Acknowledgement
The authors would like to thank the anonymous referees for their helpful comments that improved the quality of this paper.
References
- R.B. Stein, Some models of neuronal variability, Biophysical journal 7 (1967), 37-68. https://doi.org/10.1016/S0006-3495(67)86574-3
- M.W. Derstine, H.M. Gibbs, F.A. Hopf and D.L. Kaplan, Bifurcation gap in a hybrid optically bistable system, Phys. Rev. A 26 (1982), 3720-3722. https://doi.org/10.1103/PhysRevA.26.3720
- M. Wazewska-Czyzewska and A. Lasota, Mathematical models of the red cell system, Mat. Stos. 6 (1976), 25-40.
- A. Longtin and J.G. Milton, Complex oscillations in the human pipil light reflex with mixed and delayed feedback, Math. Biosci. 90 (1988), 183-189. https://doi.org/10.1016/0025-5564(88)90064-8
- V.Y. Glizer, Asymptotic solution of a boundary-value problem for linear singularly perturbed functional differential equations arising in optimal control theory, J. Optim. Theory Appl. 106 (2000), 309-335. https://doi.org/10.1023/A:1004651430364
- V. Gupta, M. Kumar and S. Kumar, Higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations, Numer. Methods Partial Differential Eq. 34 (2018), 357-380. https://doi.org/10.1002/num.22203
- M. Musila and P. Lansky, Generalized Stein's model for anatomically complex neurons, BioSystems 25 (1991), 179-191. https://doi.org/10.1016/0303-2647(91)90004-5
- V.P. Ramesh and M.K. Kadalbajoo, Upwind and midpoint difference methods for time dependent differential difference equations with layer behavior, Applied Mathematics and Computational 202 (2008), 453-471. https://doi.org/10.1016/j.amc.2007.11.033
- D. Kumar and M.K. Kadalbajoo, A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations, Applied Mathematical Modelling 35 (2011), 2805-2819. https://doi.org/10.1016/j.apm.2010.11.074
- K. Bansal, P. Rai and K.K. Sharma, Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments, Differ. Equ. Dyn. Syst. 25 (2017), 326-346.
- K. Bansal and K.K. Sharma, Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments, Numer. Algor. 75 (2017), 113-145. https://doi.org/10.1007/s11075-016-0199-3
- D. Kumar, An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience, Numerical Methods for Partial Differential Equations 34 (2018), 1933-1952. https://doi.org/10.1002/num.22269
- V.P. Ramesh and B. Priyanga, Higher order uniformly convergent numerical algorithm for time-dependent singularly perturbed differential-difference equations, Differ. Equ. Dyn. Syst. 29 (2021), 239-263. https://doi.org/10.1007/s12591-019-00452-4
- M.M. Woldaregay and G.F. Duressa, Parameter uniform numerical method for singularly perturbed parabolic differential difference equations, Journal of the Nigerian Mathematical Society 38 (2019), 223-245.
- I.T. Daba and G.F. Duressa, Extended cubic B-spline collocation method for singularly perturbed parabolic differential-difference equation arising in computational neuroscience, International Journal for Numerical Methods in Biomedical Engineering 37 (2020), e3418.
- V.A. Solonnikov, O.A. Ladyzenskaja, and N.N. Uralceva, Linear and quasi linear equations of parabolic type, American Mathematical Society Providence, Rhode Island, 1988.
- R.E. O'Malley, Introduction to singular perturbations, North-Holland Series in Applied Mathematics & Mechanics, Academic Press Inc., New York, 1974.
- R.B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comput. 32 (1978), 1025-1039. https://doi.org/10.1090/S0025-5718-1978-0483484-9
- H. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded layer, Journal of Mathematical Analysis and Applications 270 (2002), 143-149. https://doi.org/10.1016/S0022-247X(02)00056-2
- R. Ranjan and H.S. Prasad, A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts, Journal of Applied Mathematics and Computing 65 (2021), 403-427. https://doi.org/10.1007/s12190-020-01397-6
- E.P. Doolan, J.J.H. Miller and W.H.A. Schilders , Uniform numerical methods for problems with initial and boundary layers, Doole Press, Dublin, 1980.