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Research Article

The solution for singularly perturbed differential-difference equation with boundary layers at both ends by a numerical integration method

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Pages 119-137 | Received 26 Feb 2022, Accepted 26 Jan 2024, Published online: 06 Feb 2024

References

  • M. Adilaxmi, D. Bhargavi, and K. Phaneendra, Numerical solution of singularly perturbed differential-difference equations using multiple fitting factors, Commun. Math. Appl. 10(4) (2019), pp. 681–691.
  • E. Angel, and R. Bellman, Dynamic Programming and Partial Differential Equations, Academic Press, New York, 1972.
  • R.E. Bellman, and K.L. Cooke, Differential-Difference Equations, Academic Press, London, 1963.
  • C.M. Bender, and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
  • E. Cimen, Uniformly convergent numerical method for a singularly perturbed differential difference equation with mixed type, Bull. Belg. Math. Soc. Simon Stevin 27(5) (2020), pp. 755–774. DOI: 10.36045/j.bbms.200128.
  • 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, Ireland, 1980.
  • R.D. Driver, Ordinary and Delay Differential Equations, Springer, New York, 1977.
  • L.E. El’sgol’ts, and S.B. Norkin, Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Academic Press, New York, 1973.
  • G. File, and Y.N. Reddy, Numerical integration of a class of singularly perturbed delay differential equations with small shift, Int. J. Differ. Eqn. (2012), Article ID 572723, pp. 1–12.
  • F.Z. Geng, and S.P. Qian, Improved reproducing kernel method for singularly perturbed differential-difference equations with boundary layer behaviour, Appl. Math. Comput 252 (2015), pp. 58–63.
  • J.K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977.
  • P. Hammachukiattikul, E. Sekar, A. Tamilselvan, R. Vadivel, N. Gunasekaran, and P. Agarwal, Comparative study on numerical methods for singularly perturbed advanced-delay differential equations, J. Math. 2021 (2021), pp. 6636607.
  • M.K. Kadalbajoo, and D. Kumar, A computational method for singularly perturbed nonlinear differential-difference equations with small shift, Appl. Math. Model. 34 (2010), pp. 2584–2596.
  • M.K. Kadalbajoo, and Y.N. Reddy, A non-asymptotic method for general linear singular perturbation problems, J. Optim. Theory. Appl. 55 (1986), pp. 257–269.
  • M.K. Kadalbajoo, and K.K. Sharma, Numerical analysis of boundary-value problems for singularly perturbed differential-difference equations with small shifts of mixed type, J. Optim. Theory. Appl. 115(1) (2002), pp. 145–163.
  • M.K. Kadalbajoo, and K.K. Sharma, Numerical analysis of singularly perturbed delay differential equations with layer behaviour, Appl. Math. Comput. 157 (2004), pp. 11–28.
  • C.G. Lange, and R.M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations. VI. small shifts with rapid oscillations, SIAM J Appl. Math. 54 (1994), pp. 273–283.
  • C.G. Lange, and R.M. Miura, Singular perturbation analysis of boundary-value problems for differential difference equations, SIAM J. Appl. Math. 42 (1982), pp. 502–531.
  • C.G. Lange, and R.M. Miura, Singular perturbation analysis of boundary-value problems for differential difference equations II. rapid oscillations and resonances, SIAM J. Appl. Math. 45 (1985), pp. 687–707.
  • B.J. McCartin, Exponential fitting of delayed recruitment/ renewal equation, J. Comput. Appl. Math. 136 (2001), pp. 343–356.
  • J.J.H. Miller, E. O’Riordan, and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, Russ Acad Sci, Russia, 2012.
  • R.K. Mohanty, and N. Jha, A class of variable mesh spline in compression methods for singularly perturbed two-point singular boundary value problems, Appl. Math. Comput. 168 (2005), pp. 704–716.
  • J. Mohapatra, and S. Nateshan, Uniformly convergent numerical method for singularly perturbed differential-difference equation using grid equi-distribution, Int. J. Numer. Methods Biomed. Eng. 27 (2011), pp. 1427–1445.
  • A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1972.
  • R.E. O’Malley, Introduction to Singular Perturbations, Academic, New York, 1974.
  • Y.N. Reddy, and A.T. Awoke, Solving singularly perturbed differential difference equations via fitted method, Appl Appl Math.: Int J. 8(1) (2013), pp. 318–332.
  • S.M. Roberts, A boundary-value technique for singular perturbation problems, J. Math. Anal. Appl. 87 (1982), pp. 489–508.
  • H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, Berlin Heidelberg, 2008. DOI:10.1007/978-3-540-34467-4 (Revised edition).
  • L. Sirisha, K. Phaneendra, and Y.N. Reddy, Mixed finite difference method for singularly perturbed differential difference equations with mixed shifts via domain decomposition, Ain Shams Eng. J. 9 (2018), pp. 647–654.
  • D.K. Swami, K. Phaneendra, and Y.N. Reddy, Accurate numerical method for singularly perturbed differential difference equations with mixed shifts, Khayyam J. Math. 4(2) (2018), pp. 110–122.
  • M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964.

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