Comparative Study on Numerical Methods for Singularly Perturbed Advanced-Delay Differential Equations

Department of Mathematics, Phuket Rajabhat University, 83000 Phuket, ailand Department of Mathematics, SASTRA Deemed to be University, anjavur, Tamilnadu 613401, India Department of Mathematics, Bharathidasan University, Tiruchirappalli-620 024, Tamilnadu, India Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama 337-8570, Japan Department of Mathematics, Anand International College of Engineering, Jaipur, India


Introduction
Differential equations depend both on past and future values (mixed delay) called functional differential equations. It attains many application problems such as optimal control problems [1], nerve conduction theory [2], economic dynamics [3], traveling waves in a spatial lattice [4] and has discussed both linear and nonlinear functional differential equations. e functional differential equation has been multiplied by small parameter (0 < ε < 1) in the highest order derivative term called the singularly perturbed mixed delay differential equations. e main determination for such a problem is the study of biological science, epidemics, and population [5][6][7][8][9][10]. e authors in [11] have considered functional differential equation in singularly perturbed problems, such as and considered the problem of determining the expected time for the generation of action potentials in nerve cells by random synaptic inputs in the dendrites. e general linear second-order functional differential equation with the boundary-value problem arises in the modeling of neuron activation, where σ and μ are the variance and drift parameters and y is the expected first-exit time. e first-order derivative term − xy′(x) corresponds to exponential decay between synaptic and inputs. e undifferentiated terms correspond to excitatory and inhibitory synaptic inputs modeled as a Poisson process with mean rates λ E and λ I ; they produce jumps in the membrane potential of amounts a E and a I , which are small quantities and could depend on the voltage. e boundary condition is where the values x � x 1 and x � x 2 correspond to inhibitory reversal potential and the threshold value of membrane potential for action potential generation. is biological problem motivates the investigation of boundary-value problems for differential-difference equations with mixed shifts. In this biological model, using the Taylor series for the small delay term, provided the delay is of order ε, the small delay problem has oscillatory solution that has been discussed in [12]. e same authors discussed the signal transmission problem in [13]. e authors in [14,15] have considered the singularly perturbed problem with derivative depending on small delay term such as to solve the boundary-value problem using the following numerical method such as the finite difference scheme [14,16], fitted mesh B-spline collocation method [17], and hybrid difference scheme [15]. e authors in [18,19] investigated various concepts of singularly perturbed differential equation with derivative depending on both past and future small variables, also proposed a finite difference scheme to solve singular perturbation problems in [18,20,21]. e authors in [19] have been proposed to solve the singular perturbation problem with mixed small shifts using the fitted operator method. In recent years, the authors in [22][23][24][25] considered singular perturbation problem with derivative depending on large delay (τ � 1) variable, such as (5) has been developed various numerical schemes are finite and hybrid difference method [22], iterative scheme [26], finite element method [27,28]. e study in [23] proposed solving singularly perturbed delay differential equation with integral boundary condition using finite difference method. roughout the literature, the researcher concentrates on solving the singular perturbation problem with a small delay or mixed small delay or large delay using finite or hybrid or finite element methods on uniform meshes or nonuniform mesh. To the best of the author's knowledge, up to now, no theoretical results are given for comparative study on numerical methods for singularly perturbed advanced-delay differential equations. Moreover, we proposed two numerical methods such as the finite and hybrid difference scheme on nonuniform meshes, to solve the singular perturbation problem with mixed large delay using the finite difference scheme and hybrid difference scheme on Shishkin mesh.
is paper is structured as follows: Section 2 describes the problem statement. Section 3 proves the maximum principle and stability result. Moreover, it introduces the terminology for Shishkin decomposition and proves many inequalities. In Section 4, we introduce the numerical methods to discretize the continuous problem. Error analysis for finite and hybrid difference scheme approximate solution is given in Sections 5 and 6. Finally, Section 7 presents numerical results. roughout our analysis, we use the following notations: e parameter ε and mesh points 3N are independent of C and C 1 are positive constants. e norm is ‖y‖ Γ � sup r∈Γ |y(r)|.

Statement of the Problem
Consider the following singularly perturbed mixed delay differential equation: where ϕ(r) and φ(r) are history function on [− 1, 0] and [3,4]. Assume that a(r)
If μ > 0, then the function ψ(r) nonnegative is not possible. e following cases are easy to prove the contradiction if μ > 0.
Lemma 2 (stability result). If y(r) is a solution of problems (7)- (9), then Lemma 3. If y(r) is a solution of problems (7)- (9), then Proof. First, to prove y ′ (r) is bound on Γ 1 , Integrating the above equation on both sides, we have erefore,
Next, to prove y ′ (r) is bound on Γ 3 , Integrating the above equation on both sides, we have erefore, Using Mean Value eorem, then |εy where v 0 (r), v 1 (r), and v 2 (r) are solutions of the following differential equations.
Obtain reduced problem solution v 0 (r) ∈ X such that Also, w(r) satisfies the following problem: if the singular is boundary layer component and w I 1 (r), w I 2 (r) are interior layer components. If the boundary layer w B (r) ∈ X, If the first interior layer w I 1 (r) ∈ C 0 (Γ) ∩ C 2 (Γ * ), If the second interior layer w I 2 (r) ∈ C 0 (Γ) ∩ C 2 (Γ * ),
Proof. e smooth component derivative bound is easy to prove by using stability result and integrating (30a), (30b), and (31). Next, to prove (42), consider that By Lemma 1, Integration of (34) yields the estimates of |w B ′ (r)|. From the differential equations (33), one can derive the rest of the derivative estimates (42).
Inequalities (43) and (44) can be proved, using eorem 1 and maximum principle for the barrier functions: Hence, it is proved.

□
Remark. e following inequalities are easy to prove, using eorem 1 and Lemma 4:

Finite Difference Method.
e discrete scheme corresponding to the original problems (7)-(9) is as follows:

Hybrid Difference Scheme.
e hybrid scheme corresponding to the original problems (7)-(9) is as follows: where 8 Journal of Mathematics (57)

Numerical Estimates for the Finite Difference Method
Proof.
Proof. e proof of eorem 5 follows from y k � v k + w k , Y k � V k + W k , and eorems 3 and 4.

Numerical Estimates for the Hybrid Difference Method
Assume the following inequality:

Lemma 8. If Ψ(r i ) is discrete solution of problems (53)-(55), then
6.1. Error Estimate. To decompose the numerical solution Y(r i ) into V(r i ) and W(r i ), satisfy the following equations, respectively: Lemma 9. Derive the error estimation of discretization original problems (53)-(56) and regular problem (84) solutions: Proof. e proof of Lemma 9 has the same idea in Lemma 7: Proof. Utilizing the method adopted in [30], Using ε ≤ CN − 1 and the above equation, the bounds on the derivatives of v can be written as □ Lemma 11. Derive the error estimates for singular components bounded by CN − 2 log 2 N: Proof. Note that Now, Consider the mesh functions Clearly, Φ ± (r (5N/2) ) ≥ 0 and Φ ± (r 3N ) ≥ 0, for a suitable choice of C 1 > 0.
e proof of eorem 6 follows from y k � v k + w k and Y k � V k + W k and using eorems 3 and 4 □

Numerical Experiments
In this section, consider two examples for constant and variable coefficient problems and apply both of the numerical methods to find error and rate of convergence. e exact solution is not easy to find in these problems. erefore, we use the double mesh principle: We compute the uniform error and the rate of convergence as To solve the following numerical examples, we use two computational methods such as finite and hybrid difference scheme on the nonuniform mesh.

(102)
We proved that the error is of order O(N − 1 ln N) and O(N − 2 ln 2 N). e theory has been validated with two examples; referring to these numerical results, it can be observed that the proposed method has been effective and applicable.

Discussion
In the literature, many authors have considered singular perturbation problem mixed delay (τ ≪ 1) differential equation. In this paper, we consider a singular perturbation problem with mixed delay (τ � 1) differential equation. We suggested two computational methods such as finite and hybrid difference scheme. We proved that the error is of order O(N − 1 ln N) and O(N − 2 ln 2 N). Finally, two numerical examples are also presented to validate the theoretical results of this study. Maximum pointwise errors and order of convergence of Examples 1 and 2 are given in Tables 1 and 2, respectively.

Data Availability
No data were used to support this study

Conflicts of Interest
e authors declare that they have no conflicts of interest.