Uniformly convergent fitted operator method for singularly perturbed delay differential equations

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia.

2 Department of Mathematics, Jimma University, Jimma, Ethiopia.

Abstract

This paper deals with the numerical treatment of singularly perturbed delay differential equations having a delay on the first derivative term. The solution of the considered problem exhibits boundary layer behavior on the left or right side of the domain depending on the sign of the convective term. The term with the delay is approximated using Taylor series approximation, resulting in an asymptotically equivalent singularly perturbed boundary value problem. The uniformly convergent numerical scheme is developed using exponentially fitted finite difference method. The stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. Numerical examples are considered to validate the theoretical analysis. 

Keywords

References

  • [1]          Z. Ai-qing and Y. Xu-ming, Wavelet numerical detection for singularly perturbed elliptic boundary value problems, Wuhan University Journal of Natural Sciences, 5(4),(2000), 397-409.
  • [2]          A. Ashyralyev, D. Agirseven, and RP. Agarwal, Stability estimates for delay parabolic differential and difference equations, Appl. Comput. Math. 19(2)(2020), 175-204.
  • [3]          C.TH. Baker, GA. Bocharov, and FA. Rihan, A report on the use of delay differential equations in numerical modeling in the biosciences, Citeseer, 1999.
  • [4]          IT. Daba and GF. Duressa, A Robust computational method for singularly perturbed delay parabolic convection- diffusion equations arising in the modeling of neuronal variability, Computational Methods for Differential Equa- tions, 9 (2021), DOI:10.22034/cmde.2021.44306.1873. In press.
  • [5]          G. File and YN Reddy, Terminal boundary-value technique for solving singularly perturbed delay differential equations, Journal of Taibah University for Science, 8 (2014) 289–300.
  • [6]          G. File and YN Reddy, Computational Method for Solving Singularly Perturbed Delay Differential Equations with Negative Shift, International Journal of Applied Science and Engineering, 11(1) (2013), 101-113.
  • [7]          G. File and YN Reddy, A non-asymptotic method for singularly perturbed delay differential equations, J. Appl. Math. & Informatics, 32(1-2) (2014), 39–53.
  • [8]          G. File and YN Reddy, Numerical Integration of a Class of  Singularly  Perturbed  Delay  Differential  Equations  with Small Shift, International Journal of Differential Equations, 2012 (2012), 1-12.
  • [9]          K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, volume 74. Springer Science & Business Media, 2013.
  • [10]        MK. Kadalbajoo and KC. Patidar, A survey of numerical techniques for solving singularly perturbed ordinary differential equations, Applied Mathematics and Computation, 130(2-3) (2002), 457–510.
  • [11]        MK. Kadalbajoo and V. Gupta, A brief survey on numerical methods for solving singularly perturbed problems, Applied mathematics and computation, 217(8) (2010), 3641–716.
  • [12]        MK. Kadalbajoo and VP. Ramesh, Hybrid method for numerical solution of singularly perturbed delay differential equations, Applied Mathematics and Computation, 187 (2007), 797–814.
  • [13]        MK. Kadalbajoo and VP. Ramesh, Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy, Appl. Math. Comput., 188 (2007), 1816–1831.
  • [14]        Y. Kuang, Delay differential equations: with applications in population dynamics, Academic Press, 1993.
  • [15]        D. Kumar, A computational technique for solving boundary value problem with two small parameters, Electronic Journal of Differential Equations, 2013(30) (2013), 1–10.
  • [16]        JJ.H. Miller, E. O’Riordan, and GI. Shishkin, Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions, World Scientific, 2012.
  • [17]        O. Ocalan, N. Kilic, UM. Ozkan, and S. Ozturk, Oscillatory behavior for nonlinear delay differential equation with several non-monotone arguments, Computational Methods for Differential Equations, 8(1) (2020), 14-27.
  • [18]        R. E. O’Malley, Introduction to Singular Perturbations, Academic Press, New York, 1974.
  • [19]        K. Phaneendra and M. Lalu, Numerical solution of singularly perturbed delay differential equations using gaussion quadrature method, IOP Conf. Series: Journal of Physics: Conf. Series 1344 (2019) 012013.
  • [20]        V. Ravi Kanth and P. Murali Mohan Kumar, Computational results and analysis for a class of linear and nonlinear singularly perturbed convection delay problems on Shishkin mesh, Hacettepe Journal of Mathematics & Statistics, 49(1) (2020), 221–235.
  • [21]        YN. Reddy, GBSL. Soujanya, and K. Phaneendra, Numerical Integration Method for Singularly Perturbed Delay Differential Equations, International Journal of Applied Science and Engineering, 10 (2012), 249–261.
  • [22]        A. Raza and A. Khan, Treatment of singularly perturbed differential equations with delay and shift using Haar wavelet collocation method, Tamkang Journal of Mathematics, 53, (2022), In press.
  • [23]        HG. Roos, M. Stynes, and LR. Tobiska, Numerical methods for singularly perturbed differential equations, Springer Science & Business Media, 2008.
  • [24]        FA. Shah, R. Abass, and J. Iqbal, Numerical solution of singularly perturbed problems using Haar wavelet collo- cation method, Cogent Mathematics & Statistics, 3(1) (2016), 1202504.
  • [25]        FA. Shah and R. Abass, An Efficient Wavelet-Based Collocation Method for Handling Singularly Perturbed Boundary-Value Problems in Fluid Mechanics, International Journal of Nonlinear Sciences and Numerical Simu- lation, 18(6) (2017), 485-94.
  • [26]        H. Tian, The exponential asymptotic stability of singularly perturbed  delay  differential  equations  with  a  bounded  lag, J. Math. Anal. Appl., 270(1) (2002), 143–149.
  • [27]        MM. Woldaregay, WT. Aniley, and GF. Duressa, Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation, Advances in Mathematical Physics, 2021 (2021), 1-13.
  • [28]        MM. Woldaregay and GF. Duressa, Higher-Order Uniformly Convergent Numerical Scheme for Singularly Per- turbed Differential Difference Equations with Mixed Small Shifts, International Journal of Differential Equations, 2020 (2020), 1-15.
Computational Methods for Differential Equations
Volume 10, Issue 2
April 2022
Pages 502-518

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History

  • Receive Date: 14 August 2020
  • Revise Date: 09 June 2021
  • Accept Date: 20 June 2021

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