Restricted accessResearch articleFirst published online 2026
A robust Simpson’s 3/8 numerical integration method to solve singularly perturbed delay differential and differential–difference equations having boundary layer at one end
In this study, we develop a numerical integration method to solve singularly perturbed delay differential and differential–difference equations both characterized by a boundary layer at one end. Applying Taylor’s series, we approximate the second-order delay differential and differential–difference equations with an asymptotically equivalent first-order differential equation. To solve this equation, we employ the composite Simpson’s 3/8 rule, which leads to the formulation of a three-term recurrence relation. The resulting tri-diagonal system of equations is efficiently solved using the Thomas algorithm. We perform numerical experiments to investigate the effect of small shifts in the delay parameter on the boundary layer solution, focusing on cases where the boundary layer forms either at the left or right boundary. In addition, we carry out a thorough error analysis to assess the accuracy of the method and provide a discussion of its stability and convergence properties. Our numerical results show that the proposed method yields highly accurate approximations, closely matching the exact solutions.
DerstineMGibbsHHopfF, et al. Bifurcation gap in a hybrid optically bistable system. Phys Rev A1982; 26(6): 3720.
2.
GlizerV. Asymptotic analysis and solution of a finite-horizon H infinity control problem for singularly-perturbed linear systems with small state delay. J Optim Theory Appl2003; 117: 295–325.
RobertsS. Further examples of the boundary value technique in singular perturbation problems. J Math Anal Appl1988; 133(2): 411–436.
5.
BenderCMOrszagSA. Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Cham: Springer, 2013.
6.
O’MalleyR. Introduction to singular perturbations. New York: Academic Press, 1974.
7.
MillerJO’RiordanEShishkinG. Fitted numerical methods for singular perturbation problems. Moscow, Russia: Russian Academy of Sciences, 2012.
8.
KadalbajooMKSharmaKK. Numerical analysis of singularly perturbed delay differential equations with layer behavior. Appl Math Comput2004; 157(1): 11–28.
9.
KadalbajooMKKumarD. A computational method for singularly perturbed nonlinear differential-difference equations with small shift. Appl Math Model2010; 34(9): 2584–2596.
10.
SirishaLPhaneendraKReddyY. Mixed finite difference method for singularly perturbed differential difference equations with mixed shifts via domain decomposition. Ain Shams Eng J2018; 9(4): 647–654.
11.
AdilaxmiMBhargaviDPhaneendraK. Numerical solution of singularly perturbed differential-difference equations using multiple fitting factors. Commun Math Appl2019; 10(4): 681.
12.
FileGReddyY. Numerical integration of a class of singularly perturbed delay differential equations with small shift. Int J Differ Equ2012; 2012(1): 572723.
13.
SwamyDKSinghRPReddyY. Numerical integration method for a class of singularly perturbed differential-difference equations. Math Stat Eng Appl2022; 71(4): 5771–5795.
14.
DurmazME. A numerical approach for singularly perturbed reaction diffusion type Volterra-Fredholm integro-differential equations. J Appl Math Comput2023; 69(5): 3601–3624.
15.
DuressaGFDabaITDeressaCT. A systematic review on the solution methodology of singularly perturbed differential difference equations. Mathematics2023; 11(5): 1108.
16.
RanjanRPrasadHS. A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts. J Appl Math Comput2021; 65(1): 403–427.
17.
SinghRPReddyY. The solution for singularly perturbed differential-difference equation with boundary layers at both ends by a numerical integration method. Int J Comput Math2024; 101(2): 119–137.
18.
LangeCGMiuraRM. Singular perturbation analysis of boundary value problems for differential-difference equations. SIAM J Appl Math1982; 42(3): 502–531.
19.
LangeCGMiuraRM. Singular perturbation analysis of boundary value problems for differential-difference equations III. turning point problems. SIAM J Appl Math1985; 45(5): 708–734.
20.
LangeCGMiuraRM. Singular perturbation analysis of boundary-value problems for differential-difference equations. VI. Small shifts with rapid oscillations. SIAM J Appl Math1994; 54(1): 273–283.
El’sgol’tsLENorkinSBCastiJL. Introduction to the theory and application of differential equations with deviating arguments. New York: Academic Press, 1973.
23.
McCartinBJ. Exponential fitting of the delayed recruitment/renewal equation. J Comput Appl Math2001; 136(1–2): 343–356.
24.
CookeKL. Differential–difference equations. In: LasalleJ (ed.) International symposium on nonlinear differential equations and nonlinear mechanics, 1963, pp. 155–171. Amsterdam: Elsevier.
25.
AngelEBellmanR. Dynamic programming and partial differential equations. Academic Press, 1972.
26.
KadalbajooMReddyY. A nonasymptotic method for general linear singular perturbation problems. J Optim Theory Appl1987; 55: 257–269.
27.
KadalbajooMSharmaK. Numerical analysis of boundary-value problems for singularly-perturbed differential-difference equations with small shifts of mixed type. J Optim Theory Appl2002; 115: 145–163.
28.
MohantyRJhaN. A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems. Appl Math Comput2005; 168(1): 704–716.
29.
SirishaLReddyY. Fitted second order scheme for singularly perturbed differential-difference equations. Am J Appl Math Stat2014; 2(5): 136–143.
