Abstract
Purpose
The purpose of this study is to develop stable, convergent and accurate numerical method for solving singularly perturbed differential equations having both small and large delay.
Design/methodology/approach
This study introduces a fitted nonpolynomial spline method for singularly perturbed differential equations having both small and large delay. The numerical scheme is developed on uniform mesh using fitted operator in the given differential equation.
Findings
The stability of the developed numerical method is established and its uniform convergence is proved. To validate the applicability of the method, one model problem is considered for numerical experimentation for different values of the perturbation parameter and mesh points.
Originality/value
In this paper, the authors consider a new governing problem having both small delay on convection term and large delay. As far as the researchers' knowledge is considered numerical solution of singularly perturbed boundary value problem containing both small delay and large delay is first being considered.
Keywords
Citation
Garoma Debela, H. (2022), "Robust numerical method for singularly perturbed differential equations having both large and small delay", Arab Journal of Mathematical Sciences, Vol. 28 No. 1, pp. 87-99. https://doi.org/10.1108/AJMS-09-2020-0058
Publisher
:Emerald Publishing Limited
Copyright © 2020, Habtamu Garoma Debela
License
Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
A differential equation is said to be singularly perturbed delay differential equation, if it includes at least one delay term, involving unknown functions occurring with different arguments, and also, the highest derivative term is multiplied by a small parameter. Such type of delay, differential equations play a very important role in the mathematical models of science and engineering, such as the human pupil light reflex with mixed delay type [1], variational problems in control theory with small state problem [2], models of HIV infection [3] and signal transition [4].
Any system involving a feedback control almost involves time delay. The delay occurs because a finite time is required to sense the information and then react to it. Finding the solution of singularly perturbed delay differential equations, whose application mentioned above, is a challenging problem. In response to these, in recent years, there has been a growing interest in numerical methods on singularly perturbed delay differential equations. The authors of [5–7] have developed various numerical schemes on uniform meshes for singularly perturbed second-order differential equations having small delay on convection term. The authors of [8–12] have presented second-order differential equations having large delay.
In this paper, we consider a new governing problem having both small delay on convection term and large delay. Additionally, in recent years the correlative physical phenomena in-depth of the problem under consideration have been done by the authors [13–17]. As far as the researchers' knowledge is considered numerical solution of singularly perturbed boundary value problem containing both small delay and large delay is first being considered. Thus, the purpose of this study is to develop stable, convergent and accurate numerical method for solving singularly perturbed differential-difference equations having both small and large delay.
Throughout our analysis C is generic positive constant that are independent of the parameter
2. Statement of the problem
Consider the following singularly perturbed problem
The boundary value problem 1–2 exhibits strong boundary layer at
As we observed from Eqns (3) and (4), the values of
with boundary conditions
3. Properties of continuous solution
(Maximum principle) Let
Proof. For the proof refer [8] ▪
Proof. For the proof refer [8] ▪Proof. For the proof refer [8] ▪4. Numerical scheme formulation
We divide the interval
Consider a uniform mesh with interval
We can rewrite (3) as
For each segment
To determine the unknown coefficients in (10) in terms of
The coefficients in (10) are determined as
Using the continuity condition of the first derivative at
Reducing indices of Eqn (12) by one and substituting into Eqn (13), we obtain
If
Using
Using Taylorâ™s series expansions of
Substituting Eqn (16) in to Eqn (15), we obtain
Substituting Eqn (17) into Eqn (14) and rearranging, we get
From the theory of singular perturbations described in [18] and the Taylorâ™s series expansion of
Introducing a fitting factor
Multiplying Eqn (19) by h and taking a limit as
Thus, we consider two cases of the boundary layers.
Case 1: For
(Left-end boundary layer), we have
Substituting Eqn (21) into Eqn (20) and simplifying, we get
Case 2: For
(Right-end boundary layer), we have
Substituting Eq. (22) into Eq. (20) and simplifying, we obtain
In general, we take a variable fitting parameter as
Thus, Eqn (19) can be written as
Simplifying Eqn (25), for the domain
Similarly, if we consider the domain
Using
Substituting Eq. (16) in to Eq. (28), we obtain
Substituting Eqn (29) in to Eqn (14), introducing fitting factor and rearranging, we get
Therefore, on the whole domain
5. Stability and convergence analysis
5.1 Truncation error
Let expand the terms
The local truncation error
Substituting the series of
But from the values of
This establishes that the developed method is second order accurate or its order of convergence is
5.2 Convergence analysis
Local truncation errors refer to the differences between the original differential equation and its finite difference approximation at a mesh points. Finite difference scheme is called consistent if the limit of truncation error
Thus, the proposed scheme is consistent.
5.3 Stability analysis
Consider the developed scheme in Eqn (26),
Considering Eqn (37) into Eqn (26) the one which is multiplied by
Here, the coefficient matrix A is a tri-diagonal matrix with size
Again we can see that all
As discussed in [19] the eigenvalues of a tri-diagonal matrix A are given by
Hence, the eigenvalues of matrix A in Eqn (38) are
But from trigonometric identity, we have
A finite difference method for the boundary value problems is stable if A is nonsingular and
Since A is real and symmetric it follows that
6. Numerical examples and results
In this section, one example is given to illustrate the numerical method discussed above. The exact solutions of the test problem is not known. Therefore, we use the double mesh principle to estimate the error and compute the experiment rate of convergence to the computed solution. For this we put
The numerical results are presented for the values of the perturbation parameter
Consider the model singularly perturbed boundary value problem:
Subject to the boundary conditions
7. Discussion and conclusion
This study introduces a fitted nonpolynomial spline method for singularly perturbed differential equations having both small and large delay. The numerical scheme is developed on uniform mesh using fitted operator in the given differential equation. The stability of the developed numerical method is established and its uniform convergence is proved. To validate the applicability of the method, one model problem is considered for numerical experimentation for different values of the perturbation parameter and mesh points. The numerical results are tabulated in terms of maximum absolute errors, numerical rate of convergence and uniform errors (see Table 1). Further, behavior of the numerical solution (Figure 1), point-wise absolute error (Figure 2) and the
Figures

Figure 1
The behavior of the numerical solution for Example 6.1 at

Figure 2
Point wise absolute error of Example 6.1 at

Figure 3
ε-uniform convergence with fitted operator in log-log scale for Example 6.1
Maximum absolute errors and rate of convergence for Example 6.1 at different perturbation parameter and number of mesh points
ε | N = 32 | N = 64 | N = 128 | N = 256 | N = 512 |
---|---|---|---|---|---|
1.9799e-04 | 1.0004e-04 | 5.0281e-05 | 2.5206e-05 | 1.2619e-05 | |
1.9799e-04 | 1.0004e-04 | 5.0281e-05 | 2.5206e-05 | 1.2619e-05 | |
1.9799e-04 | 1.0004e-04 | 5.0281e-05 | 2.5206e-05 | 1.2619e-05 | |
1.9799e-04 | 1.0004e-04 | 5.0281e-05 | 2.5206e-05 | 1.2619e-05 | |
1.9799e-04 | 1.0004e-04 | 5.0281e-05 | 2.5206e-05 | 1.2619e-05 | |
1.9799e-04 | 1.0004e-04 | 5.0281e-05 | 2.5206e-05 | 1.2619e-05 | |
0.9849 | 0.9925 | 0.9962 | 0.9982 |
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Acknowledgements
The authors wish to express their thanks to Jimma University, College of Natural Sciences, for technical support and the authors of the literature for the provided scientific aspects and idea for this work.