Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

,is paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. ,e highest-order derivative term in the equation is multiplied by a small perturbation parameter ε taking arbitrary values in the interval (0, 1]. For small values of ε, the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large.,e terms having the shifts are treated using Taylor’s series approximation. ,e resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. ,e formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation.,e theoretical analysis of the scheme is validated using numerical test examples for different values of ε and mesh number N.


Introduction
Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Currently different authors are working on analytical and numerical solutions of differential equations using different techniques [1,2]. Differential difference equations (DDEs) are differential equations where the evolution of the system not only depends on the present state of the system but also depends on the past history. Singularly perturbed differential difference equations are differential equations in which the highest-order derivative term is multiplied by a small perturbation parameter ε and involves at least one term with delay. In general, when the perturbation parameter tends to zero, the smoothness of the solution of the singularly perturbed differential difference equations (SPDDEs) deteriorates and it forms boundary layer [3]. Such type of equations has applications in the study of variational problems in control theory [4] and in modelling of neuronal variability [5]. e presence of singular perturbation parameter ε in the equation leads to oscillation in the computed solution, while using standard numerical methods like FDM, FEM, and spline method [6]. To avoid this oscillation, an unacceptably large number of mesh points are required when ε is very small. is is not practical and leads to round-off error. So, to overcome the drawbacks associated with standard numerical methods, different authors have developed schemes that converge uniformly.
Numerical treatments of a class of SPDDEs have received a great deal of attention recently because of their wide applications. It is of theoretical and practical interest to consider numerical methods for such problems. Owing to this, here we present some prior studies on numerical solution of the considered problem. Lange and Miura in [7][8][9][10] studied a class of second-order DDEs in which the second derivative term is multiplied by a small parameter. e authors extend the method of matched asymptotic expansions initially developed for solving boundary value problems to obtain approximate solution for SPDDEs. In a series of papers [11][12][13][14], Kadalbajoo and Sharma developed uniformly convergent numerical methods using fitted mesh FDMs techniques. Swamy et al. [15][16][17] considered the problem and developed a numerical scheme using fitted operator finite difference techniques. Melesse et al. [18] applied initial value technique for treating the considered SPDDEs. Ranjan and Prasad [19] used modified fitted operator FDM for solving the problem. Sirisha et al. [20] developed fitted operator finite difference scheme using the procedure of domain decomposition. A number of authors have developed numerical scheme using exponentially fitted method for solving SPDDEs. To the authors' knowledge, none of them show the uniform convergence of their schemes. is motivates to treat the considered SPDDEs and formulate the uniform convergence analysis of the scheme. Our contribution in this paper is to develop higher-order uniformly convergent numerical scheme using exponentially fitted FDM and to analyse the uniform convergence of the proposed scheme.

Notation 1.
e symbol C is used to denote positive constant independent of ε and N. e norm ‖.‖ denotes the maximum norm.
Setting c ε � 0 in equations (5)- (6) gives the reduced problem. For the case where p(x) > 0, it is given by and, for the case where p(x) < 0, it is given by It is a first-order initial value problem; for small values of c ε , the solution of (5)-(6) is very close to the solution of (8) or (9).
On the boundary points, we have On the differential operator, which implies that Lϑ ± (x) ≥ 0. Hence, using the maximum principle in Lemma 1, we obtain ϑ ± (x) ≥ 0, ∀x ∈ Ω. □ Lemma 3. e derivatives of the solutions of (5)-(6) satisfy the bound for left boundary layer problems and for right boundary layer problems.

Numerical Scheme
First, let us discretize the domain Ω � [0, 1] into N equal number of subintervals with mesh length h � (1/N) as Let u(x) be smooth function on the domain Ω � [0, 1]; then, using Taylor series approximation, we have Taking the difference in (15), we obtain Differentiating (16) two times gives Now, multiplying (17) by − (h 2 /12) and adding with (16) to eliminate the term with u (4) i gives International Journal of Differential Equations where τ � O(h 6 ). Evaluating (5) at x i− 1 , x i , and x i+1 , respectively, we obtain Next, approximate the first derivative terms u i− 1 ′ , u i ′ , and u i− 1 ′ in (19), using the right shifted, central, and left shifted finite difference approximations as Substituting (20) into (19) and then (19) into (18) gives where p i− 1 , p i , and p i+1 are denoted for p(x i− 1 ), p(x i ), and p(x i+1 ), respectively. We denote U i for the approximate solution of u(x i ) in the above discretization.
To get small truncation error in boundary layer region, we apply exponentially fitted operator finite difference method (FOFDM). For developing the FOFDM, we use the theory developed in asymptotic method for treating singularly perturbed BVPs. Let us consider and treat the left and the right boundary layer cases separately.

Left Boundary Layer Problems.
In this case, the boundary layer occurs near x � 0. From the theory of singular perturbation given in [24], the zeroth-order asymptotic solution of (5)-(6) is given by where u 0 is the solution of the reduced problem. Using Taylor's series approximation for u 0 (x), p(x), and q(x) centred at x i � ih up to first order and considering c ε ⟶ 0, the discretized form of (22) becomes To handle the effect of the perturbation parameter, exponential fitting factor σ 1 (ρ) is multiplied on the term containing the perturbation parameter as

International Journal of Differential Equations
Multiplying both sides of (25) by h and substituting ρ for h/c ε and taking the limit as h ⟶ 0 give From (23) and (24), we obtain Substituting (27) into (26) and simplifying give e exponential fitting factor is obtained as Hence, the required finite difference scheme becomes where International Journal of Differential Equations 5 with the boundary values U 0 � ϕ(0) and U N � ψ(1).

Right Boundary Layer Problems.
In this case, the boundary layer occurs near x � 1. From [24], the zerothorder asymptotic solution of (5)-(6) is given by where u 0 is the solution of the reduced problem. Using Taylor's series approximation for u 0 (x), p(x), and q(x) centred at x i � ih up to first order and considering c ε ⟶ 0, the discretized form of (24) becomes Using similar procedures as the left boundary layer case, the exponential fitting factor is obtained as Hence, the required finite difference scheme becomes where

Convergence Analysis.
In this section, we show the stability and convergence analysis for the right boundary layer problems. In similar manner, it is proved for the left boundary layer case. First, we need to prove the discrete comparison principle for the scheme in (35) for guaranteeing existence of unique discrete solution.

Lemma 4 (Discrete comparison principle). Assume that, for mesh function U i there exists a comparison function
Proof.
e matrix associated with operator L h R is of size (N + 1) × (N + 1) and satisfies the property of M-matrix. See the detailed proof in [23].
is lemma gives guarantee for the existence of unique discrete solution. In the next lemma, we discuss the uniform stability of the discrete solution. e solution U i of the discrete scheme in (35) satisfies the following bound: On the boundary points, we obtain International Journal of Differential Equations On the discretized spatial domain By the discrete comparison principle in Lemma 4, we obtain ϑ ± i ≥ 0, ∀x i ∈ Ω N . Hence, the required bound is satisfied. Now, let us denote the right shifted, centred, and left shifted finite differences, respectively, as Using Taylor's series approximation, we obtain the bound where Now, for ρ > 0, C 1 and C 2 are constants, and we have |ρ coth(ρ) − 1| ≤ C 1 ρ 2 , for ρ ≤ 1. For ρ ⟶ ∞, since lim ρ⟶∞ coth(ρ) � 1, |ρ coth(ρ) − 1| ≤ C 1 ρ is given. In general, for all ρ > 0, we write implying that e following theorem gives truncation error bound of the proposed scheme. □ Theorem 1. Let u(x i ) and U i be solutions of (5)- (6) and (35), respectively. en the following error estimate holds: Proof. Consider the truncation error that is given by Using the bounds in (44), (41), and (42) gives Substituting the bounds for the derivatives of the solution in Lemma 3, we obtain Since □ Lemma 6. For c ε ⟶ 0, and for given fixed N, we obtain where x j � jh, h � 1/N, ∀j � 1, 2, . . . , N − 1.

Theorem 2. Under the hypothesis of boundedness of discrete solution, the solution of the discrete schemes in (30) satisfies the following uniform error bound:
Proof. Substituting the results in Lemma 6 into eorem 1, applying Lemma 5 gives the required bound.

Richardson Extrapolation.
We apply the Richardson extrapolation technique to accelerate the rate of convergence of the proposed scheme. Richardson extrapolation is a convergence acceleration technique that involves combination of two computed approximations of solution. Interested reader can see the details of Richardson extrapolation in [28]. From (34) and Lemma 6, we obtain where u(x i ) and U i are the exact and approximate solutions of (5)-(6), respectively. Applying Lemma 5 in (52) gives (54) Combining (53) and (54) for removing the term CN − 1 results in giving as the extrapolated solution. e error bound for the extrapolated solution in (56) becomes

Examples and Numerical Results
In this section, we consider numerical examples to illustrate the theoretical findings of the developed schemes.
Example 4. Consider the problem is given by where . e exact solutions of the variable coefficient problems are not known. So, we use the procedure of the double mesh technique to calculate maximum absolute error. e maximum absolute error is defined as where U N i denotes the solution of the problem on N number of mesh points and U 2N i denotes the numerical solution on 2N number of mesh points by including the mid-points x (i+1)/2 into the mesh numbers. e uniform error estimate is defined as e rate of convergence of the scheme is given by and the uniform rate of convergence is given as In Tables 1-8, the maximum absolute error of Examples 1-4 using the proposed scheme is given. In Tables 1, 3, 5, and 7, the maximum absolute error before the Richardson extrapolation is given, and in Tables 2, 4, 6, and 8, the maximum absolute error after the Richardson extrapolation is given. As one observes in the tables, for each number of mesh interval N as ε ⟶ 0, the maximum absolute error becomes stable and uniform.
is indicates that the proposed scheme convergence is independent of the perturbation parameter ε. In the last two rows of each table, we observe the ε-uniform error and the ε-uniform rate of convergence of the scheme. e scheme before the extrapolation gives first-order uniform convergence and the extrapolated scheme gives second-order uniform convergence. In Tables 9 and 10, we compare the maximum absolute error of the proposed scheme with recently published papers in [14,15,19]. As one observes, the proposed scheme gives more accurate result.       Figure 1, the influence of the delay parameter on the behaviour of the solution of Examples 3 and 4 is shown for ε � 2 − 3 and δ � 0.1ε, 0.5ε, and 0.9ε. From Figure 2, we observe the numerical solution of Examples 1-4 for different values of the perturbation parameter ε � 2 − 5 , 2 − 6 , and 2 − 7 . As observed in the figures, for ε going small, strong boundary layer is created.     International Journal of Differential Equations 13

Conclusion
In this paper, singularly perturbed differential difference equations having mixed small shifts on reaction terms of the equation are considered. e considered problem exhibits boundary layer for small values of the perturbation parameter. e bounds and the behaviour of the continuous solution are discussed. Numerical scheme is developed using the technique of exponentially fitted finite difference method. Stability of the scheme is investigated using comparison principle and solution bound. e proposed scheme converges uniformly with rate of convergence of one before Richardson extrapolation and of two after Richardson extrapolation is applied. Test examples exhibiting boundary layers are considered to validate the theoretical finding. e finding in the computation agrees well with the theoretical findings. e proposed scheme gives more accurate results than existing research findings in the literature. In future works, we extend the proposed scheme for singularly perturbed parabolic differential difference equations and singularly perturbed problems with degenerate coefficients.

Data Availability
No data were used to support the study.

Conflicts of Interest
e authors declare no conflicts of interest.