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

Abstract

In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).

1. 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.

The functional differential equation has been multiplied by small parameter in the highest order derivative term called the singularly perturbed mixed delay differential equations. The main determination for such a problem is the study of biological science, epidemics, and population [5–10].

The authors in [11] have considered functional differential equation in singularly perturbed problems, such asand considered the problem of determining the expected time for the generation of action potentials in nerve cells by random synaptic inputs in the dendrites. The general linear second-order functional differential equation with the boundary-value problem arises in the modeling of neuron activation, where and are the variance and drift parameters and is the expected first-exit time. The first-order derivative term corresponds to exponential decay between synaptic and inputs. The undifferentiated terms correspond to excitatory and inhibitory synaptic inputs modeled as a Poisson process with mean rates and ; they produce jumps in the membrane potential of amounts and , which are small quantities and could depend on the voltage. The boundary condition iswhere the values and correspond to inhibitory reversal potential and the threshold value of membrane potential for action potential generation. This 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 , the small delay problem has oscillatory solution that has been discussed in [12]. The same authors discussed the signal transmission problem in [13].

The authors in [14, 15] have considered the singularly perturbed problem with derivative depending on small delay term such asto 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].

The 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].

The 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 [22–25] considered singular perturbation problem with derivative depending on large delay variable, such ashas been developed various numerical schemes are finite and hybrid difference method [22], iterative scheme [26], finite element method [27, 28]. The study in [23] proposed solving singularly perturbed delay differential equation with integral boundary condition using finite difference method.

Throughout 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.

This 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.

Throughout our analysis, we use the following notations: , , , , , . , , , . The parameter and mesh points are independent of and are positive constants. The norm is .

2. Statement of the Problem

Consider the following singularly perturbed mixed delay differential equation:where and are history function on and . Assume that , , , , , and the coefficients are smooth function on . The above problem solution satisfies Problem (1) is rewritten as , wherewith boundary conditions

3. Analytical Results

Lemma 1 (maximum principle). If such that , , , , , , , , , and , then , .
Proof. Let

Clearly, , , , , , and . Consider that ; then, there exists such that and implies that obtain minimum at . If , then .

If , then the function nonnegative is not possible. The following cases are easy to prove the contradiction if . Case (i): : Case (ii): : Case (iii): : Case (iv): : Case (v): : Case (vi): : Case (vii): :

All the cases are contradiction.

Lemma 2 (stability result). Ifis a solution of problems (7)–(9), then

Lemma 3. Ifis a solution of problems (7)–(9), then

Proof. First, to prove is bound on , Integrating the above equation on both sides, we haveTherefore,Using Mean Value Theorem, then , for some and . Then, we have .
To prove is bound on , Integrating the above equation on both sides, we haveTherefore,Using Mean Value Theorem, then , for some and . Then, we have .
Next, to prove is bound on ,Integrating the above equation on both sides, we haveTherefore,Using Mean Value Theorem, then , for some and . Then, we have, .
Hence,

3.1. Shishkin Decomposition

The solution is decomposed into smooth component and -layer component. Furthermore, , where , , and are solutions of the following differential equations.

Obtain reduced problem solution such that

If ,

If ,

If ,

Also, satisfies the following problem: if the singular component ,

Furthermore, we decompose as , where the function is boundary layer component and , are interior layer components.

If the boundary layer ,

If the first interior layer ,

If the second interior layer ,

Theorem 1. Ifandare solutions of problems (7)–(9) and (29a)-(29b), then

Proof. ConsiderClearly, . Note that , , for a suitable choice of .
If ,If ,Following the same process, we have .
Using Lemma 1, then . Therefore, .

Lemma 4. Ifandare the solution of regular and singular component problems (32) and (33), thenwhere k = 0, 1, 2, 3, 4.

Proof. The smooth component derivative bound is easy to prove by using stability result and integrating (30a), (30b), and (31). Next, to prove (42), consider thatNote that , , andBy Lemma 1,Integration of (34) yields the estimates of . From the differential equations (33), one can derive the rest of the derivative estimates (42).
Inequalities (43) and (44) can be proved, using Theorem 1 and maximum principle for the barrier functions:Hence, it is proved.

Remark. The following inequalities are easy to prove, using Theorem 1 and Lemma 4:

4. The Discrete Problem

4.1. Shishkin Mesh

Problems (7)–(9) are convection-diffusion type containing delay term. Then, the layers occur in boundary at and interior at and .

The intervals , , and are partitioned into , , , , , and for each interval mesh points and is transition parameter.

The interior of points is denoted by . Then, the mesh widths are

4.2. Finite Difference Method

The discrete scheme corresponding to the original problems (7)–(9) is as follows:with

4.3. Hybrid Difference Scheme

The hybrid scheme corresponding to the original problems (7)–(9) is as follows:where

5. Numerical Estimates for the Finite Difference Method

Lemma 5 (discrete maximum principle). Ifsatisfies,,,,,, and, then,.

Proof. It is easy to see that , , , , , and . Let .
Then, there exists such that and . Then, attains its maximum at . If , then . Suppose . Case (i): : Case (ii): : Case (iii): : Case (iv): : Case (v): : Case (vi): : Case (vii): :All the cases are a contradiction.

Lemma 6. The discrete solution of (51) and (52) is bounded:

Proof. Consider , where .
Observe and :Using Lemma 5, ,
To decompose numerical solution into and satisfy the following equations, respectively:

Theorem 2. Ifandare a solution of discretization problem (51), (52), and (68), then