Uniform numerical method for singularly perturbed parabolic time delay reaction-diffusion problems arising in control theory
DOI: https://doi.org/10.3846/mma.2026.24165Abstract
This paper introduces a uniform numerical scheme to find approximate solutions for singularly perturbed parabolic time delay reaction-diffusion problems. The scheme utilizes the Crank-Nicolson method for approximating time derivatives, combined with a novel finite difference method for spatial discretization. The stability and uniform convergence of the proposed scheme are investigated. The primary objective of this work is to demonstrate that the proposed scheme achieves a parameter-free error bound of order O(k2 + N−2). To validate the theoretical results, various numerical experiments have been conducted, showing that the proposed scheme yields superior results compared to some existing methods in the literature.
Keywords:
singular perturbation problem, reaction-diffusion equation, Crank-Nicolson method, time delayHow to Cite
Share
License
Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
G. Adomian and R. Rach. Nonlinear stochastic differential delay equations. J. Math. Anal. Appl., 91(1):94–101, 1983. https://doi.org/10.1016/0022-247X(83)90094-X
C. Clavero and J.C. Jorge. Efficient numerical methods for semilinear one dimensional parabolic singularly perturbed convection-diffusion systems. Appl. Numer. Math., 198:461–473, 2024. https://doi.org/10.1016/j.apnum.2024.02.006
I.T. Daba, N.E. Kebede and G.G. Gonfa. Extended B-spline collocation method for singularly perturbed time delay parabolic differentialdifference problems. Int. J. Comput. Methods, 22(01):2450046, 2025. https://doi.org/10.1142/S0219876224500464
E.P. Doolan, J.J.H. Miller and W.H.A. Schilders. Uniform numerical methods for problems with initial and boundary layers. Boole Press, Dublin, 1980.
I.R. Epstein. Delay effects and differential delay equations in chemical kinetics. Int. Rev. Phys. Chem., 11(1):135–160, 1992. https://doi.org/10.1080/01442359209353268
F.W. Gelu, I.T. Daba, W.G. Melesse and G.D. Kebede. Fitted mesh numerical method for two-parameter singularly perturbed partial differential equations with large time lag. Partial Differ. Equ. Appl. Math., 11:100844, 2024. https://doi.org/10.1016/j.padiff.2024.100844
M.J. Huntul. Recovering a source term in the higher-order pseudoparabolic equation via cubic spline functions. Phys. Scr., 97(3):035004, 2022. https://doi.org/10.1088/1402-4896/ac54d0
M.J. Huntul, A.T. Ramazanova and Y.T. Mehraliyev. On the solvability of an inverse boundary problem for the equation of transverse oscillation of the rod. Math. Methods Appl. Sci., 47(18):14629–14638, 2024. https://doi.org/10.1002/mma.10294
K. Khari and V. Kumar. Finite element analysis of the singularly perturbed parabolic reaction-diffusion problems with retarded argument. Numer. Methods Partial Differential Equations, 38(4):997–1014, 2022. https://doi.org/10.1002/num.22785.
P.M.M. Kumar and A.S.V. Ravi Kanth. Computational study for a class of time-dependent singularly perturbed parabolic partial differential equation through tension spline. Comput. Appl. Math., 39:1–19, 2020. https://doi.org/10.1007/s40314-020-01278-5
S. Kumar and M. Kumar. High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay. Comput. Math. Appl., 68(10):1355–1367, 2014. https://doi.org/10.1016/j.camwa.2014.09.004
B.J. McCartin. Discretization of the semiconductor device equations. In New Problems and New Solutions for Device and Process Modelling, pp. 72–82, Dublin, 1985. Boole Press.
E.A. Megiso, M.M. Woldaregay and T.G. Dinka. Fitted tension spline method for singularly perturbed time delay reaction diffusion problems. Math. Probl. Eng., 2022(1):8669718, 2022. https://doi.org/10.1155/2022/8669718
Y.T. Mehraliyev, M.J. Huntul and E.I. Azizbayov. Simultaneous identification of the right-hand side and time-dependent coefficients in a twodimensional parabolic equation. Math. Model. Anal., 29(1):90–108, 2024. https://doi.org/10.3846/mma.2024.17974
J.J.H. Miller, E. O’Riordan and G.I. Shishkin. Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific, Singapore, 1996.
N.T. Negero. A robust fitted numerical scheme for singularly perturbed parabolic reaction–diffusion problems with a general time delay. Results Phys., 51:106724, 2023. https://doi.org/10.1016/j.rinp.2023.106724
R.N. Rao and P.P. Chakravarthy. A fitted Numerov method for singularly perturbed parabolic partial differential equation with a small negative shift arising in control theory. Numer. Math. Theory Methods Appl., 7(1):23–40, 2014. https://doi.org/10.4208/nmtma.2014.1316nm
S. Sultan and Z. Zhang. A stable time-dependent mesh method for generalized credit rating migration problem. J. Nonlinear Math. Phys., 30(4):1774–1803, 2023. https://doi.org/10.1007/s44198-023-00157-x
A.N. Tikhonov and A.A. Samarskii. Equations of Mathematical Physics. Courier Corporation, North Chelmsford, MA, 2013. ISBN 9780486641140. Reprint of the 1963 Pergamon Press edition.
A. Van Harten and J.M. Schumacher. On a class of partial functional differential equations arising in feed-back control theory. In North-Holland Mathematics Studies, volume 31, pp. 161–179. Elsevier, 1978. https://doi.org/10.1016/S0304-0208(08)70556-5
P.K.C. Wang. Asymptotic stability of a time-delayed diffusion system. J. Appl. Mech., 30(4):500–504, 1963. https://doi.org/10.1115/1.3636609
J. Wu. Theory and applications of partial functional differential equations, volume 119. Springer Science & Business Media, New York, 2012.
View article in other formats
CrossMark check
Published
Issue
Section
Copyright
Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
License
This work is licensed under a Creative Commons Attribution 4.0 International License.