High Order Methods for State-Dependent Delay Differential Equations with Nonsmooth Solutions

Feldstein, Alan; Neves, Kenneth W.

doi:10.1137/0721055

Cited by 51 publications

(29 citation statements)

References 11 publications

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“…Thus, itt) has a jump discontinuity at to = O. The effect of this discontinuity propagates to the points t = 1,2,3, ... due to the delayed term x(t -1) in (5). Differentiating (5) k times leads to X(k+l)(t) = X(k)(t -1), which can be written as x(k+l)(t) = i(t -k).…”

Section: Then There Exists An Unique Solution For the Problem (2) Anmentioning

confidence: 95%

“…Thus, it is sufficient to understand how discontinuities propagate through the solution. In the following, the main results from [5,8] concerning the propagation of jump discontinuities within the solution of a delay differential equation are presented.…”

Section: Then There Exists An Unique Solution For the Problem (2) Anmentioning

confidence: 99%

“…has to be given. Hereby, the set [t, tol is called the initial set, <I> (t) is called the initial function and <I> (to) is called the initial value [5,14]. The term solution is defined in the following.…”

Section: Delay Differential Equationsmentioning

confidence: 99%

“…(1) where itt) denotes the time derivative of x. The functions exilt, x(t)) ~ t, i = 1, ... , r are called the retarding arguments, retarding junctions or lag junctions [5].…”

Section: Delay Differential Equationsmentioning

confidence: 99%

“…Therefore, the remainder of this article is splitted into three parts: The first, dealing with an introduction to DDEs and discussing special properties of DDEs, the second, presenting requirements for the numerical simulation of DDEs, and the third, where difficulties within parameter estimation for DDEs are discussed. In the Sections 2 and 4 the most important parts of [5,8], [12,13] and [1,2] are cited.…”

Section: Introductionmentioning

confidence: 99%

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Differential Equations with State-Dependent Delays

Hofer

Tibken

Lehn

2001

Online Optimization of Large Scale Systems

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In this article an introduction to the wide field of retarded or delay differential equations with state-dependent delays is given. Hereby, the most important features of this type of differential equations which are profoundly different from ordinary differential equations are discussed. The presence of discontinuities in higher derivatives of the solution of delay differential equations require suitable integration methods. Thus, an efficient numerical code for the integration of delay differential equations is presented. This code is based on a fourth order RK one step algorithm. Furthermore, the presence of discontinuities may lead to severe problems when parameters are to be estimated using higher order optimization techniques. These problems are pointed out and a systematic way for the analysis of the smoothness of a least squares objective function is presented.

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Section: Then There Exists An Unique Solution For the Problem (2) Anmentioning

confidence: 95%

Section: Then There Exists An Unique Solution For the Problem (2) Anmentioning

confidence: 99%

Section: Delay Differential Equationsmentioning

confidence: 99%

“…(1) where itt) denotes the time derivative of x. The functions exilt, x(t)) ~ t, i = 1, ... , r are called the retarding arguments, retarding junctions or lag junctions [5].…”

Section: Delay Differential Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Differential Equations with State-Dependent Delays

Hofer

Tibken

Lehn

2001

Online Optimization of Large Scale Systems

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The Numerical Integration of Neutral Functional‐Differential Equations by Fully Implicit One‐Step Methods

Jackiewicz

1995

Z Angew Math Mech

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An algorithm for the numerical solution of neutral functional‐differential equations is described. This algorithm is based on divided difference representation of fully implicit one‐step methods. The resulting systems of nonlinear equations are solved using the predictor‐corrector approach for nonstiff equations and by the modified Newton method for stiff equations. The step control strategy is based on local error estimation by comparing two approximations of successive orders. The details of implementations are described for systems of neutral delay‐differential equations with state dependent delays, for Volterra integro‐differential equations and for stiff delay‐differential equations. The results of some numerical experiments on four test examples are presented and discussed.

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