A Numerical Approach for Singularly Perturbed Nonlinear Delay Differential Equations Using a Trigonometric Spline

In this paper, a computational procedure for solving singularly perturbed nonlinear delay differentiation equations (SPNDDEs) is proposed. Initially, the SPNDDE is reduced into a series of singularly perturbed linear delay differential equations (SPLDDEs) using the quasilinearization technique. A trigonometric spline approach is suggested to solve the sequence of SPLDDEs. Convergence of the method is addressed. The efficiency and applicability of the proposed method are demonstrated by the numerical examples.

The proposed equation usually plays an important role in illustrating different applications, such as theory of nonpremixed combustion [2], geodynamics [3], oceanic and atmosphere circulation [4], and chemical reactions [5]. More attention has been given in the past to the computational analysis of SPDDEs [6][7][8][9]. However, motivation for the research and solution of the SPNDDE has been increasing in the last few years. These problems may have steep exponential boundary layers as a solution. Classical methods for solving such types of problems are ineffective since a boundary layer structure is present when the perturbation parameter goes to zero. For these equations, effective numerical methods should be established, the accuracy of which does not depend on ε. Hence, in this work, we proposed a higher order numerical scheme using a trigonometric spline which gives more accuracy with a smaller number of mesh points. The existence and originality of the solutions of a SPNDDE with shift were studied by Lange and Miura [10]. The authors in [11] presented a fixed-point strategy to solve a second order SPDDE. The authors in [12] assemble two methodical spectral Legendre's derivative methods to solve numerically the Lane-Emden, Bratu's, and singularly perturbed type equations. For generating numerical spectrum solutions to linear and nonlinear second-order boundary value problems, a new operational matrix approach based on shifted Legendre polynomials is introduced and studied in [13].
In [14], the authors proposed schemes with finite differences for solving the system of SPNDDE. In [15], a B-spline collocation method is constructed to solve Equations (1) and (2). In [16], the authors used shifted Legendre polynomials for studying the spectral collocation approach to solve neutral functional-differential equations with proportional delays. In [17], the Legendre spectral collocation approach is suggested by the authors for handling multipantograph delay boundary value problems. In [18], a new numerical method is proposed for solving a class of delay timefractional partial differential equations. The fractional partial differential equations are reduced into an associated system of algebraic equations that may be solved by some robust iterative solvers using the localization method, which is based on space-time collocation in some appropriate points. In [19], the authors developed a numerical technique for nonlinear singly perturbed two-point boundary value problems based on a noniterative integration method with a modest deviation argument.
The following is a concise summary of the contents of the paper. In Section 2, the approach of quasilinearization and the analysis of convergence are discussed. The continuous problem is discussed in Section 3. In Section 4, the procedure using a trigonometric spline for the solution of the problem is derived. Error estimates of the proposed scheme are discussed in Section 5. Numerical examples and computational results are shown in Section 6. Finally, the Section 7 ends with the conclusion.

The Method of Quasilinearization
Using the method of quasilinearization [20], the given nonlinear differential Equations (1) and (2) are reduced into a sequence of SPLDDEs. We take the initial approximation θ 0 ðsÞ which serves as a starting point for the function θðsÞ in F and expand Fðs, θðsÞ, θ′ðs − δÞÞ, around the function θ 0 ðsÞ; we get In general, we can write for ν = 0, 1, 2, ⋯ Using the quasilinearization technique, Equations (1) and (2) become and F ðνÞ = Fðs, θ ðνÞ , θ′ ðνÞ ðs − δÞÞ. Thus, Equation (6) with Equation (7) is linear in θ ðν+1Þ ðsÞ. Now, we solve the problems given by Equations (6) and (7) using the nonpolynomial spline method. Theoretically, the solution to the nonlinear problem satisfies where θ * ðsÞ is the solution of the nonlinear problem. Computationally, we require Here, Tol. is a prescribed small tolerance. Once the tolerance test is achieved, the iteration is terminated.

Convergence Analysis
The convergence of the sequence of solutions hθ ðνÞ i is obtained as follows. For convenience purpose, we refer Fðs , θ, θ ′ ðs − δÞÞ as FðθÞ in the entire convergence part. Consider the problem with After quasilinearization, we have a sequence hθ ðνÞ i of linear equations defined by the following recurrence 2 Computational and Mathematical Methods relation: where F ′ ðθÞ = ∂FðθÞ/∂θ: Let θ ð0Þ ðsÞ be an initial approximation; then using Equation (12), we have Using Equations (12) and (14), we have Equation (15) is a differential equation of second order in ðθ ðν+1Þ ðsÞ − θ ðνÞ ðsÞÞ. Thus, by using Green's function, the integral form of Equation (15) is where the Gðs, tÞ is the Green's function and determined by [21] G s, t where max s,t jGðs, tÞj = 1/4 . By using the mean value theorem, where θ ðν−1Þ ðsÞ ≤ t ≤ θ ðνÞ ðsÞ. Substituting Equation (18) into Equation (16), we get On both sides of Equation (18), taking the maximum of the moduli over the region of interest, we get A simplification yields where K 1 = ða 2 /ð8εð1 − a 1 /4εÞÞÞ < 1. This shows that, given K 1 < 1, the sequence hθ ðνÞ ðsÞi of linear equations converges quadratically. As a result, to get the approximate solution of Equation (1) with Equation (2), it is required to estimate the solution of the sequence of SPLDDEs of the form where

Continuous Problem
When the delay argument δ is oðεÞ, sequential expanding for the term θ ′ ðν+1Þ ðs − δÞ in Equation (22) yields with θ ν+1 The boundary layer appears on the left or right side of the interval depending on the sign of the coefficient p ν ðsÞ, i.e., as p ν ðsÞ > 0 or p ν ðsÞ < 0, respectively.
where c 1 is constant and M is positive constant independent of h and ε.

Trigonometric Spline
The integration domain [0, 1] with mesh size h = 1/N is decomposed into N equal subintervals, so that s i = ih, i = 0

Computational and Mathematical Methods
where a i , b i , c i , and d i are constants and τ is a free parameter.

Method of Solution
At the grid points s i , Equation (25) may be discretised by Using Equation (39) in Equation (38) and utilising the first derivatives of θ using the following estimations: we get Using Equation (41), we have the following tridiagonal system:

Computational and Mathematical Methods
Here,

Error Estimate
The truncation error in the proposed numerical scheme is given by Thus, for different values of ω, α and β in the approach (Equation (42)), the following different orders are indicated: for α = 1/12, β = 5/12, ω = −1/20ε. Here, k 1 , k 2 , M are positive constants, independent of h and ε.
Proof. Using Lemma 3, we have Therefore

Computational and Mathematical Methods
Similarly, Now, ☐ The matrix form of the system Equation (42) is where A is the matrix of the system Equation (42), θ ðν+1Þ and B are the corresponding vectors, and μ i ðθ ðν+1Þ Þ is the local truncation error. Thus,

Numerical Examples
To show the relevance and validity of the approach, it was implemented for the following problems. The maximum pointwise errors (MAEs) (E K N,ε ) are determined by using the double mesh principle [3]:

Conclusion
To solve a singularly perturbed nonlinear delay differentiation equation, a computational technique is proposed using a trigonometric spline. The SPNDDE is reduced into a series of linear SPDDEs using quasilinearization. A trigonometric spline approach is suggested to solve the sequence of linear SPDDEs. The scheme was implemented on two problems. The values of the maximum absolute errors produced by the suggested scheme are compared to the results in [15,23] presented in Tables 1-4. Comparisons reveal that the suggested scheme outperforms the methods given in [15,23] in terms of maximum error. Results of simulation have shown that as we increase the value of the parameter N, the accuracy of the computed approximate solutions is significantly improved. In addition, while the error values generally increase as the perturbation parameter ε decreases, they are usually within reasonable limits even for small values of it. It is also worth noting that the approach works well even when h ≥ ε is used. Figures 1-4 depict the layer behaviour at various δ values. It has been noticed that when the delay value increases, the thickness of the boundary layer increases as well. The simulation results show that the computational method proposed in this study is capable of giving accurate results for SPNDDE.

Data Availability
The proposed equations usually play an important role in illustrating different applications, such as theory of nonpremixed combustion, geodynamics, oceanic and atmosphere circulation, and chemical reactions.