The numerical solution of neutral functional differential equations by Adams predictor- corrector methods

Jackiewicz, Zdzisław; Lo, Edisanter

doi:10.1016/0168-9274(91)90110-l

Cited by 20 publications

(7 citation statements)

References 22 publications

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“…It is worthwhile to remark that there are codes which solve neutral problems without formulating them as implicit systems (e.g. DDE-STRIDE by Baker, Butcher and Paul [4] and SNDDELM by Jackiewicz and Lo [15]). …”

Section: Introductionmentioning

confidence: 99%

Computing breaking points in implicit delay differential equations

Guglielmi

Hairer

2007

Adv Comput Math

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Systems of implicit delay differential equations, including state-dependent problems neutral and differential-algebraic equations, singularly perturbed problems, and small or vanishing delays are considered. The numerical integration of such problems is very sensitive to jump discontinuities in the solution or in its derivatives (so-called breaking points). In this article we discuss a new strategy - peculiar to implicit schemes - that allows codes to detect automatically and then to compute very accurately those breaking points which have to be inserted into the mesh to guarantee the required accuracy. In particular for state-dependent delays, where breaking points are not known in advance, this treatment leads to a significant improvement in accuracy. As a theoretical result we obtain a general convergence theorem which was missing in the literature (see cite{BZ03}). Furthermore, as a useful by-product, we design strategies that are able to detect points of non-uniqueness or non-existence of the solution so that the code can terminate when such a situation occurs. A new version of the code RADAR5 together with drivers for some real-life problems is available on the homepages of the authors

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Section: Introductionmentioning

confidence: 99%

Computing breaking points in implicit delay differential equations

Guglielmi

Hairer

2007

Adv Comput Math

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“…having f as an independent function. Neves [1], Bellen and Zennaro [2] and Al-Mutib [3] discussed algorithms based on one-step methods, while Jackiewicz and Lo [4] and Tavernini [5] discussed algorithms based on liner multistep methods. Fabiano and Payne [6] extended the spline approximation technique for Neutral Delay Differential Equations.…”

Section: ( ) Dy F X Y a X B Dxmentioning

confidence: 99%

A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for Solving Neutral Delay Differential Equations

Yashkun

Aziz

2019

SJCMS

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A modified 3-point Adams block method of order six (3ABM6) to solve neutral delay differential equations (NDDEs) using variable step size strategy is developed. The approximate solution of the retarded () yx  − and the neutral terms () yx   − of the neutral delay differential equations at the grid points is obtained using the Newton divided difference interpolation technique. The proposed method will approximate the solution in each step using the three-points concurrently. To determine the performance of the proposed method, the maximum errors (MAXERR) and total number of function calls (FNC) will be compared with the method of 2-point order six predictor-corrector. The numerical results show that the 3ABM6 reduces the number of function calls and better accuracy in term of MAXERR.

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“…It follows from the results of 131 (see also [12]) that if the method (2.1) applied to ordinary differential equations i ", A-stable, then it is also NP-stable. Therefore, the method (2.1) is NP-stable if the points by, j = 0, 1, ..,, p , are chosen td be, for example, the Lobatto points for [0, 11, since, as it is known the corresponding method for ordinary differentid equations is A-stable.…”

Section: Stiff Delay Differential Equationsmentioning

confidence: 99%

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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The numerical solution of neutral functional differential equations by Adams predictor- corrector methods

Cited by 20 publications

References 22 publications

Computing breaking points in implicit delay differential equations

Computing breaking points in implicit delay differential equations

A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for Solving Neutral Delay Differential Equations

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

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