Asymptotic and numerical methods for solving singularly perturbed differential difference equations with mixed shifts

Document Type : Research Article

Authors

• S. Priyadarshana ¹
• S.R. Sahu ²
• J. Mohapatra ¹

¹ Department of Mathematics, National Institute of Technology Rourkela-769008, Odisha, India.

² Department of Mathematics, The ICFAI University Tripura-799210, Tripura, India

10.22067/ijnao.2021.70731.1038

Abstract

This article deals with an effcient approximation method named successive complementary expansion method (SCEM) for solving singularly perturbed differential-difference equations with mixed shifts. It is compared with the method of matched asymptotic expansion (MMAE) and the parameter uniform upwind finite difference scheme for solving such a model. The comparison shows, unlike the MMAE, the SCEM method requires no matching procedure. It requires less computation when compared to the upwind finite difference scheme on the Shishkin mesh. The error analysis is carried out to prove the robustness of the method. Some numerical experiments are provided, which show the effectiveness of the proposed method.

Keywords

• Singular perturbation
• Mixed shifts
• Asymptotic expansion
• MMAE
• SCEM
• Upwind scheme

20.1001.1.24236977.2022.12.1.3.7

Main Subjects

• Numerical methods for differential equations and its applications

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References

1. Amiraliyev, G.M., Amiraliyeva, I.G. and Kudu, M. A numerical treatment for singularly perturbed differential equations with integral boundary condition, Appl. Math. Comput. 185(1) (2007), 574–582.
2. Arora, G. and Kaur, M. Numerical simulation of singularly perturbed differential equation with small shift, AIP Conf. Proc. 1860 (1) (2017), 020047.

 3. Bush, A.W. Perturbation methods for engineers and scientists, CRC Press, London, 1992.
4. Cengizci, S., Natesan, S. and Atay, M.T. An asymptotic-numerical hybrid method for singularly perturbed system of two-point reaction-diffusion boundary-value problems, Turk. J. Math. 43, (2019), 460–472.
5. Chakravarthy, P.P. and Gupta, T. Numerical solution of a weakly coupled system of singularly perturbed delay differential equations via cubic spline in tension, Natl. Acad. Sci. Lett. 43(3), (2020), 259–262.
6. Chakravarthy, P.P. and Kumar, K. A novel method for singularly perturbed delay differential equations of reaction-diffusion type, Differ. Equ. Dyn. Syst. (2017), 1–12.
7. Cousteix, J. and Mauss, J. Asymptotic analysis and boundary layers, Springer Science and Business Media, (2007).
8. Duressa, G.F. and Reddy, Y.N. Domain decomposition method for singularly perturbed differential difference equations with layer behavior, Int. J. Eng. Appl. Sci. 7 (1), (2015), 86–102.
9. Kadalbajoo, M.K. and Sharma, K.K. Numerical analysis of singularly perturbed delay differential equations with layer behavior, APPL. MATH. Comput. 157(1), (2004), 11–28.
10. Kadalbajoo, M.K. and Sharma, K.K. Numerical treatment of a mathematical model arising from a model of neuronal variability, J. MATH. ANAL. APPL. 307(2), (2005), 606–627.
11. Kellogg, R.B. and Tsan, A. Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comput. 32(144), (1978), 1025–1039.
12. Kokotovic, P., Khali, H.K. and O’reilly, J. Singular perturbation methods in control: analysis and design, SIAM Books, 25, (1999).
13. Kudu, M., Amirali, I. and Amiraliyev, G.M. A layer analysis of parameterized singularly perturbed boundary value problems, Int. J. Appl. Math, 29(4), (2016), 439–449.
14. Kudu, M., Amirali, I. and Amiraliyev, G.M. Uniform numerical approximation for parameter dependent singularly perturbed problem with integral boundary condition, Miskolc Mathematical Notes, 19(1), (2018), 337–353, DOI: 10.18514/MMN.2018.2455
15. Lange, C.G. and 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), (1994), 249–272.

16. Lange, C.G. and Miura, R.M. Singular perturbation analysis of boundaryvalue problems for differential-difference equations. VI. small shifts with rapid oscillations, SIAM J. Appl. Math. 54(1), (1994), 273–283.
17. Mauss, J. and Cousteix, J. Uniformly valid approximation for singular perturbation problems and matching principle, Comptes Rendus Mecanique, 330(10), (2002), 697–702.
18. Melesse, W.G.,Tiruneh, A.A. and Derese, G.A. Solving linear secondorder singularly perturbed differential difference equations via initial value method, Int. J. Differ. Equ. (2019).
19. Mohapatra, J. and Natesan, S. Uniformly convergent second order numerical method for a singularly perturbed delay differential equation using a Shishkin mesh, Neural, Parallel and Sci. Comput. 16(3), (2008), 353–370.
20. Mohapatra, J. and Natesan, S. Uniformly convergent numerical method for singularly perturbed differential-difference equation using grid equidistribution, Int. J. Numer. Method Biomed. Eng. 27(9), (2011), 1427–1445.
21. Mohapatra, J. and Natesan, S. Parameter-uniform numerical methods for singularly perturbed mixed boundary value problems using grid equidistribution, J. Appl. Math. Comput. 37(1), (2011), 247–265.
22. Mushahary, P., Sahu, S.R. and Mohapatra, J. A parameter uniform numerical scheme for singularly perturbed differential-difference equations with mixed shifts, J. Appl. Comput. Mech. 6(2), (2020), 344–356.
23. Rao, R.N. and 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), (2014), 1351–1360.
24. Reddy, N.R. and Mohapatra, J. An effcient numerical method for singularly perturbed two point boundary value problems exhibiting boundary layers, Natl. Acad. Sci. Lett. 38(4), (2015), 355–359.
25. Saberi-Nadjafi, J. and Ghassabzade, F.A. The numerical solution of the singularly perturbed differential-difference equations based on the meshless method, Int. J. Appl. Math. Res. 3(2), (2014), 116.
26. Sirisha, L., Phaneendra, K. and 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), (2018), 647–654.
27. Stein, R.B. Some models of neuronal variability, Biophys. J. 7(1), (1967), 37–68.