Singularly perturbed delay differential equations (SPDDEs) are differential equations that involve diffusion parameter, known as the perturbation parameter ε, and at least a delay term. SPDDEs arise in many practical phenomena, such as epidemiology [1], population growth [2], chemostat models [3], circadian rhythms [4], the respiratory system [5], and tumor growth [6]. The following is a typical example of SPDDEs, which is a mathematical model of the overall control system [7] which models a furnace used to process metal sheets.

defined on a one-dimensional spatial domain 0 < x < 1, where u(x, t) is the temperature distribution in metal sheet moving at a velocity υ depending on a prescribed spatial average of the time-delayed temperature distribution u(x, t−τ) and f represents a distributed temperature source function depending on u(x, t−τ). The spatial temperature distribution in the incoming and out coming material within the furnace is given as u(0, t) and u(1, t), respectively. A controlling device monitoring the current temperature dynamically adapts both υ and f. A fixed delay of length τ is induced by the finite speed of the controller. Another typical example of SPDDEs is the following logistic equation [8]:

which arise in mathematical ecology for the evolution of a population with density u(x, t). A population u(x, t) depends on the population at an earlier time, t − τ rather than t. The delay τcan arise from a great variety of causes, such as duration of gestation, hatching period, and slow replacement of food supplies. Thus, u(x, t) depends on average past population u(x, t−τ). The initial and final spatial spread of the favored population are given by u(0, t) and u(1, t), respectively. A set of examples is available in Wu et al. [9] to illustrate the wide range of existing delay differential equation models.

When the highest order term coefficient ε tends to zero, the solution of SPDDE possesses a boundary layer. This layer is a thin region at the right side of the domain where the solution has a steep gradient. As the solution gets steeper, the classic methods on the uniform mesh are unable to solve SPDDEs without using an inadequately small mesh size, which is not practicable. This motivated various researchers to develop ε-uniform non-classical numerical methods. When the delay parameter τ is smaller than ε, the given delay differential equation is reduced by means of Taylor series expansion, but if the delay parameter is higher, this approach does not work [10]. In such case, during developing the scheme the point t − τ must coincide with a mesh point. In this study, we have a numerical scheme for the problem with the large delay.

In the last few years, several numerical approaches have been developed for the solution of time dependent singularly perturbed differential equations [8]. However, in the case of these equations with time delay, few numerical methods have been developed, and studies are still at an early stage [8]. Some of them, which treat time dependent singularly perturbed delay parabolic convection-diffusion initial boundary value problems are listed here. Das and Natesan [11] had proposed to a numeric scheme consisting of an implicit-Euler scheme in the temporal direction, accompanied by a hybrid scheme in the space direction. Negero and Duressa [12] developed a second-order ε-uniform convergent scheme on a uniform mesh. They discretize time and spatial derivatives by the implicte Euler rule and Micken's non-standard method with extrapolation, respectively. In the same year, Negero and Duressa [13] designed an efficient numerical approach which is uniformly convergent of second order of convergent. They discretize time and spatial derivatives, respectively, by Crank-Nicolson and the exponentially fitted spline method. Woldaregay et al. [14] developed a novel numerical scheme using by an exponentially fitted operator which is ε-uniform convergent of linear order of convergent. Kumar et al. [15] developed a graded mesh refinement approach. The approach shows parameter uniform convergent linear order. Abdelhakem and Youssri [16] proposed two spectral Legendre's derivative algorithms for Lane-Emden, Bratu equations, and singular perturbed problems. They have shown the numerical schemes are stable, convergent, and accurate. Abd-Elhameed et al. [17] suggested and analyzed a new operational matrix method based on shifted Legendre polynomials for obtaining numerical spectral solutions of linear and non-linear second-order boundary value problems. The authors showed that the method has the following advantages. The method can be applied for both linear and non-linear second-order boundary value problems including some important singular perturbed equations and also a Bratu-type equation. This method computes highly accurate approximate solutions. Authors [18–24] also developed ε-uniform convergent numerical methods for singularly perturbed delay partial differential equation.

Nowadays, the use of the spline based approach has become very popular among different numerical methods to solve SPDDEs. Daba and Duressa [25] proposed a uniform convergent numerical scheme for singularly perturbed parabolic convection- diffusion equation with a small delay and advance parameter in the spatial variable of reaction term using an extended cubic B-spline method. They also suggested uniformly convergent numerical scheme for this problem using cubic B-spline [26] on a uniform mesh. Kumar and Kadalbajoo [27] proposed the parameter-uniform numerical method for the problem using cubic B-spline on a Shishkin mesh. Kumar [8] and Negero and Duressa [28] developed a parameter uniform convergent method to solve time dependent singularly perturbed delay parabolic convection-diffusion initial boundary value problems using the cubic B-spline collocation method on a piecewise uniform Shishkine mesh and uniform mesh, respectively. The extended cubic B-spline collocation method is a generalization of the cubic B-spline collocation method. It introduces a free parameter to allow the cubic B-spline's shape to alter while maintaining the continuity in the order of three. Kumar and Kumari [29] designed ε-uniform convergent numerical methods using an extended cubic B-spline collocation method for singularly perturbed delay parabolic convection-diffusion initial boundary value problems on a piecewise uniform Shishkine mesh.

The main contribution of this study is to construct a parameter uniform numerical method for singularly perturbed delay parabolic convection-diffusion problems. The method consists of the implicit Euler rule for temporal discretization and the extended cubic B-spline collocation method for spatial discretization on a uniform mesh using artificial viscosity. In this method, we use artificial viscosity σ(x, ε) to substitute the perturbation parameter ε that affects the highest derivative.

The rest of this article is organized as follows: in Section 2, we discuss the continuous problem and show the boundedness of the exact solution. The numerical scheme of the problem is discussed in Section 3. In this section, the implicit Euler method, the extended cubic B-spline collocation method, and the derivation of artificial viscosity are briefly discussed. Section 4 describes the convergence analysis of the proposed method. In Section 5, numerical illustrations are given to confirm the theoretical investigation. Lastly, Section 6 gives conclusion of the article.

Let Ω_x = (0, 1), Ω_t* = [−τ, 0] and Ω_t = (0, T] for some fixed positive time T. We define D = Ω_x × Ω_t and ∂D = D₀ ∪ D_b ∪ D₁, where D0={(0,t),t∈Ωt¯}, D1={(1,t),t∈Ωt¯}, and Db=Ωx¯×Ωt*. We consider the following singularly perturbed delay parabolic initial boundary value problem (IBVP):

with

and

where τ and 0 < ε ≪ 1 are delay and singular perturbation parameter, respectively. The differential operator 𝔏_ε in Eq. (1) is defined as

The functions a(x), b(x, t), c(x, t), f(x, t), ψ₀(t), ψ₁(t), and ψ_b(x, t) are assumed to be sufficiently smooth, bounded, and independent of ε. It is also assumed that

and terminal T, satisfy T = kτ for some positive integer k. We also assume that the data satisfy the following compatibility conditions:

Under the above assumptions and conditions, the solution of IBVP Eq. (1) is unique which has a parabolic boundary layer of width O(ε) along x = 1. Compatibility conditions [30] are relationships between the data of the problem and the differential operator that ensure that derivatives of u(x, t) up to a desired order are continuous on the closed domain D. They do not result from the problem's singularly perturbed nature because they only appear at corners.

Lemma 2.1 (Continuous Maximum Principle). Suppose the function φ(x,t)∈C2,1(D¯) satisfies (∂∂t+𝔏ε)φ(x,t)≥0,∀(x,t)∈D and φ(x, t) ≥ 0, ∀(x, t) ∈ ∂D, then φ(x,t)≥0,∀(x,t)∈D¯.

Proof. Let (x~,t~)∈D¯, such that φ(x~,t~)=min(x,t)∈D¯φ(x,t) and suppose that φ(x~,t~)<0. Clearly (x~,t~)∉∂D. Then, it follows from calculus that ∂φ(x~,t~)∂x=0,∂φ(x~,t~)∂t=0 and ∂2φ(x~,t~)∂x2≥0. Therefore, from Eq (1), we have

which is a contradiction to the assumption made. Thus, φ(x~,t~)≥0 which leads to φ(x,t)≥0,∀(x,t)∈D¯.     □

Lemma 2.2. [11] The solution u(x, t) of the IBVP Eq. (1) satisfies the estimate

Lemma 2.3. The solution u(x, t) of the IBVP Eq. (1) satisfies the estimate

Proof. Since t ∈ (0, T], by Lemma 2.2, we get |u(x, t)−ψ_b(x, 0)| ≤ Ct ≤ CT. It follows that

Since |ψb(x,0)|∈C2,1(D¯), CT+|ψ_b(x, 0)| is bounded by some positive constant C, |u(x,t)|≤C,∀(x,t)∈D¯.     □

Lemma 2.4 (Uniform stability estimate). Let u(x, t) be solution of the IBVP Eq. (1). Then, we obtain the bound:

Proof. Let define barrier functions

Ψ±(x,t)=β-1||(∂∂t+𝔏ε)u(x,t)||+max{|ψ0(t)|,|ψ1(t)|,|ψb(x,t)|}±u(x,t). By applying the continuous maximum principle Lemma 2.1, we obtain the required result.     □

Theorem 2.1. The solution u(x, t) of Eq (1) and its derivatives satisfy the following bounds :

where i and j are non-negative integers such that 0 ≤ i + j ≤ 5.

Proof. See details of the proof in Mbroh et al. [31].     □

This section describes the semi-discretization and extended cubic B-spline method on the uniform mesh by introducing artificial viscosity.

In this section, we establish the parameter-uniform convergence of the extended cubic B-spline collocation method. The following lemma will be used in the convergence analysis.

Lemma 4.1. The extended cubic B-spline set Λ = {Q₋₁, Q₀, Q₁, …, Q_N+1} defined in Eq. (15), satisfy the inequality

Proof. We know that

Let x = x_i be a nodal point. Then, by the definition of Q_j(x, η), we have