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Numerical and asymptotic studies of ...

Adhikari, Mohit Hemchandra.

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Numerical and asymptotic studies of delay differential equations.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Numerical and asymptotic studies of delay differential equations./
Author:	Adhikari, Mohit Hemchandra.
Description:	156 p.
Notes:	Advisers: J. K. McIver; E. A. Coutsias.
Contained By:	Dissertation Abstracts International69-01B.
Subject:	Physics, Theory. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3298165
ISBN:	9780549424468

Numerical and asymptotic studies of delay differential equations.
Adhikari, Mohit Hemchandra.

Numerical and asymptotic studies of delay differential equations. - 156 p.

Advisers: J. K. McIver; E. A. Coutsias.

Thesis (Ph.D.)--The University of New Mexico, 2007.

Two classes of differential delay equations exhibiting diverse phenomena are studied. The first one is a singularly perturbed delay differential equation which is used to model selected physical systems involving feedback where relaxation effects are combined with nonlinear driving from the past. In the limit of fast relaxation, the differential equation reduces to a difference equation or a map, due to the presence of the delay. A basic question in this field is how the behavior of the map is reflected in the behavior of the solutions of the delay differential equation. In this work, a generic logistic form is used for the underlying map and the above question is studied in the first period-doubling regime of the map. Using an efficient numerical algorithm, the shape and the period of the corresponding asymptotically stable periodic solution is studied first, for various values of the delay. In the limit of large delay, these solutions resemble square-waves of period close to twice the value of the delay, with sharp transition layers joining flat plateau-like regions. A Poincare-Lindstedt method involving a two-parameter perturbation expansion is applied to solve equations representing these layers and accurate expressions for the shape and the period of these solutions, in terms of Jacobi elliptic functions, are obtained. A similar approach is used to obtain leading order expressions for sub-harmonic solutions of shorter periods, but it is shown that while they are extremely long-lived for large values of delay, they eventually decay to the fundamental solutions mentioned above. The spectral algorithm used for the numerical integration is tested by comparing its accuracy and efficiency in obtaining stiff solutions of linear delay equations, with that of a current state-of-the-art time-stepping algorithm for integrating delay equations.

ISBN: 9780549424468Subjects--Topical Terms:

1019422
Physics, Theory.

Numerical and asymptotic studies of delay differential equations.
LDR:03569nam 2200289 a 45 001 942192
005 20110519
008 110519s2007 ||||||||||||||||| ||eng d
020 $a 9780549424468
035 $a (UMI)AAI3298165
035 $a AAI3298165
040 $a UMI $c UMI
100 1 $a Adhikari, Mohit Hemchandra. $3 1266288
245 1 0 $a Numerical and asymptotic studies of delay differential equations.
300 $a 156 p.
500 $a Advisers: J. K. McIver; E. A. Coutsias.
500 $a Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0393.
502 $a Thesis (Ph.D.)--The University of New Mexico, 2007.
520 $a Two classes of differential delay equations exhibiting diverse phenomena are studied. The first one is a singularly perturbed delay differential equation which is used to model selected physical systems involving feedback where relaxation effects are combined with nonlinear driving from the past. In the limit of fast relaxation, the differential equation reduces to a difference equation or a map, due to the presence of the delay. A basic question in this field is how the behavior of the map is reflected in the behavior of the solutions of the delay differential equation. In this work, a generic logistic form is used for the underlying map and the above question is studied in the first period-doubling regime of the map. Using an efficient numerical algorithm, the shape and the period of the corresponding asymptotically stable periodic solution is studied first, for various values of the delay. In the limit of large delay, these solutions resemble square-waves of period close to twice the value of the delay, with sharp transition layers joining flat plateau-like regions. A Poincare-Lindstedt method involving a two-parameter perturbation expansion is applied to solve equations representing these layers and accurate expressions for the shape and the period of these solutions, in terms of Jacobi elliptic functions, are obtained. A similar approach is used to obtain leading order expressions for sub-harmonic solutions of shorter periods, but it is shown that while they are extremely long-lived for large values of delay, they eventually decay to the fundamental solutions mentioned above. The spectral algorithm used for the numerical integration is tested by comparing its accuracy and efficiency in obtaining stiff solutions of linear delay equations, with that of a current state-of-the-art time-stepping algorithm for integrating delay equations.
520 $a Effect of delay on the synchronization of two nerve impulses traveling along two parallel nerve fibers, is the second question studied in this dissertation. Two coupled nonlinear diffusion equations of the Fitzhugh-Nagurno type are used to model this system with the delay introduced in the coupling term. A multiple-scale perturbation approach is used for the analysis of these equations in the limit of weak coupling. In the absence of delay, it is shown that two pulses with identical speeds can synchronize. However, as the delay is increased beyond a critical value, this synchrony is destroyed. A quantitative estimate for the actual values of delay at which this can occur in the case of squid giant neurons is found and compared with the relevant time-scales involved.
590 $a School code: 0142.
650 4 $a Physics, Theory. $3 1019422
690 $a 0753
710 2 $a The University of New Mexico. $3 1018024
773 0 $t Dissertation Abstracts International $g 69-01B.
790 $a 0142
790 1 0 $a Coutsias, E. A., $e advisor
790 1 0 $a McIver, J. K., $e advisor
791 $a Ph.D.
792 $a 2007
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3298165