A robust numerical method for a two-parameter singularly perturbed time delay parabolic problem
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A robust numerical method for a two-parameter singularly perturbed time delay parabolic problem Sumit1 · Sunil Kumar1 · Kuldeep1 · Mukesh Kumar2 Received: 16 January 2020 / Revised: 26 May 2020 / Accepted: 23 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this article, we consider a class of singularly perturbed two-parameter parabolic partial differential equations with time delay on a rectangular domain. The solution bounds are derived by asymptotic analysis of the problem. We construct a numerical method using a hybrid monotone finite difference scheme on a rectangular mesh which is a product of uniform mesh in time and a layer-adapted Shishkin mesh in space. The error analysis is given for the proposed numerical method using truncation error and barrier function approach, and it is shown to be almost second- and first-order convergent in space and time variables, respectively, independent of both the perturbation parameters. At the end, we present some numerical results in support of the theory. Keywords Singular perturbation · Delay differential equation · Shishkin mesh · Hybrid scheme · Uniform convergence Mathematics Subject Classification 65M06 · 65M12 · 65L11
Communicated by José R Fernández.
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Sunil Kumar [email protected] Sumit [email protected] Kuldeep [email protected] Mukesh Kumar [email protected]
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Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, Uttar Pradesh, India
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Department of Mathematics, College of Charleston, Charleston, SC 29424, USA 0123456789().: V,-vol
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Sumit et al.
1 Introduction Singularly perturbed delay differential equations often arise in modeling of various physical, biological and chemical systems such as in population dynamics, variational problems in control theory, epidemiology, circadian rhythms, respiratory system, chemostat models, tumor growth and neural networks. The delay terms in these models enable us to include some past behavior to get more practical models for the phenomena. For example, in population ecology, time delay represents the hatching period or duration of gestation; in genetic repression modeling, time delays play an important role in processes of transcription and translation as well as spatial diffusion of reactants and in control systems, delay terms account for the time delay in actuation and in information transmission and processing. Many other examples can be found in Wu (2012). In this paper, we consider a singularly perturbed delay initial-boundary value problem in one space dimension with two small parameters. Defining G¯ = G ∪ ∂G, where G = (0, 1) × (0, T ] and ∂G = b ∪ r ∪ l with b = [0, 1] × [−τ, 0], l = {0} × (0, T ], and r = {1} × (0, T ], we consider ⎧ Lu ⎪ ≡ L ε,μ u − u t = −cu(x, t − τ ) + f (x, t) in G, ⎪ ⎨ u = ϕb (x, t), b (1.1) = ϕl (t), u ⎪ ⎪ ⎩ l u = ϕr (t), r
where L ε,μ u := εu x x +μau x −bu with parameters ε and μ such that 0 < ε ≤ 1, 0 ≤ μ ≤ 1. The coefficients are s
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