DOI QR코드

DOI QR Code

COMPUTATIONAL METHOD FOR SINGULARLY PERTURBED PARABOLIC REACTION-DIFFUSION EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

  • Received : 2021.05.01
  • Accepted : 2021.06.02
  • Published : 2022.01.30

Abstract

In this study, the non-standard finite difference method for the numerical solution of singularly perturbed parabolic reaction-diffusion subject to Robin boundary conditions has presented. To discretize temporal and spatial variables, we use the implicit Euler and non-standard finite difference method on a uniform mesh, respectively. We proved that the proposed scheme shows uniform convergence in time with first-order and in space with second-order irrespective of the perturbation parameter. We compute three numerical examples to confirm the theoretical findings.

Keywords

References

  1. S. Kumar and S.C.S. Rao, A robust overlapping Schwarz domain decomposition algorithm for time-dependent singularly perturbed reaction-diffusion problems, J. Comput. Appl. Math. 261 (2014), 127-138. https://doi.org/10.1016/j.cam.2013.10.053
  2. S.C.S. Rao, S. Kumar and J. Singh, A discrete Schwarz waveform relaxation method of higher order for singularly perturbed parabolic reaction-diffusion problems, J. Math. Chem. 58 (2020), 574-594. https://doi.org/10.1007/s10910-019-01086-1
  3. P. Mishra, K.K. Sharma, A.K. Pani, G. Fairweather, High-order discrete-time orthogonal spline collocation methods for singularly perturbed 1D parabolic reaction-diffusion problems, Numer Methods Partial Differential Eq. 36 (2019), 495-523. https://doi.org/10.1002/num.22438
  4. C. Clavero and J.L. Gracia, A higher order uniformly convergent method with Richardson extrapolation in time for singularly perturbed reaction-diffusion parabolic problems, J. Comput. Appl. Math. 252 (2013), 75-85. https://doi.org/10.1016/j.cam.2012.05.023
  5. J.L. Gracia and E. O'Riordan, Numerical approximation of solution derivatives in the case of singularly perturbed time dependent reaction-diffusion problems, J. Comput. Appl. Math. 273 (2015), 13-24. https://doi.org/10.1016/j.cam.2014.05.023
  6. S. Natesan and S. Gowrisankar, Robust numerical scheme for singularly perturbed parabolic initial-boundary-value problems on equidistributed mesh, CMES 88 (2012), 245-267.
  7. T.A. Bullo, G.F. Duressa and G.A. Delga, Accelerated fitted operator finite difference method for singularly perturbed parabolic reaction-diffusion problems, Computational Methods for Differential Equations 3 (2021). 886-898.
  8. R. Ishwariya, J.J.H. Miller and S.A. Valarmathi, parameter uniform essentially first order convergent numerical method for a parabolic singularly perturbed differential equation of reaction-diffusion type with initial and Robin boundary conditions, arXiv:1906.01598v1 [math.NA] 4 Jun 2019.
  9. K. Sunil, Sumit and H. Ramos, Parameter-uniform approximation on equidistributed meshes for singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions, Applied Mathematics and Computation 392 (2021), DOI:10.1016/j.amc.2020.125677.
  10. F.W. Gelu and G.F. Duressa, A uniformly convergent collocation method for singularly perturbed delay parabolic reaction-diffusion problem, Abstract and Applied Analysis 2021 (2021), 11 pages.
  11. M.M. Woldaregay, W.T. Aniley and G.F. Duressa, Novel numerical scheme for singularly perturbed time delay convection-diffusion equation, Advances in Mathematical Physics 2021 (2021), 13 pages.
  12. D.A. Turuna, M.M. Woldaregay and G.F. Duressa, Uniformly convergent numerical method for singularly perturbed convection-diffusion problems, Kyungpook National University 60 (2020), 629-645.
  13. M.M. Woldaregay and G.F. Duressa, Higher-order uniformly convergent numerical scheme for singularly perturbed differential difference equations with mixed small shifts, International Journal of Differential Equations 2020 (2020), 15 pages.
  14. M.M. Woldaregay and G.F. Duressa, Fitted numerical scheme for solving singularly perturbed parabolic delay partial differential equations, Tamkang Journal of Mathematics 53 (2022), DOI:10.5556/j.tkjm.53.2022.3638.
  15. R.E. Mickens, Nonstandard finite difference models of differential equations, World Scientific, Singapore, 1994.
  16. R.E. Mickens, Advances in the applications of nonstandard finite difference schemes, World Scientific, Singapore, 2005.
  17. K.C. Patidar and K.K. Sharma, Uniformly convergent nonstandard finite difference methods for singularly perturbed differential-difference equations with delay and advance, Int. J. Numer. Methods Eng. 66 (2006), 272-296. https://doi.org/10.1002/nme.1555
  18. R.E. Mickens, Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numer Methods Partial Differential Eq. 23 (2007), 672-691. https://doi.org/10.1002/num.20198
  19. J.M.S. Lubuma and K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 191 (2006), 228-238. https://doi.org/10.1016/j.cam.2005.06.039
  20. J.B. Munyakazi and K.C. Patidar, A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems, Computat. Appl. Math. 32 (2013), 509-519. https://doi.org/10.1007/s40314-013-0033-7