A fourth-order optimal numerical approximation and its convergence for singularly perturbed time delayed parabolic problems

Iranian Journal of Numerical Analysis and Optimization

مقاله 1، دوره 12، شماره 2 - شماره پیاپی 22، آذر 2022، صفحه 250-276   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2021.71891.1053

نویسندگان

J. Mohapatra^{* 1}؛ L. Govindarao²

¹Department of Mathematics, National Institute of Technology Rourkela, 769008, India,

²Department of Mathematics, Amrita School of Engineering Coimbatore, Amrita Vishwa Vidyapeetham, Amrita University, Ettimadai, Tamilnadu, India,

چکیده

This paper presents a numerical solution for a time delay parabolic problem (reaction-diffusion) containing a small parameter. The numerical method combines the implicit Crank–Nicolson scheme for the time  derivative on the uniform mesh and the central difference scheme for the spatial derivative on the Shishkin type meshes. It is shown to be second-order uniformly convergent in time and space. Then Richardson extrapolation technique is applied to enhance the accuracy from second-order to fourth-order. The error analysis is carried out, and the method is proved to be uniformly convergent. These two methods are applied to two test examples in support of the theoretical results.
 

کلیدواژه‌ها

Time delayed parabolic problem؛ Boundary layer؛ Post processing technique؛ Singular perturbation

مراجع

1. Ansari, A.R., Bakr, S.A., and Shishkin, G.I. A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. Comput. Appl. Math. 205(1) (2007), 552–566.
2. Chandru, M., Prabha, T. and Shanthi, V. A hybrid difference scheme for a second-order singularly perturbed reactiondiffusion problem with non-smooth data, Int. J. Appl. Comput. Math. 1(1) (2015), 87–100.
3. Clavero, C., Gracia, J.L. and Jorge, J.C. High-order numerical methods for one-dimensional parabolic singularly perturbed problems with regular layers, Numer. Methods Partial Differential Equations 21 (2005), 149–169.
4. Constantinou, P. and Xenophontos, C. Finite element analysis of an exponentially graded mesh for singularly perturbed problems, Comput. Methods Appl. Math. 15(2) (2015), 135–143.
5. Das, P. and Natesan, S. Richardson extrapolation method for singularly perturbed convection-diffusion problems on adaptively generated mesh, CMES Comput. Model. Eng. Sci. 90(6) (2013), 463–485.
6. Das, A. and Natesan, S. Second-order uniformly convergent numerical method for singularly perturbed delay parabolic partial differential equa-tions, Int. J. Comput. Math. 95(3) (2018), 490–510.
7. Geetha, N., Tamilselvan, A. and Subburayan, V. Parameter uniform numerical method for third order singularly perturbed turning point problems exhibiting boundary layers, Int. J. Appl. Comput. Math. 2(3) (2016), 349–364.
8. Govindarao L. and Mohapatra J. A second-order numerical method for singularly perturbed delay parabolic partial differential equation, Eng. Comput. 36(2) (2019), 420–444.
9. Govindarao, L., Mohapatra, J. and Das, A. A fourth-order numerical
scheme for singularly perturbed delay parabolic problem arising in population dynamics, J. Appl. Math. Comput. 63 (2020), 171–195.
10. Govindarao, L., Sahu, S.R. and Mohapatra, J. Uniformly convergent numerical method for singularly perturbed time delay parabolic problem with two small parameters, Iran. J. Sci. Technol. Trans. A Sci. 43(5) (2019), 2373–2383.
11. Gupta, V., Kumar, M. and Kumar, S. Higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations, Numer. Methods Partial Differential Equations 34(1) (2018), 357–380.
12. Keller, H.B. Numerical Methods for Two-point Boundary Value Problems, Dover, New York, 1992.
13. Kuang, Y. Delay Differential Equations with Applications to Population Biology, Academic Press, New York, 1993.
14. Kudu, M. A parameter uniform difference scheme for the parameterized singularly perturbed problem with integral boundary condition, Adv. Difference Equ. 2018(1), (2018), 1–12.
15. Kumar, D., A parameter-uniform scheme for the parabolicsingularly perturbed problem with a delay in time, Numer. Methods Partial Differ. Equ., 37(1) (2021), 626–642.
16. Kumar, D. and Kumari, P. A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay, Appl. Math. Comput. 59(1) (2019), 179–206.
17. Kumar, D. and Kumari, P. Parameter-uniform numerical treatment of singularly perturbed initial-boundary value problems with large delay, Appl.Numer. Math. 153 (2020), 412–429.
18. Kumar, M. and Sekhara Rao, S.C. High order parameter-robust numerical method for time dependent singularly perturbed reaction-diffusion problems, Computing 90(1-2) (2010), 15–38.
19. Ladyzenskaja, O.A., Solonnikov, V.A. and Uraluceva N.N. Linear and Quasilinear Equations of Parabolic type, (Vol. 23), American Mathematical Soc, 1968.
20. Mohapatra, J. Equidistribution grids for two-parameter convection–diffusion boundary-value problems, J. Math. Model. 2(1) (2014), 1–21.
21. Mohapatra, J. and Natesan, S. Uniformly convergent second-order numerical method for singularly perturbed delay differential equations, Neural Parallel Sci. Comput. 30 (2008), 353–370.
22. Miller, J.J.H., O’Riordan, E. and Shishkin, G.I. Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
23. Salama, A. A. and Al-Amerya, D.G. A higher order uniformly convergent method for singularly perturbed delay parabolic partial differential equations, Int. J. Comput. Math. 12(94) (2017), 2520–2546.
24. Raji Reddy, N. and Mohapatra, J. An efficient numerical method for singularly perturbed two point boundary value problems exhibiting boundary layers, Natl. Acad. Sci. Lett. 4(38) (2015), 355–359.
25. Selvi, P.A. and Ramanujam, N. An iterative numerical method for singularly perturbed reaction-diffusion equations with negative shift, J. Appl. Comput. Math. 296 (2016), 10–23.
26. Sedighi, H.M. and Bozorgmehri, A. Dynamic instability analysis of doubly clamped cylindrical nanowires in the presence of Casimir attraction and surface effects using modified couple stress theory, Acta Mechanica 227(6) (2016), 1575–1591.
27. Shishkin, G.I. and Shishkina, L.P. Difference Methods for Singular Perturbation Problems, Chapman and Hall/CRC, New York, 2008.
28. Shishkin, G.I. and Shishkina, L.P. A richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation, Comp. Math. and Math. Phy. 50(12) (2010), 2003–2022.
29. Stein, R.B., Some models of neuronal variability, Biophys. J. 7 (1967), 37–68.
30. Zheng, Q., Li, X. and Gao, Y. Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs, Appl. Numer. Math. 871 (2015), 46–59.
31. Zheng, Q. and Wang, Y. Y., The midpoint upwind scheme on the Bakhvalov-Shishkin mesh for parabolic singularly perturbed problems, In Journal of Physics: Conference Series, 2012(1) (2021), P. 012047, IOP Publishing.