Numerical Treatment on Parabolic Singularly Perturbed Differential Difference Equation via Fitted Operator Scheme

This paper proposes a new ﬁ tted operator strategy for solving singularly perturbed parabolic partial di ﬀ erential equation with delay on the spatial variable. We decomposed the problem into three piecewise equations. The delay term in the equation is expanded by Taylor series, the time variable is discretized by implicit Euler method, and the space variable is discretized by central di ﬀ erence methods. After developing the ﬁ tting operator method, we accelerate the order of convergence of the time direction using Richardson extrapolation scheme and obtained O ð h 2 + k 2 Þ uniform order of convergence. Finally, three examples are given to illustrate the e ﬀ ectiveness of the method. The result shows the proposed method is more accurate than some of the methods that exist in the literature.


Introduction
The spatial delay parabolic singularly perturbed differential equation is a differential equation in which the perturbation parameter multiply the highest order derivative, and it has at least one retarded term on the spatial variable. Some mathematical problems can be treated as singularly perturbed problems such as Navier-Stokes equations, atmospheric pollution, turbulent transport, groundwater flow and solute transport, and Black-Scholes model, as presented in the survey paper by Kadalbajoo and Gupta [1]. There are two principle approaches for solving singular perturbation problems: numerical approach and asymptotic approach by Sharma et al. [2]. The numerical solution of the model problem is challenging due to the existence of a boundary layer. Furthermore, when using the classical finite difference method, we are unable to achieve an accurate solution, and the system becomes unstable. To tackle this issue, we need to create a fitted method for uniform or nonuniform meshes. As a result, ε-uniformly convergent numerical methods were constructed, with the order of convergence and error constant being independent of the perturbation parameter ε. For solving singular perturbation problems, some ε-uni-form numerical schemes have been developed in the literature.
Most of the scholars studied on time delayed singularly perturbed problems. Mbroh et al. [3] proposed parameter uniform method for solving a time delay nonautonomous singularly perturbed parabolic differential equation. Clavero and Gracia [4] studied singularly perturbed time-dependent problem of reaction-diffusion type using Richardson extrapolation technique. Woldaregay and Duressa [5] considered a numerical method for both small time delay and large time delay singularly perturbed boundary value problem. Kumar and Kumari [6] show that the influence of a small delay on the solution is extremely sensitive and that a small change in the delay can have a significant impact on the solution. Chakravarthy et al. [7] using the fitted technique, singularly perturbed differential equations with large delays were investigated.
Nowadays, a few scholars have examined numerical solution for spatial delay singularly perturbed parabolic partial differential equations. Bansal and Sharma [8] numerical solutions for a large delay reaction-diffusion problem has been developed. Gupta et al. [9] examined spatial delay parabolic singularly perturbed partial differential equations and its solution using higher order fitted mesh method. Exponentially fitted operator method was developed to solve differential-difference singularly perturbed problem by Woldaregay and Duressa [10]. Das and Natesan [11] presented the solution of delay singularly perturbed partial differential equation using second-order convergent method. Singh and Srinivasan [12] developed Richardson extrapolation method for solving convection-diffusion equations with retarded term. Chahravarthy and Kumar [13] presented adaptive grid method for solving singularly perturbed convectiondiffusion problems with spatial delay. Bansal et al. [14] designed numerical scheme for solving general shift singularly perturbed parabolic convectional diffusion problems.
In this study, a singularly perturbed delayed partial differential equation with small spatial shift right boundary layer problem is decomposed into three piecewise equations which are treated using fitted operator difference methods. To accelerate the order of accuracy in the time variable, the Richardson extrapolation method is applied. The order of convergence of the present method is shown to be second order in both time and spatial variable, whereas the rate of convergence is two. Furthermore, the numerical results of the examples considered shows that the present method has better accuracy compared to some results that appear in the literature.
Proof. Let the barrier function ψ ± and defined as applying Lemma 1, and we get the above bound.☐ Lemma 3. The derivatives of the exact solution uðx, tÞ of the Equation (1) fulfill the following bound for v = 0, 1, and 2.
where M ∈ ℝ is independent of ε.
Proof. The proof of this lemma can be found in [7].☐ Theorem 4. The analytical solution of Equation (1) satisfies Proof. The proof for the bounds of its derivatives is given in [4].☐

Numerical Method
In this section, we develop an exponentially fitted operator difference method to solve Equation (1).

Time Discretization.
A uniform mesh with a time step of k is used to discretize the time domain ½0, T as follows: where k = T/n and in the interval ½0, T, n is the number of subintervals in the time direction. We utilize the implicit Euler's approach to approximate the time derivative term of Equation (1), which results in a system of boundary value problems.
Using Equations (3) and (14), we have where EðxÞ = BðxÞ + ð1/kÞ: The spatial domain ½0, 1 is subdivided as follows using a uniform mesh with a step length of h : where h = 1/m and m is the number of subintervals in spatial direction in the interval ½0, 1. Using expansion of Taylor series expansion, we have Substituting Equation (17) into Equation (15) and rearranging gives with the boundary conditions: and initial condition where Abstract and Applied Analysis Applying central difference approximation for spatial variable of Equation (18), we obtain To obtain more accurate and ε-uniform solution for Equation (22), introducing fitting factor ðσÞ as follows: To obtain the value of the fitting factor, multiply both sides of the equations in (23) by h and evaluating limit as where ρ = h/ε. The solution of Equation (18) comes from the theory of singular perturbation [16] given as where u 0 ðx, tÞ is outer solution.
This implies the approximate solution at ðx i , t j Þ is given as: Using Equation (26), we have and 5 Abstract and Applied Analysis Thus, using Equation (24), (27a), and (27b), we get Substituting Equation (28) into Equation (23), we get the following tri-diagonal system of equations that can be solved using Thomas algorithm.
Since q v ðx i Þ is nonnegative, then the system of Equation (29) becomes diagonally dominant.
Thus, the present method have convergent solution.
Proof. Define the barrier function ψ ± i,j+1 as In the barrier function at the boundary condition, we obtain The barrier function at spatial domain After simplifying the terms, the truncation error becomes which gives Thus, lim ðh,kÞ⟶ð0,0Þ T:E: = 0, which shows that the present method is consistent. Since the method is consistent and stable, then using Lax equivalence theorem, the present method is convergent.

Theorem 7.
Let the exact solution and numerical solution of Equation (1), respectively, are u and U. Then, Proof. For the proof, one can refer [17].☐

Richardson Extrapolation Approach
The Richardson extrapolation approach has been described, and it is designed to improve the accuracy of the computed solutions in the basic scheme.
Let D n 2 ⊆ D 2n 2 , where D 2n 2 is the mesh obtained from bisecting the step size k. Denote the numerical solution obtained from D 2n 2 by U j ðxÞ, we have where R n j ðxÞ and R 2n j ðxÞ are the remainder terms of the error. Now, subtracting the inequality (42b) from (42a) to obtain the extrapolation formula.
which gives that is an approximate solution [12].

Theorem 8.
Let uðx i , t j+1 Þ and U ext i,j+1 be the solution of problems in (1) and (44), respectively, then the proposed scheme satisfies the following error estimate Proof. Using the error for the temporal and spatial discretization gives the required bound.☐

Numerical Examples
To determine the efficacy of the current scheme, we looked at model problems that had been addressed in the literature and had approximate solutions that could be compared. We used the double-mesh principle to estimate the absolute maximum error of the current approach when the exact solution for the given problem was unknown. We use the following formula to approximate the absolute maximum error at the selected mesh points: Case 1. If the exact solution is known, Case 2. If the exact solution is unknown,

Discussions and Results
We have presented the method for solving spatial delayed singularly perturbed parabolic partial differential equation. The basic mathematical procedures are defining the model problem, decomposing into three equations, approximating time variable using implicit Euler's method, approximating the delay term using Taylor series expansion of order two, approximating the spatial variable using the central difference method, and finding fitting factor. Finally, apply Richardson extrapolation method to accelerate the accuracy of the method.
Three model examples are used to exemplify the performance of the proposed method. The maximum error and rate of convergence are shown in Tables 1-3 with different values of ε, delay parameters, and mesh length. The physical    behavior of the solution are shown in Figures 1-4. We examined the suggested numerical scheme for stability, consistency, and ε uniform convergence. As shown in the result, the current method is second-order convergent with respect to time and spatial variables, the rate of convergence is two and more accurate than some of the methods that appear in the literature.

Data Availability
No data were used to support the study.

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