Abstract
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.
1. Introduction
Consider a nonlinear singularly perturbed delay differential equation in the form under the interval and boundary conditions where is a perturbation parameter and is a delay parameter of . The solution of Equations (1) and (2) must be continuous on [0,1] and differentiable continuously on (0,1). Suppose is smooth function and follows the conditions (1) is a positive constant(2)The growth condition and all real and .
Under the above conditions, Equations (1) and (2) have a unique solution [1] for .
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 [6–9]. 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 time-fractional 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.
2. 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 which serves as a starting point for the function in and expand , around the function ; we get
In general, we can write for
Using the quasilinearization technique, Equations (1) and (2) become with and . Thus, Equation (6) with Equation (7) is linear in . 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 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.
3. Convergence Analysis
The convergence of the sequence of solutions is obtained as follows. For convenience purpose, we refer as in the entire convergence part. Consider the problem with
After quasilinearization, we have a sequence of linear equations defined by the following recurrence relation: where
Let 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 . Thus, by using Green’s function, the integral form of Equation (15) is where the is the Green’s function and determined by [21] where . By using the mean value theorem, we have where . Substituting Equation (18) into Equation (16), we get
Let . On both sides of Equation (18), taking the maximum of the moduli over the region of interest, we get
A simplification yields where . This shows that, given , the sequence 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 with where
4. Continuous Problem
When the delay argument is , sequential expanding for the term in Equation (22) yields where with
The boundary layer appears on the left or right side of the interval depending on the sign of the coefficient , i.e., as or , respectively.
The operator in Equation (25) satisfies Lemma 1.
Lemma 1. Assume is a smooth function satisfying , . Then, implies
Proof. Let be such that and Clearly, ; therefore, and .☐
Now, consider which is opposite to our assumption.
Hence, .
Lemma 2. Let be the solution of Equation (25) with Equation (27); then, we have where is the norm given by .
Proof. Let be two barrier functions specified by ☐
Then, this implies
As implies and since , we have Using Lemma 1, the required estimate is obtained.
Lemma 3. If satisfies Equations (25)–(27), then where where is constant and is positive constant independent of and .
5. Trigonometric Spline
The integration domain [] with mesh size is decomposed into equal subintervals, so that are the nodes with . Let be the exact solution and be an approximation to by the trigonometric spline passing through the points and . Here, meets the interpolatory conditions at and ; also the continuity of first derivatives at the common nodes is fulfilled. For each subinterval, the trigonometric spline function has the form where , and are constants and is a free parameter.
To get the values of the coefficients of Equation (35) in terms of the following are defined:
Using simple calculations, the following expressions are obtained for the coefficients: where , for . Using the first derivative continuity at , that is , we get the following relation:
Here, , and .
6. Method of Solution
At the grid points , Equation (25) may be discretised by
Using Equation (39) in Equation (38) and utilising the first derivatives of using the following estimations: we get
Using Equation (41), we have the following tridiagonal system:
Here,
7. Error Estimate
The truncation error in the proposed numerical scheme is given by
Thus, for different values of , in the approach (Equation (42)), the following different orders are indicated: (1)The scheme of Equation (44) yields the second order approach for any arbitrary choice of α and with and for any value of (2)For , the scheme Equation (44) gives the sixth order
Theorem 4. Let N be an approximate solution to of Equation (1) obtained by the proposed method. Then, the required estimate is for . Here, are positive constants, independent of and .
Proof. Using Lemma 3, we have Therefore Similarly, Now, ☐
The matrix form of the system Equation (42) is where is the matrix of the system Equation (42), and are the corresponding vectors, and is the local truncation error. Thus,
By using a result in [22], we have giving .
Therefore, for . Here, are positive constants, independent of and .
8. Numerical Examples
To show the relevance and validity of the approach, it was implemented for the following problems. The maximum pointwise errors (MAEs) () are determined by using the double mesh principle [3]:
Example 1. with the interval and boundary constraints
Example 2. with the interval and boundary constraints
9. 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 , the accuracy of the computed approximate solutions is significantly improved.
In addition, while the error values generally increase as the perturbation parameter 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 is used. Figures 1–4 depict the layer behaviour at various 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.