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AN EXPONENTIALLY FITTED METHOD FOR TWO PARAMETER SINGULARLY PERTURBED PARABOLIC BOUNDARY VALUE PROBLEMS

  • Received : 2022.02.01
  • Accepted : 2022.06.28
  • Published : 2023.01.31

Abstract

This article devises an exponentially fitted method for the numerical solution of two parameter singularly perturbed parabolic boundary value problems. The proposed scheme is able to resolve the two lateral boundary layers of the solution. Error estimates show that the constructed scheme is parameter-uniformly convergent with a quadratic numerical rate of convergence. Some numerical test examples are taken from recently published articles to confirm the theoretical results and demonstrate a good performance of the current scheme.

Keywords

References

  1. P. H. Bhathawala and A. P. Verma, A two-parameter singular perturbation solution of one dimension flow through unsaturated porous media, Appl. Math. 43 (1975), 380-384.
  2. T. A. Bullo, G. F. Duressa, and G. A. Degla, Higher order fitted operator finite difference method for two-parameter parabolic convection-diffusion problems, Inter. J. Engin. App. Sci. 11 (2019), 455-467.
  3. T. A. Bullo, G. F. Duressa, and G. A. Degla, Robust finite difference method for singularly perturbed two-parameter parabolic convection-diffusion problems, Int. J. Comput. Methods 18 (2021), no. 2, Paper No. 2050034, 17 pp. https://doi.org/10.1142/S0219876220500346
  4. P. P. Chakravarthy and M. Shivhare, Numerical study of a singularly perturbed two parameter problems on a modified Bakhvalov mesh, Comput. Math. Math. Phys. 60 (2020), no. 11, 1778-1786. https://doi.org/10.1134/S0965542520110111
  5. I. T. Daba and G. F. Duressa, Extended cubic B-spline collocation method for singularly perturbed parabolic differential-difference equation arising in computational neuroscience, Int. J. Numer. Methods Biomed. Eng. 37 (2021), no. 2, Paper No. e3418, 20 pp.
  6. I. T. Daba and G. F. Dureessa, A robust computational method for singularly perturbed delay parabolic convection-diffusion equations arising in the modeling of neuronal variability, Comput. Methods Differ. Equ. 10 (2022), no. 2, 475-488. https://doi.org/10.22034/cmde.2021.44306.1873
  7. P. Das, A higher order difference method for singularly perturbed parabolic partial differential equations, J. Difference Equ. Appl. 24 (2018), no. 3, 452-477. https://doi.org/10.1080/10236198.2017.1420792
  8. P. Das and V. Mehrmann, Numerical solution of singularly perturbed convectiondiffusion-reaction problems with two small parameters, BIT 56 (2016), no. 1, 51-76. https://doi.org/10.1007/s10543-015-0559-8
  9. E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dun Laoghaire, 1980.
  10. K. Ganesh and K. Phaneendra, Computational technique for two parameter singularly perturbed parabolic convection-diffusion problem, J. Math. Comput. Sci. 10 (2020), 1251-1261.
  11. V. Gupta, M. K. Kadalbajoo, and R. K. Dubey, A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters, Int. J. Comput. Math. 96 (2019), no. 3, 474-499. https://doi. org/10.1080/00207160.2018.1432856
  12. S. Y. Hahn, J. Bigeon, and J. C. Sabonnadiere, An upwind finite element method for electromagnetic field problems in moving media, Inter. J. Numer. Meth. Engin. 24 (1987), 2071-2086. https://doi.org/10.1002/nme.1620241105
  13. M. K. Kadalbajoo and A. Awasthi, Crank-Nicolson finite difference method based on a midpoint upwind scheme on a non-uniform mesh for time-dependent singularly perturbed convection-diffusion equations, Int. J. Comput. Math. 85 (2008), no. 5, 771-790. https://doi.org/10.1080/00207160701459672
  14. M. K. Kadalbajoo and A. S. Yadaw, Parameter-uniform finite element method for twoparameter singularly perturbed parabolic reaction-diffusion problems, Int. J. Comput. Methods 9 (2012), no. 4, 1250047, 16 pp. https://doi.org/10.1142/S0219876212500478
  15. V. Kumar and B. Srinivasan, A novel adaptive mesh strategy for singularly perturbed parabolic convection diffusion problems, Differ. Equ. Dyn. Syst. 27 (2019), no. 1-3, 203-220. https://doi.org/10.1007/s12591-017-0394-2
  16. T. Linss and H.-G. Roos, Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters, J. Math. Anal. Appl. 289 (2004), no. 2, 355-366. https://doi.org/10.1016/j.jmaa.2003.08.017
  17. T. B. Mekonnen and G. F. Duressa, Computational method for singularly perturbed two-parameter parabolic convection-diffusion problems, Cogent Math. & Stat. 7 (2020), 1829277.
  18. T. B. Mekonnen and G. F. Duressa, Uniformly convergent numerical method for twoparametric singularly perturbed parabolic convection-diffusion problems, J. Appl. Comput. Mech. 7 (2021), 535-545.
  19. T. B. Mekonnen and G. F. Duressa, A fitted mesh cubic spline in tension method for singularly perturbed problems with two parameters, Int. J. Math. Math. Sci. 2022 (2022), Art. ID 5410754, 11 pp. https://doi.org/10.1155/2022/5410754
  20. J. B. Munyakazi, A robust finite difference method for two-parameter parabolic convection-diffusion problems, Appl. Math. & Info. Sci. 9 (2015), 2877.
  21. N. K. Nichols, On the numerical integration of a class of singular perturbation problems, J. Optim. Theory Appl. 60 (1989), no. 3, 439-452. https://doi.org/10.1007/BF00940347
  22. R. E. O'Malley, Jr., Singular perturbations of boundary value problems for linear ordinary differential equations involving two parameters, J. Math. Anal. Appl. 19 (1967), 291-308. https://doi.org/10.1016/0022-247X(67)90124-2
  23. R. E. O'Malley, Jr., Topics in singular perturbations, Adv. in Math. 2 (1968), 365-470. https://doi.org/10.1016/0001-8708(68)90023-6
  24. R. E. O'Malley, Jr., Singular perturbation methods for ordinary differential equations, Applied Mathematical Sciences, 89, Springer-Verlag, New York, 1991. https://doi. org/10.1007/978-1-4612-0977-5
  25. E. O'Riordan and M. L. Pickett, Numerical approximations to the scaled first derivatives of the solution to a two parameter singularly perturbed problem, J. Comput. Appl. Math. 347 (2019), 128-149. https://doi.org/10.1016/j.cam.2018.08.004
  26. E. O'Riordan, M. L. Pickett, and G. I. Shishkin, Singularly perturbed problems modeling reaction-convection-diffusion processes, Comput. Methods Appl. Math. 3 (2003), no. 3, 424-442. https://doi.org/10.2478/cmam-2003-0028
  27. E. O'Riordan, M. L. Pickett, and G. I. Shishkin, Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems, Math. Comp. 75 (2006), no. 255, 1135-1154. https://doi.org/10.1090/S0025-5718-06-01846-1
  28. R. Ranjan and H. S. Prasad, A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts, J. Appl. Math. Comput. 65 (2021), no. 1-2, 403-427. https://doi.org/10.1007/s12190-020-01397-6
  29. H. G. Roos, Layer-adapted meshes: Milestones in 50 years of history, arXiv:1909.08273, 2019.
  30. M. Shivhare, P. C. Podila, and D. Kumar, A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters, J. Math. Chem. 59 (2021), no. 1, 186-215. https://doi.org/10.1007/s10910-020-01190-7
  31. S. Yuzbasi and N. Sahin, Numerical solutions of singularly perturbed one-dimensional parabolic convection-diffusion problems by the Bessel collocation method, Appl. Math. Comput. 220 (2013), 305-315. https://doi.org/10.1016/j.amc.2013.06.027
  32. J. Zhang and Y. Lv, High-order finite element method on a Bakhvalov-type mesh for a singularly perturbed convection-diffusion problem with two parameters, Appl. Math. Comput. 397 (2021), Paper No. 125953, 10 pp. https://doi.org/10.1016/j.amc.2021.125953