Volterra Systems with Realizable Kernels

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.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2004

Author dc:creator

• Nguyen, Hoan Kim Huynh

Chair dc:contributor.committeechair

• Herdman, Terry L.

Committee members dc:contributor.committeemember

• Rogers, Robert C.
• Borggaard, Jeffrey T.
• Cliff, Eugene M.
• Burns, John A.

• Volterra Integro-Differential Equations
• Runge-Kutta Method
• Realization Theory
• Delay Equations
• Internal State

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

etd-04292004-143629

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/11153