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A Higher Order Finite Difference Method for a Singularly Perturbed Boundary Value Problem with a Small Negative Shift

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Abstract

This study proposes a uniformly convergent finite difference scheme on a uniform mesh to solve singularly perturbed boundary value problems for second-order ordinary differential-difference equation of the convection-diffusion type. Error estimates are produced for the proposed numerical scheme. The theoretical results are supported by numerical simulations of test problems.

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References

  1. Nayfeh, A.H., Chin, C., Pratt, J.: Perturbation methods in nonlinear dynamics-applications to machining dynamics. J. Manuf. Sci. Eng. 119(4A), 485–493 (1997)

    Article  Google Scholar 

  2. Rihan, F.A.: Delay Differential Equations and Applications to Biology. Springer Nature, Singapore (2021)

    Book  MATH  Google Scholar 

  3. Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary value problems for differential-difference equations. V. small shifts with layer behavior. SIAM J. Appl. Math. 54(1), 249–272 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary-value problems for differential-difference equations. VI. small shifts with rapid oscillations. SIAM J. Appl. Math. 54(1), 273–283 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kadalbajoo, M.K., Sharma, K.K.: Numerical analysis of singularly perturbed delay differential equations with layer behavior. Appl. Math. Comput. 157(1), 11–28 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Kadalbajoo, M.K., Sharma, K.K.: Parameter-uniform fitted mesh method for singularly perturbed delay differential equations with layer behavior. Electron. Trans. Numer. Anal. 23, 180–201 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Kadalbajoo, M.K., Sharma, K.K.: A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations. Appl. Math. Comput. 197(2), 692–707 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Kadalbajoo, M.K., Ramesh, V.P.: Hybrid method for numerical solution of singularly perturbed delay differential equations. Appl. Math. Comput. 187(4), 797–814 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Kadalbajoo, M.K., Kumar, D.: Fitted mesh B-spline collocation method for singularly perturbed differential-difference equations with small delay. Appl. Math. Comput. 204(1), 90–98 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Mohapatra, J., Natesan, S.: Uniform convergence analysis of finite difference scheme for singularly perturbed delay differential equation on an adaptively generated grid. Numer. Math. Theory Methods Appl. 3(1), 1–22 (2020)

    MathSciNet  MATH  Google Scholar 

  11. Rao, R.N., Chakravarthy, P.P.: A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior. Appl. Math. Modell. 37(8), 5743–5755 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rao, R.N., Chakravarthy, P.P.: An exponentially fitted tridiagonal finite difference method for singularly perturbed differential-difference equations with small shift. Ain Shams Eng. J. 5(4), 1351–1360 (2014)

    Article  Google Scholar 

  13. Sahihi, H., Abbasbandy, S., Allahviranloo, T.: Computational method based on reproducing kernel for solving singularly perturbed differential-difference equations with a delay. Appl. Math. Comput. 361, 583–598 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Swamy, D.K., Phaneendra, K., Babu, A.B., Reddy, Y.N.: Computational method for singularly perturbed delay differential equations with twin layers or oscillatory behaviour. Ain Shams Eng. J. 6(1), 391–398 (2015)

    Article  Google Scholar 

  15. Kiltu, G.G., Duressa, G.F., Bullo, T.A.: Computational method for singularly perturbed delay differential equations of the reaction-diffusion type with negative shift. J. Ocean Eng. Sci. 6(3), 285–291 (2021)

    Article  Google Scholar 

  16. Sirisha, L., Phaneendra, K., Reddy, Y.N.: Mixed finite difference method for singularly perturbed differential difference equations with mixed shifts via domain decomposition. Ain Shams Eng. J. 9(4), 647–654 (2018)

    Article  Google Scholar 

  17. Kanth, A.R., Murali, M.K.P.: A numerical technique for solving nonlinear singularly perturbed delay differential equations. Math. Modell. Anal. 23(1), 64–78 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rai, P., Sharma, K.K.: Numerical approximation for a class of singularly perturbed delay differential equations with boundary and interior layer(s). Numer. Algor. 85, 305–328 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chakravarthy, P.P., Kumar, K.: A novel method for singularly perturbed delay differential equations of reaction-diffusion type. Differ. Eq. Dyn. Syst. 29, 723–734 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Subburayan, V., Ramanujam, N.: Uniformly convergent finite difference schemes for singularly perturbed convection diffusion type delay differential equations. Differ. Eq. Dyn. Syst. 29, 139–155 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Woldaregay, M.M., Duressa, G.F.: Robust mid-point upwind scheme for singularly perturbed delay differential equations. Comput. Appl. Math. 40(178), 1–2 (2021)

    MathSciNet  MATH  Google Scholar 

  22. Doolan, E.P., Miller, J.J.H., Schilderr, W.H.A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin (1980)

    Google Scholar 

  23. R.E. O\(^\prime \)Malley, Introduction to Singular Perturbations, Academic Press, New York, (1974)

  24. Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods For Singular Perturbation Problems. World Scientific Publishing Ltd, New Jersey (1996)

    Book  MATH  Google Scholar 

  25. Chawla, M.M.: A sixth-order tridiagonal finite difference method for general non-linear two-point boundary value problems. IMA J. Appl. Math. 24(1), 35–42 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall Inc, Englewood Cliffs, New Jersey (1962)

    MATH  Google Scholar 

  27. Young, D.M.: Iterative Solution of Large Linear Systems. Academic Press, New York (1971)

    MATH  Google Scholar 

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Acknowledgements

The authors wish to thank the National Board for Higher Mathematics, Department of Atomic Energy, Government of India, for their financial support under the project No. 02011/8/2021 NBHM(R.P)/R &D II/7224, dated 24.06.2021.Authors are grateful to the anonymous referees for their valuable suggestions and comments that improved the quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, 632 014, India

    T. Prathap & R. Nageshwar Rao

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  1. T. Prathap
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  2. R. Nageshwar Rao
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Both the authors RNR, PT, have contributed equally to this work.

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Correspondence to R. Nageshwar Rao.

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Prathap, T., Rao, R.N. A Higher Order Finite Difference Method for a Singularly Perturbed Boundary Value Problem with a Small Negative Shift. Int. J. Appl. Comput. Math 9, 101 (2023). https://doi.org/10.1007/s40819-023-01578-4

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