ASYMPTOTIC SOLUTION FOR A SYSTEM OF SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS

Tao Feng; Mingkang Ni

doi:10.11948/20250097

2026 Volume 16 Issue 1

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Tao Feng, Mingkang Ni. ASYMPTOTIC SOLUTION FOR A SYSTEM OF SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(1): 458-478. doi: 10.11948/20250097

Citation:

Tao Feng, Mingkang Ni. ASYMPTOTIC SOLUTION FOR A SYSTEM OF SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(1): 458-478. doi: 10.11948/20250097

ASYMPTOTIC SOLUTION FOR A SYSTEM OF SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS

Tao Feng^1, ,,
Mingkang Ni^2,

1.
School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, 233030 Bengbu, China
2.
School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, 200241 Shanghai, China

Author Bio: Email: xiaovikdo@163.com(M. Ni)
Corresponding author: Email: tfeng@aufe.edu.cn(T. Feng)
Fund Project: The first author is supported by the Major Natural Science Foundation of Universities in Anhui Province (No. 2024AH040001) and the Talent Foundation (No. 85106) as well as Scientific Research Foundation (No. ACKYC22085) of Anhui University of Finance & Economics; The second author is supported by the National Natural Science Foundation of China (No. 12371168), and the Science and Technology Commission of Shanghai Municipality (No. 18dz2271000)

Abstract

Abstract

This paper is mainly aimed to investigate the asymptotic solution for a system of singularly perturbed delay differential equations. In order to establish the step—like asymptotic solution, the basic framework of contrast structure theory is employed. Some sufficient criteria for the existence of asymptotic solution will be proposed. After that, by means of the boundary layer functions method and sewing method, the expansion for the asymptotic solution will be constructed and the existence of a uniformly valid continuous solution is proved. Finally, the effectiveness of the established results is validated via a concrete example.
- Asymptotic solution /
- singular perturbation /
- delay /
- boundary layer functions /
- contrast structure
MSC: 34E20, 34B15

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References

[1]	K. Aarthika, V. Shanthi and H. Ramos, A non–uniform difference scheme for solving singularly perturbed 1D–parabolic reaction–convection–diffusion systems with two small parameters and discontinuous source terms, J. Math. Chem., 2020, 58(3), 663–685. doi: 10.1007/s10910-019-01094-1 CrossRef Google Scholar
[2]	V. F. Butuzov, Asymptotic behavior and stability of a stationary boundary–layer solution to a partially dissipative system of equations, Comput. Math. Math. Phys., 2019, 59(7), 1148–1171. doi: 10.1134/S0965542519070145 CrossRef Google Scholar
[3]	V. F. Butuzov, Asymptotic expansion of the solution to a partially dissipative system of equations with a multizone boundary layer, Comput. Math. Math. Phys., 2019, 59(10), 1672–1692. doi: 10.1134/S0965542519100051 CrossRef Google Scholar
[4]	F. Candelier and O. Vauquelin, Matched asymptotic solutions for turbulent plumes, J. Fluid. Mech., 2012, 699, 489–499. doi: 10.1017/jfm.2012.134 CrossRef Google Scholar
[5]	P. P. Chakravarthy and K. Kumar, A novel method for singularly perturbed delay differential equations of reaction–diffusion type, Differ. Equ. Dyn. Syst., 2021, 29(3), 723–734. doi: 10.1007/s12591-017-0399-x CrossRef Google Scholar
[6]	H. G. Debela and G. F. Duressa, Parametric uniform numerical method for singularly perturbed differential equations having both small and large delay, Acta Univ. Sapientiae Math., 2023, 15(1), 54–69. doi: 10.2478/ausm-2023-0004 CrossRef Google Scholar
[7]	S. Elango, Second order singularly perturbed delay differential equations with non–local boundary condition, J. Comput. Appl. Math., 2023, 417, 114498. doi: 10.1016/j.cam.2022.114498 CrossRef Google Scholar
[8]	F. Erdogan and G. M. Amiraliyev, Fitted finite difference method for singularly perturbed delay differential equations, Numer. Algorithms, 2012, 59(1), 131–145. doi: 10.1007/s11075-011-9480-7 CrossRef Google Scholar
[9]	T. Feng and M. K. Ni, Internal layers for a quasi–linear singularly perturbed delay differential equation, J. Appl. Anal. Comput., 2020, 10(4), 1666–1682. Google Scholar
[10]	Z. Gajic, D. Petkovski and N. Harkara, The recursive algorithm for the optimal static output feedback control problem of linear singularity perturbed systems, IEEE T. Automat. Control, 1989, 34(4), 465–468. doi: 10.1109/9.28027 CrossRef Google Scholar
[11]	V. Y. Glizer, Asymptotic analysis and solution of a finite–horizon H_∞ control problem for singularly–perturbed linear systems with small state delay, J. Optim. Theory Appl., 2003, 117(2), 295–325. doi: 10.1023/A:1023631706975 CrossRef Google Scholar
[12]	K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer, Dordrecht, 1992. Google Scholar
[13]	D. Joy and S. D. Kumar, Numerical method for solving weakly coupled systems of singularly perturbed delay differential equations, Comput. Appl. Math., 2024, 43(8), 419. doi: 10.1007/s40314-024-02932-y CrossRef Google Scholar
[14]	D. Joy and S. D. Kumar, Liouville–Green transformation technique to solve singularly perturbed delay differential equations of reaction diffusion type, Math. Meth. Appl. Sci., 2025, 48(10), 10314–10331. doi: 10.1002/mma.10247 CrossRef Google Scholar
[15]	D. Joy, S. D. Kumar and F. A. Rihan, Advancing numerical solutions for a system of singularly perturbed delay differential equations at linear rate, Bound. Value Probl., 2025, 2025, 15. doi: 10.1186/s13661-025-02000-2 CrossRef Google Scholar
[16]	T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Anal. Real World Appl., 2012, 13(4), 1802–1826. doi: 10.1016/j.nonrwa.2011.12.011 CrossRef Google Scholar
[17]	M. Kamenskii and P. Nistri, An averaging method for singularly perturbed systems of semilinear differential inclusions with C₀–semigroups, Set–Valued Anal., 2003, 11(4), 345–357. doi: 10.1023/A:1025604317289 CrossRef Google Scholar
[18]	A. Kaushik, Singular perturbation analysis of bistable differential equation arising in the nerve pulse propagation, Nonlinear Anal. Real World Appl., 2008, 9(5), 2106–2127. doi: 10.1016/j.nonrwa.2007.06.014 CrossRef Google Scholar
[19]	I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Modelling, 2002, 35(3–4), 295–301. doi: 10.1016/S0895-7177(01)00166-2 CrossRef Google Scholar
[20]	C. Kuehn, Multiple Time Scale Dynamics, Springer, Switzerland, 2015. Google Scholar
[21]	C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary value problems for differential–difference equations, SIAM J. Appl. Math., 1982, 42(3), 502–531. doi: 10.1137/0142036 CrossRef Google Scholar
[22]	A. Longtin and J. G. Milton, Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback, Math. Biosci., 1988, 90(1–2), 183–199. doi: 10.1016/0025-5564(88)90064-8 CrossRef Google Scholar
[23]	A. Longtin and J. G. Milton, Insight into the transfer function, gain, and oscillation onset for the pupil light reflex using nonlinear delay–differential equations, Biol. Cybernet., 1989, 61(1), 51–58. doi: 10.1007/BF00204759 CrossRef Google Scholar
[24]	R. A. Mariya and S. D. Kumar, Fitted mesh cubic spline tension method for singularly perturbed delay differential equations with integral boundary condition, Comput. Methods Differ. Equ., 2025, 13(2), 618–633. Google Scholar
[25]	A. A. Melnikova and M. Chen, Existence and asymptotic representation of the autowave solution of a system of equations, Comput. Math. Math. Phys., 2018, 58(5), 680–690. doi: 10.1134/S0965542518050147 CrossRef Google Scholar
[26]	J. D. Murray, Mathematical Biology: I. An Introduction, Springer, New York, 2002. Google Scholar
[27]	A. H. Nayfeh, Perturbation Methods, Wiley, New York, 2000. Google Scholar
[28]	M. K. Ni, On the internal layer for a singularly perturbed system of second–order delay differential equations, Differ. Equ., 2013, 49(8), 941–954. doi: 10.1134/S001226611308003X CrossRef Google Scholar
[29]	M. K. Ni, N. N. Nefedov and N. T. Levashova, Asymptotics of the solution of a singularly perturbed second–order delay differential equation, Differ. Equ., 2020, 56(3), 290–303. doi: 10.1134/S0012266120030027 CrossRef Google Scholar
[30]	Y. F. Pang, M. K. Ni and N. T. Levashova, Internal layer for a system of singularly perturbed equations with discontinuous right–hand side, Differ. Equ., 2018, 54(12), 1583–1594. doi: 10.1134/S0012266118120054 CrossRef Google Scholar
[31]	L. Prandtl, Über flüssigkeitsbewegung bei sehr kleiner reibung (in German), Verhandlg. Ⅲ, Intern. Math. Kongr., Heidelberg, 1904, 484–491. Google Scholar
[32]	N. Sun, S. Y. Shi and W. L. Li, Singular renormalization group approach to SIS problems, Discrete Contin. Dyn. Syst. Ser. B, 2020, 25(9), 3577–3596. Google Scholar
[33]	S. K. Tesfaye, T. G. Dinka, M. M. Woldaregay and G. F. Duressa, Numerical analysis for a singularly perturbed parabolic differential equation with a time delay, Comput. Math. Math. Phys., 2024, 64(3), 537–554. doi: 10.1134/S096554252403014X CrossRef Google Scholar
[34]	R. Vallée, P. Dubois, M. Coté and C. Delisle, Second–order differential–delay equation to describe a hybrid bistable device, Phy. Rev. A, 1987, 36(3), 1327–1332. doi: 10.1103/PhysRevA.36.1327 CrossRef Google Scholar
[35]	A. B. Vasil'eva, On contrast structures of step type for a system of singularly perturbed equations, Comput. Math. Math. Phys., 1994, 34(10), 1215–1223. Google Scholar
[36]	A. B. Vasil'eva and V. F. Butuzov, Asymptotic Expansions of the Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973. Google Scholar
[37]	A. B. Vasil'eva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM, Philadelphia, 1995. Google Scholar
[38]	C. Wang and X. Zhang, Canards, heteroclinic and homoclinic orbits for a slow–fast predator–prey model of generalized Holling type Ⅲ, J. Differential Equations, 2019, 267(6), 3397–3441. doi: 10.1016/j.jde.2019.04.008 CrossRef Google Scholar
[39]	K. Wang, Z. J. Du and J. Liu, Traveling pulses of coupled FitzHugh–Nagumo equations with doubly–diffusive effect, J. Differential Equations, 2023, 374, 316–338. doi: 10.1016/j.jde.2023.07.027 CrossRef Google Scholar
[40]	L. M. Wu, M. K. Ni and H. B. Lu, Internal layer solution of singularly perturbed optimal control problem with integral boundary condition, Qual. Theory Dyn. Syst., 2018, 17(1), 49–66. doi: 10.1007/s12346-017-0261-0 CrossRef Google Scholar
[41]	L. Xu, Z. G. Xu, W. L. Li and S. Y. Shi, Renormalization group approach to a class of singularly perturbed delay differential equations, Commun. Nonlinear Sci. Numer. Simul., 2021, 103, 106028. doi: 10.1016/j.cnsns.2021.106028 CrossRef Google Scholar
[42]	H. Yüce and C. Y. Wang, A boundary perturbation method for circularly periodic plates with a core, J. Sound Vibration, 2009, 328(3), 345–368. doi: 10.1016/j.jsv.2009.08.006 CrossRef Google Scholar