Title page for ETD etd-04292004-143629


Type of Document Dissertation
Author Nguyen, Hoan Kim Huynh
Author's Email Address honguye5@vt.edu
URN etd-04292004-143629
Title Volterra Systems with Realizable Kernels
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Herdman, Terry L. Committee Chair
Borggaard, Jeffrey T. Committee Member
Burns, John A. Committee Member
Cliff, Eugene M. Committee Member
Rogers, Robert C. Committee Member
Keywords
  • Delay Equations
  • Realization Theory
  • Runge-Kutta Method
  • Volterr Integro-Differential Equations
  • Internal State
Date of Defense 2004-04-28
Availability unrestricted
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.
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