Analysis of an adaptive collocation solution for retarded and neutral delay systems
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Analysis of an adaptive collocation solution for retarded and neutral delay systems Mohammad Maleki1
· Ali Davari1
Received: 21 September 2020 / Accepted: 8 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper introduces an adaptive collocation method to solve retarded and neutral delay differential equations (RDDEs and NDDEs) with constant or time-dependent delays. The delays are allowed to be small or become vanishing during the integration. We determine the convergence properties of the proposed method for neutral equations with solutions in appropriate Sobolev spaces. It is shown that the proposed scheme enjoys the spectral accuracy. Numerical results show that the proposed method can be implemented in an efficient and accurate manner for a wide range of RDDE and NDDE model problems. Keywords Retarded and neutral delay systems · Collocation method · Convergence analysis · Spectral accuracy · Vanishing delay Mathematics Subject Classification (2010) 34K40 · 65L04 · 65L20 · 65L60 · 65L70 · 41A05
1 Introduction Delay differential equations (DDEs) are widely utilized in the modeling of population dynamics, the spread of infection diseases, and two-body problems of electrodynamics [7, 10, 12, 32, 33]. The delays (lags) can represent gestation times, incubation periods, transport delays, or they can simply lump complicated biological processes together, accounting only for the time required for these processes to occur.
Mohammad Maleki
[email protected] Ali Davari a [email protected] 1
Department of Mathematics, University of Khansar, Khansar, Iran
Numerical Algorithms
A general DDE with multiple time-dependent delays is given by d X(t) = f t, X(t), X(t − τ1 (t)), . . . , X(t − τm (t)), dt d d X(t − τm+1 (t)), . . . , X(t − τm+n (t)) , t0 t T , (1.1) dt dt d d X(t) = φ(t), t−1 t < t0 , (1.2) dt dt where f and φ are given functions with certain properties, τq (t) 0, 1 q m + n are time-dependent delays, T is a positive constant and t−1 = inf {t − X(t) = φ(t),
t0 tT
τq (t)}q . For the questions of existence and uniqueness of the solution to the model (1.1)–(1.2) refer to [11, 18, 29]. The model (1.1) with derivative delay terms is called explicit neutral delay differential equation (NDDE), otherwise it is called retarded delay differential equation (RDDE). Another well-known and widely studied class of NDDEs is implicit NDDEs, a form frequently called Hale’s form [3]. For the ease of presentation, we assume m = n = 1 without loss of generality. Thus, we consider the model problem d d (1.3) X(t) = f t, X(t), X(t − τ1 (t)), X(t − τ2 (t)) , t0 t T , dt dt d d X(t) = φ(t), t−1 t < t0 . (1.4) dt dt In the context of population dynamics, the model problem (1.1) can be obtained, e.g., from the balance laws of the age-structured population dynamics, assuming that the birth rates and death rates, as functions of age, are piecewise constant. The delay arises naturally from biology as the age-at-maturity of individuals. Th
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