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
Proof. Consider
Note that and :, , using Lemma 5; then, the theorem has been proved.
Theorem 3. The error estimates for smooth components bounded by:
Proof. The proof of Theorem 3 has the same idea in [29]:Using Lemma 6, thenTherefore, we get
Theorem 4. Derive the error estimates for singular components bounded by:
Proof. Note thatThen, by (49) and from Theorems 1, 3, we haveNow,Consider the mesh functions:Observe that and , and .
Then, by the Lemma 5, we have . Therefore,
Theorem 5. Ifandare a solution of (7)–(9) and (51), (52),
That is, the order of convergence is almost one.
Proof. The proof of Theorem 5 follows from , , and Theorems 3 and 4.
6. Numerical Estimates for the Hybrid Difference Method
Assume the following inequality:
Lemma 7. Assume (78) holds true. Letsatisfy,; the operatordefined by (53)–(55) satisfies,,; and then, .
Lemma 8. Ifis discrete solution of problems (53)–(55), then
6.1. Error Estimate
To decompose the numerical solution into and , satisfy the following equations, respectively:
Lemma 9. Derive the error estimation of discretization original problems (53)–(56) and regular problem (84) solutions:
Proof. The proof of Lemma 9 has the same idea in Lemma 7:
Lemma 10. The error estimates for smooth components are bounded by:
Proof. Utilizing the method adopted in [30],Using and the above equation, the bounds on the derivatives of can be written asThen, we have . Similarly, , , and, by Lemma 8, we have
Lemma 11. Derive the error estimates for singular components bounded by:
Proof. Note thatNow,Consider the mesh functionsClearly, and , for a suitable choice of .Then, by Lemma 7, we have . Therefore,
Theorem 6. Ifandare the solution of (7)–(9) and (53)–(56), then
Proof. The proof of Theorem 6 follows from and and using Theorems 3 and 4
7. 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. The exact solution is not easy to find in these problems. Therefore, 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.
Example 1.
Example 2. We proved that the error is of order and . The 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.
8. Discussion
In the literature, many authors have considered singular perturbation problem mixed delay differential equation. In this paper, we consider a singular perturbation problem with mixed delay differential equation. We suggested two computational methods such as finite and hybrid difference scheme. We proved that the error is of order and . 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
The authors declare that they have no conflicts of interest.