[1] Alam, M.J., Prasad, H.S. and Ranjan, R. A novel fitted method for a class of singularly
perturbed differential-difference equations with small delay exhibiting twin layer or oscillatory
behavior, Comput. Math. Math. Phys. 63(12) (2023), 2528–2550.
[2] Andreev, V.F. and Popov, A.M. Using Richardson’s method to construct high-order accurate
adaptive grids, Comput. Math. Model. 10(3) (1999), 227–238.
[3] Angasu, M.A., Duressa, G.F. and Woldaregay, M.M. Exponentially fitted numerical scheme
for singularly perturbed differential equations involving small delays, J. Appl. Math. Inform.
39(3-4) (2021), 419–435.
[4] Bala, S. and Govindarao, L. and Das, A. and Majumdar, A. Numerical scheme for partial
differential equations involving small diffusion term with non-local boundary conditions, J.
Appl. Math. Comput. 69(6) (2023), 4307–4331.
[5] Balachandran, B. and Kalmár-Nagy, T. and Gilsinn, D.E. Delay differential equations,
Springer, New York, 2009.
[6] Banasiak, J. and Lachowicz, M. Methods of small parameter in mathematical biology,
Springer, New York, 2014.
[7] Bawa, R. K. Spline based computational technique for linear singularly perturbed boundary
value problems, Appl. Math. Comput., 167(1) (2005), 225–236.
[8] Bellen, A. and Zennaro, M. Numerical methods for delay differential equations, Springer,
Cham, Switzerland, 2013.
[9] Bellman, R. E. and Kalaba, R. E. Quasilinearization and nonlinear boundary-value problems,
American Elsevier Publishing Company, New York, 1965.
[10] Bender, C. M. and Orszag, S. A. Advanced mathematical methods for scientists and engi-
neers I: Asymptotic methods and perturbation theory, Springer Science & Business Media,
New York, 2013.
[11] Cooke, K. L. Differential–difference equations, in: International Symposium on Nonlinear
Differential Equations and Nonlinear Mechanics, Elsevier, 1963, 155–171.
[12] Das, A., Govindarao, L. and Mohapatra, J. A second-order weighted monotone numerical
scheme for time-delayed parabolic initial-boundary-value problem involving a small param-
eter, Int. J. Math. Model. Numer. Optim. 12(3) (2022), 233–251.
[13] Das, A. and Natesan, S. Second-order uniformly convergent numerical method for singularly
perturbed delay parabolic partial differential equations, Int. J. Comput. Math. 95(3) (2018),
490–510.
[14] Das, P. Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear
parameterized problems, J. Comput. Appl. Math. 290 (2015), 16–25.
[15] Das, P. A higher order difference method for singularly perturbed parabolic partial differential
equations, J. Difference Equ. Appl. 24(3) (2018), 452–477.
[16] Das, P. An a posteriori based convergence analysis for a nonlinear singularly perturbed system
of delay differential equations on an adaptive mesh, Numer. Algorithms 81(2) (2019), 465–
487.
[17] Das, P. and Natesan, S. A uniformly convergent hybrid scheme for singularly perturbed
system of reaction-diffusion Robin type boundary-value problems, J. Appl. Math. Comput.
41 (2013), 447–471.
[18] Doolan, E.P. and Miller, J.J.H. and Schilders, W.H.A. Uniform numerical methods for
problems with initial and boundary layers, Boole Press, Dublin, 1980.
[19] Driver, R.D. Ordinary and delay differential equations, Springer, New York, 1977.
[20] Duressa, G.F., Daba, I.T. and Deressa, C.T. A systematic review on the solution methodology
of singularly perturbed differential difference equations, Mathematics 11(5) (2023), 1108.
[21] Duressa, G.F. and Debela, H.G. Numerical solution of singularly perturbed differential dif-
ference equations with mixed parameters, J. Math. Model. 9(4) (2021), 691–705.
[22] Fouque, J.-P., Papanicolaou, G. and Sircar, R. and Solna, K. Singular perturbations in
option pricing, SIAM J. Appl. Math. 63(5) (2003), 1648–1665.
[23] Hek, G. Geometric singular perturbation theory in biological practice, J. Math. Biol. 60(3)
(2010), 347–386.
[24] Hutchinson, G.E. Circular causal systems in ecology, Ann. N. Y. Acad. Sci. 50(4) (1948),
221–246.
[25] Joy, D. and Kumar, S.D. Liouville–Green transformation technique to solve singularly per-
turbed delay differential equations of reaction diffusion type, Math. Methods Appl. Sci.
48(10) (2025), 10314–10331.
[26] Joy, D., Kumar, S.D. and Rihan, F.A. Advancing numerical solutions for a system of
singularly perturbed delay differential equations at linear rate, Bound. Value Probl. 2025(1)
(2025), 15.
[27] Joy, D. and Regal, A.M. Tension spline approach to singularly perturbed delay differential
equations involving large delay, Gulf J. Math. 15(2) (2023), 166–174.
[28] Kadalbajoo, M.K. and Gupta, V. A brief survey on numerical methods for solving singularly
perturbed problems, Appl. Math. Comput. 217(8) (2010), 3641–3716.
[29] Kadalbajoo, M.K. and Patidar, K.C. Singularly perturbed problems in partial differential
equations: A survey, Appl. Math. Comput. 134(2–3) (2003), 371–429.
[30] Kadalbajoo, M.K. and Patidar, K.C. and Sharma, K.K. ε-uniformly convergent fitted meth-
ods for the numerical solution of the problems arising from singularly perturbed general
DDEs, Appl. Math. Comput. 182(1) (2006), 119–139.
[31] Kadalbajoo, M.K. and Reddy, Y.N. Numerical solution of singular perturbation problems
via deviating arguments, Appl. Math. Comput. 21(3) (1987), 221–232.
[32] Kadalbajoo, M.K. and Yadaw, A.S. and Kumar, D. Comparative study of singularly per-
turbed two-point BVPs via: Fitted-mesh finite difference method, B-spline collocation method
and finite element method, Appl. Math. Comput. 204(2) (2008), 713–725.
[33] Kaur, J. and Kaur, B. A Review on singularly perturbed problems, IJRAR (2023).
[34] Kuang, Y. Delay differential equations, Academic Press, New York, 1993.
[35] Kumar, D. and Kadalbajoo, M.K. Numerical treatment of singularly perturbed delay differ-
ential equations using B-spline collocation method on Shishkin mesh, J. Numer. Anal. Ind.
Appl. Math. 7(3–4) (2012), 73–90.
[36] Kumar, M. and Singh, P. and Mishra, H.K. A recent survey on computational techniques for
solving singularly perturbed boundary value problems, Int. J. Comput. Math. 84(10) (2007),
1439–1463.
[37] Lakshmikantham, V. and Wen, L. and Zhang, B. Theory of differential equations with
unbounded delay, Springer, New York, 2013.
[38] Lange, C.G. and Miura, R.M. Singular perturbation analysis of boundary value problems for
differential-difference equations, SIAM J. Appl. Math. 42(3) (1982), 502–531.
[39] Lange, C.G. and Miura, R.M. Singular perturbation analysis of boundary-value problems
for differential-difference equations. II. Rapid oscillations and resonances, SIAM J. Appl.
Math. 45(5) (1985), 687–707
[40] Lange, C.G. and Miura, R.M. Singular perturbation analysis of boundary-value problems for
differential-difference equations. III. Turning point problems, SIAM J. Appl. Math. 45(5)
(1985), 708–734.
[41] Lange, C.G. and Miura, R.M. Singular perturbation analysis of boundary-value problems for
differential-difference equations. V. small shifts with layer behavior, SIAM J. Appl. Math.
54(1) (1994), 249–272.
[42] Lange, C.G. and Miura, R.M. Singular perturbation analysis of boundary-value problems
for differential-difference equations. VI. small shifts with rapid oscillations, SIAM J. Appl.
Math. 54(1) (1994), 273–283.
[43] Linß, T. An upwind difference scheme on a novel Shishkin-type mesh for a linear convection–
diffusion problem, J. Comput. Appl. Math. 110(1) (1999), 93–104.
[44] Linß, T. Solution decompositions for linear convection-diffusion problems, Z. Anal. Anwend.
21(1) (2002), 209–214.
[45] Linß, T. and Stynes, M. A hybrid difference scheme on a Shishkin mesh for linear convection–
diffusion problems, Appl. Numer. Math. 31(3) (1999), 255–270.
[46] Longtin, A. and Milton, J.G. Complex oscillations in the human pupil light reflex with mixed
and delayed feedback, Math. Biosci. 90(1–2) (1988), 183–199.
[47] Miller, J.J.H. Singular perturbation problems in chemical physics: Analytic and computa-
tional methods, Wiley, New York, 1997.
[48] Miller, J.J.H. and O’Riordan, E. and Shishkin, G.I. Fitted numerical methods for singular
perturbation problems, World Scientific, Singapore, 1996.
[49] Mukherjee, K. and Natesan, S. Richardson extrapolation technique for singularly perturbed
parabolic convection–diffusion problems, Computing 92(1) (2011), 1–32.
[50] Munyakazi, J.B. and Patidar, K.C. Limitations of Richardson’s extrapolation for a high-
order fitted mesh method for self-adjoint singularly perturbed problems, J. Appl. Math.
Comput. 32(1) (2010), 219–236.
[51] Munyakazi, J.B. and Patidar, K.C. Performance of Richardson extrapolation on some nu-
merical methods for a singularly perturbed turning point problem whose solution has boundary
layers, J. Korean Math. Soc. 51(4) (2014), 679–702
[52] Mushahary, P. and Sahu, S.R. and Mohapatra, J. A parameter uniform numerical scheme
for singularly perturbed differential-difference equations with mixed shifts, J. Appl. Comput.
Mech. 6(2) (2020), 344–356.
[53] Norkin, S.B. Differential equations of the second-order with retarded argument: some prob-
lems of the theory of vibrations of systems with retardation, American Mathematical Society
1972.
[54] Norkin, S.B. Introduction to the theory and application of differential equations with devi-
ating arguments, (Vol. 105), Academic Press, 1973.
[55] O’Malley Jr., R.E. Introduction to singular perturbations, J. Fluid Mech. 1974.
[56] O’Malley, Jr. R.E. Singular perturbation methods for ordinary differential equations,
Springer-Verlag, Berlin, 1991.
[57] O’Malley Jr., R.E., Kevorkian J. and Cole, J.D. Perturbation methods in applied mathemat-
ics, and Ali Hasan Nayfeh, Introduction to perturbation techniques, J. Fluid Mech. 71(4)
(1982), 827–828.
[58] Phaneendra, K. and Lalu, M. Numerical solution of singularly perturbed delay differential
equations using Gaussian quadrature method, J. Phys. Conf. Ser. 1344(1) (2019), 012013.
[59] Prasad, H.S. and Reddy, Y.N. Numerical solution of singularly perturbed differential-
difference equations with small shifts of mixed type by differential quadrature method, Am.
J. Comput. Appl. Math. 2(1) (2012), 46–52.
[60] Ranjan, R. A novel fitted spline method for the numerical treatment of singularly perturbed
differential equations having small delays, J. Appl. Math. Comput. 69(6) (2023), 4645–4664.
[61] Ranjan, R. and Prasad, H.S. A novel exponentially fitted finite difference method for a
class of 2nd order singularly perturbed boundary value problems with a simple turning point
exhibiting twin boundary layers, J. Ambient Intell. Humaniz. Comput. 13(9) (2022), 4207–
4221.
[62] Ranjan, R., Prasad, H.S. and Kiltu, G.G. A novel fitted three-term method for the numerical
treatment of singularly perturbed differential–difference equations, Mediterr. J. Math. 21(2)
(2024), 53.
[63] Raza, A., Khan, A., and Sharma, P. and Ahmad, K. Solution of singularly perturbed differ-
ential difference equations and convection delayed dominated diffusion equations using Haar
wavelet, Math. Sci. 15 (2021), 123–136.
[64] Reddy, Y.N. and Reddy, K.A. Numerical integration method for general singularly perturbed
two-point boundary value problems, Appl. Math. Comput. 133(2–3) (2002), 351–373.
[65] Regal, A.M. and Kumar, S.D. Non-polynomial spline method for singularly perturbed dif-
ferential difference equations with delay and advance terms, Bull. Karaganda Univ. Math.
Ser. 118(2) (2025), 208–221.
[66] Regal, A.M. and Kumar, S.D. Numerical approximation for singularly perturbed differential
equations exhibiting significant positive shift arising in neuronal activity, Appl. Math. Sci.
Eng. 33(1) (2025), 2498414.
[67] Sharma, N. and Kaushik, A. A uniformly convergent difference method for singularly per-
turbed parabolic partial differential equations with large delay and integral boundary condi-
tion, J. Appl. Math. Comput. 69(1) (2023), 1071–1093.
[68] Shishkin, G.I. and Shishkina, L.P. The Richardson extrapolation technique for quasilin-
ear parabolic singularly perturbed convection–diffusion equations, J. Phys. Conf. Ser. 55(1)
(2006), 203–209.
[69] Stein, R.B. Some models of neuronal variability, Biophys. J. 7(1) (1967), 37–68.
[70] Stynes, M. and Roos, H.-G. The midpoint upwind scheme, Appl. Numer. Math. 23(3) (1997),
361–374.
[71] Tian, H. The exponential asymptotic stability of singularly perturbed delay differential equa-
tions with a bounded lag, J. Math. Anal. Appl. 270(1) (2002), 143–149.
[72] Tuckwell, H.C. On the first-exit time problem for temporally homogeneous Markov processes,
J. Appl. Probab. (1976), 39–48.
[73] Villasana, M. and Radunskaya, A. A delay differential equation model for tumor growth, J.
Math. Biol. 47(3) (2003), 270–294.
[74] Wazewska-Czyzewska, M. and Lasota, A. Mathematical models of the red cell system, Mat.
Stos. 6(1) (1976), 25–40.
[75] Woldaregay, M.M. Solving singularly perturbed delay differential equations via fitted mesh
and exact difference method, Res. Math. 9(1) (2022), 2109301.
[76] Woldaregay, M.M. and Duressa, G.F. Higher-order uniformly convergent numerical scheme
for singularly perturbed differential difference equations with mixed small shifts, Int. J. Differ.
Equ. 2020(1) (2020), 6661592.
[77] Woldaregay, M.M. and Duressa, G.F. Robust mid-point upwind scheme for singularly per-
turbed delay differential equations, Comput. Appl. Math. 40(5) (2021), 178.
[78] Woldaregay, M.M. and Duressa, G.F. Robust numerical scheme for solving singularly per-
turbed differential equations involving small delays, Appl. Math. E-Notes, 21 (2021), 622–
633.