-Kutta method for solving Volterra integro-differential equations and Volterra delay differential equations. The internal state method requires the kernel of the Volterra integral to be realizable as an impulse response function. We discover that when applicable, the internal state method is orders of magnitude more efficient than the direct numerical method. However, constructing state representation for realizable kernels can be challenging at times; therefore, we propose a rational approximation approach to avoid the problem. That is, we approximate the transfer function by a rational function, construct the corresponding linear system, and then approximate the Volterra integro-differential equation. We show that our method is convergent for the case where the kernel is nuclear. We focus our attention on time-invariant realizations but the case where the state representation of the kernel is a time-variant linear system is briefly discussed.","abstract_html":"We compare an internal state method and a direct Runge-Kutta method for solving Volterra integro-differential equations and Volterra delay differential equations. The internal state method requires the kernel of the Volterra integral to be realizable as an impulse response function. We discover that when applicable, the internal state method is orders of magnitude more efficient than the direct numerical method. However, constructing state representation for realizable kernels can be challenging at times; therefore, we propose a rational approximation approach to avoid the problem. That is, we approximate the transfer function by a rational function, construct the corresponding linear system, and then approximate the Volterra integro-differential equation. We show that our method is convergent for the case where the kernel is nuclear. We focus our attention on time-invariant realizations but the case where the state representation of the kernel is a time-variant linear system is briefly discussed.","abstract_has_math":false,"creators":["Nguyen, Hoan Kim Huynh"],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Herdman, Terry L."],"committee_members":["Rogers, Robert C.","Borggaard, Jeffrey T.","Cliff, Eugene M.","Burns, John A."],"year":2004,"date_issued":"2004-04-12","date_published":"2004-04-12","updated_at":"2026-07-22T22:20:09Z","subjects":["Volterra Integro-Differential Equations","Runge-Kutta Method","Realization Theory","Delay Equations","Internal State"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-04292004-143629"],"render_values":[{"text":"etd-04292004-143629","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/11153","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Herdman, Terry L."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Rogers, Robert C.","Borggaard, Jeffrey T.","Cliff, Eugene M.","Burns, John A."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Nguyen, Hoan Kim Huynh"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2011-08-22T19:00:56Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2011-08-22T19:00:56Z","2007-04-30"]},{"key":"dc:date.issued","label":"Date","values":["2004-04-12"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. 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The internal state method requires the kernel of the Volterra integral to be realizable as an impulse response function. We discover that when applicable, the internal state method is orders of magnitude more efficient than the direct numerical method. However, constructing state representation for realizable kernels can be challenging at times; therefore, we propose a rational approximation approach to avoid the problem. That is, we approximate the transfer function by a rational function, construct the corresponding linear system, and then approximate the Volterra integro-differential equation. We show that our method is convergent for the case where the kernel is nuclear. We focus our attention on time-invariant realizations but the case where the state representation of the kernel is a time-variant linear system is briefly discussed."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. 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