Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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EXPONENTIALLY FITTED NUMERICAL SCHEME FOR SINGULARLY PERTURBED DIFFERENTIAL EQUATIONS INVOLVING SMALL DELAYS

  • Received : 2020.09.16
  • Accepted : 2021.02.18
  • Published : 2021.05.30
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Abstract

This paper deals with numerical treatment of singularly perturbed differential equations involving small delays. The highest order derivative in the equation is multiplied by a perturbation parameter 𝜀 taking arbitrary values in the interval (0, 1]. For small 𝜀, the problem involves a boundary layer of width O(𝜀), where the solution changes by a finite value, while its derivative grows unboundedly as 𝜀 tends to zero. The considered problem contains delay on the convection and reaction terms. The terms with the delays are approximated using Taylor series approximations resulting to asymptotically equivalent singularly perturbed BVPs. Inducing exponential fitting factor for the term containing the singular perturbation parameter and using central finite difference for the derivative terms, numerical scheme is developed. The stability and uniform convergence of difference schemes are studied. Using a priori estimates we show the convergence of the scheme in maximum norm. The scheme converges with second order of convergence for the case 𝜀 = O(N-1) and for the case 𝜀 ≪ N-1, the scheme converge uniformly with first order of convergence, where N is number of mesh intervals in the domain discretization. We compare the accuracy of the developed scheme with the results in the literature. It is found that the proposed scheme gives accurate result than the one in the literatures.

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Acknowledgement

This work was supported by the research grant of College of Natural Sciences, Jimma University

References

  1. M. Adilaxmi, D. Bhargavi and Y.N. Reddy, An initial value technique using exponentially fitted non standard finite difference method for singularly perturbed differential difference equations, Applications and applied Mathematics 14 (2019), 245-269.
  2. M. Bahgat and M.A. Hafiz, Numerical Solution of Singularly Perturbed Two-Parameters DDE using Numerical Integration Method, European Journal of Scientific Research 122 (2014), 36-44.
  3. M. Bestehorn and E.V. Grigorieva, Formation and propagation of localized states in extended systems, Ann. Phys. 13 (2004), 423-431. https://doi.org/10.1002/andp.200410085
  4. G.F. Duressa and Y.N. Reddy, Domain decomposition method for singularly perturbed differential difference equations with layer behaviour, International Journal of Engineering & Applied Sciences 7 (2015), 86-102. https://doi.org/10.24107/ijeas.251236
  5. D.D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys. 61 (1989), 41-73. https://doi.org/10.1103/RevModPhys.61.41
  6. M.K. Kadalbajoo and V.P. Ramesh, Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy, Applied Mathematics and Computation 188 (2007), 1816-1831. https://doi.org/10.1016/j.amc.2006.11.046
  7. T. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning point, Math. Comput. 32 (1978), 1025-1039. https://doi.org/10.1090/S0025-5718-1978-0483484-9
  8. D. Kumar and M.K. Kadalbajoo, Numerical treatment of singularly perturbed delay differential equations using B-Spline collocation method on Shishkin mesh, Journal of Numerical Analysis, Industrial and Applied Mathematics 7 (2012), 73-90.
  9. C.G. Lange and R.M. Miura, Singular perturbation analysis of boundary value problems for differential-difference equations, SIAM Journal on Applied Mathematics 42 (1982), 502-531. https://doi.org/10.1137/0142036
  10. Q. Liu, X. Wang and D.D. Kee, Mass transport through swelling membranes, Int. J. Eng. Sci. 43 (2005), 1464-1470. https://doi.org/10.1016/j.ijengsci.2005.05.010
  11. M.C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science 197 (1977), 287-289. https://doi.org/10.1126/science.267326
  12. W.G. Melesse, A.A. Tiruneh, and G.A. Derese, Solving second order singularly perturbed delay differential equations with layer behavior via initial value method, J. Appl. Math. & Informatics 36 (2018), 331-348. https://doi.org/10.14317/JAMI.2018.331
  13. W.G. Melesse, A.A. Tiruneh, and G.A. Derese, Fitted mesh method for singularly perturbed delay differential turning point problems exhibiting twin boundary layers, J. Appl. Math. & Informatics 38 (2020), 113-132. https://doi.org/10.14317/jami.2020.113
  14. 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, 2012.
  15. R.E. O'Malley, Singular perturbation methods for ordinary differential equations, Springer Science, New York, 1991.
  16. K. Phaneendra and M. Lalu, Numerical solution of singularly perturbed delay differential equations using gaussion quadrature method, IOP Journal Physics. Conference Series 1344 (2019), 1-13.
  17. H. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl. 270 (2002), 143-149. https://doi.org/10.1016/S0022-247X(02)00056-2
  18. D.Y. Tzou, Micro-to-macro scale Heat Transfer, Taylor & Francis, Washington DC, 1997.
  19. M. Wazewska-Czyzewska and A. Lasota, Mathematical models of the red cell system, Mat. Stos. 6 (1976), 25-40.
  20. M.M. Woldaregay and G.F. Duressa, Parameter uniform numerical method for singularly perturbed differential difference equations, J. Nigerian Math. Soc. 38 (2019), 223-245.
  21. M.M. Woldaregay and G.F. Duressa, Uniformly convergent numerical method for singularly perturbed delay parabolic differential equations arising in computational neuroscience, Kragujevac J. Math. 46 (2022), 65-84.
  22. M.M. Woldaregay and G.F. Duressa, Fitted Numerical Scheme for Singularly Perturbed Differential Equations Having Small Delays, Caspian Journal of Mathematical Sciences 10 (2021), in press.
  23. M.M. Woldaregay and G.F. Duressa, Robust Numerical Scheme for Solving Singularly Perturbed Differential Equations Involving Small Delays, Applied Mathematics E-Notes 2020 (2020), in press.
  24. M.M. Woldaregay and G.F. Duressa, Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts, International Journal of Differential Equations 2020 (2020), 1-15.