Almost second-order uniformly convergent numerical method for singularly perturbed convection–diffusion–reaction equations with delay

References

• Roos HG, Stynes M, Tobiska LR. Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems. Berlin: Springer Science & Business Media; 2008.
   Google Scholar
• Bellen A, Zennaro M. Numerical methods for delay differential equations. Oxford: Oxford University Press; 2013.
   Google Scholar
• Kuang Y. Delay differential equations: with applications in population dynamics. California: California University Press; 2012.
   Google Scholar
• Baker C, Bocharov GA, Rihan FA. A report on the use of delay differential equations in numerical modelling in the biosciences. Manchester: Citeseer; 1999.
   Google Scholar
• Mahaffy PR, A'Hearn MF, Atreya SK, et al. The Champollion cometary molecular analysis experiment. Adv Space Res. 1999;23(2):349–359.
   Web of Science ®Google Scholar
• Singh H, Kumar D, Singh J, et al. A reliable numerical algorithm for the fractional Klein-Gordon equation. Eng Trans. 2019;67(1):21–34.
   Google Scholar
• Singh H, Kumar D, Pandey RK. An efficient computational method for the time-space fractional Klein-Gordon equation. Front Phys. 2020;8:281.
   Web of Science ®Google Scholar
• Singh H, Pandey RK, Singh J, et al. A reliable numerical algorithm for fractional advection-dispersion equation arising in contaminant transport through porous media. Phys A: Stat Mech Appl. 2019;527:121077.
   Web of Science ®Google Scholar
• Srivastava HM, Dubey VP, Kumar R, et al. An efficient computational approach for a fractional-order biological population model with carrying capacity. Chaos Soliton Fract. 2020;138:109880.
   Web of Science ®Google Scholar
• Stein RB. Some models of neuronal variability. Biophys J. 1967;7(1):37–68.
   PubMed Web of Science ®Google Scholar
• Glizer VY. 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.
   Web of Science ®Google Scholar
• Ramesh VP, Kadalbajoo MK. Upwind and midpoint upwind difference methods for time-dependent differential difference equations with layer behavior. Appl Math Comput. 2008;202(2):453–471.
   Web of Science ®Google Scholar
• Miller JJ, O'Riordan E, Shishkin GI. Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions. Singapore: World Scientific; 2012.
   Google Scholar
• Bansal K, Sharma KK. ε-uniform numerical technique for the class of time dependent singularly perturbed parabolic problems with state dependent retarded argument arising from generalised Stein's model of neuronal variability. Differ Equ Dyn Syst. 2016;24:1–28.
   Google Scholar
• Bansal K, Sharma KK. Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments. Numer Alg. 2017;75(1):113–145.
   Web of Science ®Google Scholar
• Bansal K, Rai P, Sharma KK. Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments. Differ Equ Dyn Syst. 2017;25(2):327–346.
   Web of Science ®Google Scholar
• Chakravarthy PP, Kumar K. An adaptive mesh method for time dependent singularly perturbed differential-difference equations. Nonlinear Eng. 2019;8(1):328–339.
   Google Scholar
• Gupta V, Kumar M, Kumar S. Higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations. Numer Methods Partial Differ Equ. 2018;34:357–380.
   Web of Science ®Google Scholar
• Kumar D, Kadalbajoo MK. A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations. Appl Math Model. 2011;35(6):2805–2819.
   Web of Science ®Google Scholar
• Kumar D. An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience. Numer Methods Partial Differ Equ. 2018;34(6):1933–1952.
   Web of Science ®Google Scholar
• Pratima R, Sharma KK. Singularly perturbed parabolic differential equations with turning point and retarded arguments. IAENG Int J Appl Math. 2015;45(4):404–409.
   Google Scholar
• Ramesh VP, Priyanga B. Higher order uniformly convergent numerical algorithm for time-dependent singularly perturbed differential-difference equations. Differ Equ Dyn Syst. 2019;27:1–25.
   Google Scholar
• Rao R, Chakravarthy PP. Fitted numerical methods for singularly perturbed 1D parabolic partial differential equations with small shifts arising in the modelling of neuronal variability. Differ Equ Dyn Syst. 2017;25:1–18.
   Google Scholar
• Woldaregay MM, Aniley WT, Duressa GF. Novel numerical scheme for singularly perturbed time delay convection-diffusion equation. Adv Math Phys. 2021;2021:1–13.
   Web of Science ®Google Scholar
• Woldaregay MM, Duressa GF. Fitted numerical scheme for solving singularly perturbed parabolic delay partial differential equations. Tamkang J Math. 2022;53. In press.
   Web of Science ®Google Scholar
• Kumar R, Kumar S. Convergence of three-step Taylor Galerkin finite element scheme based monotone Schwarz iterative method for singularly perturbed differential-difference equation. Numer Funct Anal Optim. 2015;36(8):1029–1045.
   Web of Science ®Google Scholar
• Kumar S, Kumar R. A finite element domain decomposition approximation for a semi-linear parabolic singularly perturbed differential equation. Int J Nonlinear Sci Numer Simul. 2017;18(1):41–55.
   Web of Science ®Google Scholar
• Kumar S, Kumar R. A domain decomposition Taylor Galerkin finite element approximation of a parabolic singularly perturbed differential equation. Appl Math Comput. 2017;293:508–522.
   Web of Science ®Google Scholar
• Tian H. The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag. J Math Anal Appl. 2002;270(1):143–149.
   Web of Science ®Google Scholar
• Bobisud L. Second-order linear parabolic equations with a small parameter. Arch Ration Mech Anal. 1968;27(5):385–397.
   Web of Science ®Google Scholar
• Clavero C, Gracia JL, Jorge JC. Higher order numerical methods for one dimensional parabolic singularly perturbed problems with regular layers. Numer Methods Partial Differ Equ. 2005;21(1):149–169.
   Web of Science ®Google Scholar
• Kellogg T. Analysis of some difference approximations for a singular perturbation problem without turning point. Math Comput. 1978;32:1025–1039.
   Web of Science ®Google Scholar