Accelerated fitted operator finite difference method for singularly perturbed parabolic reaction-diffusion problems

Document Type : Research Paper

Authors

1 Department of Mathematics, College of Natural science, Jimma University, Ethiopia.

2 Institut De Mathematiques et de sciences physiques, Universit D’Abomey Calavi, Benin.

Abstract

This paper deals with the numerical treatment of singularly perturbed parabolic reaction-diffusion initial boundary value problems. Introducing a fitting parameter into the asymptotic solution and applying average finite difference approximation, a fitted operator finite difference method is developed for solving the problem. To accelerate the rate of convergence of the method, Richardson extrapolation technique is applied. The consistency and stability of the proposed method have been established very well to ensure the convergence of the method. Numerical experimentation is carried out on some model problems and both the results are presented in tables and graphs. The numerical results are compared with findings of some methods existing in the literature and found to be more accurate. Generally, the formulated method is consistent, stable, and more accurate than some methods existing in the literature for solving singularly perturbed parabolic reaction-diffusion initial boundary value problems.

Keywords

References

[1] A. Akgl and E. Bonyah, Reproducing kernel Hilbert space method for the solutions of generalized Kuramoto Sivashinsky equation, Journal of Taibah University forScience, 13 (2019), 661-669.
[2] A. Akgl, A. Cordero, and J. R. Torregrosa, A fractional Newton method with 20th-order of convergence and its stability, Applied Mathematics Letters, 98 (2019), 344-351.
[3] A. Akgl, Reproducing kernel Hilbert space method based on reproducing kernel functions for investigating boundary layer flow of a PowellEyring non-Newtonian fluid, Journal of Taibah University for Science, 13 (2019), 858-863, DOI: 10.1080/16583655.2019.1651988.
[4] A. Akgl, A. Cordero, and J. R. Torregrosa, Solutions of fractional gas dynamics equation by a new technique, Math Meth Appl Sci. (2019), 110, DOI: 10.1002/mma.5950.
[5] M. B. Aktas and H. M. Baskonus, New Complex and Hyperbolic Forms for AblowitzKaupNewellSegur Wave Equation with Fourth Order, Applied Mathematics and Nonlinear Sciences, 4, (2019), 105112.
[6] T. Bullo, G. Duressa, and G. DEGLA, Higher Order Fitted Operator Finite Difference Method for Two-Parameter Parabolic Convection-Diffusion Problems, International Journal of Engineering and Applied Sciences (IJEAS), 11 (2019), 455-467.
[7] T. Bullo, G. Duressa, and G. DEGLA, Fitted operator average finite difference method for solving singularly perturbed parabolic convection-diffusion problems, International Journal of Engineering and Applied Sciences (IJEAS), 11 (2019), 414-427.
[8] C. Cattani1 and Ya. Rushchitskii, Cubically nonlinear elastic waves: wave equations and methods of analysis, International Applied Mechanics, 39 (2003), 337.
[9] C. Clavero and J. L. Gracia, A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction diffusion problems, Applied Mathematics and Computation, 218 (2012), 50675080.
[10] C. Clavero and J. L. Gracia, A higher order uniformly convergent method with Richardson extrapolation in time for singularly perturbed reactiondiffusion parabolic problems, Journal of Computational and Applied Mathematics, 252 (2013), 7585.
[11] P. Das and V. Mehrmann, Numerical solution of singularly perturbed convection-diffusionreaction problems with two small parameters, BIT Numer Math, (2015), DOI:10.1007/s10543-015-0559-8.
[12] W. Gao, P. Veeresha, D. G. Prakasha, and H. M. Baskonus, Novel Dynamic Structures of 2019-nCoV with Nonlocal Operator via Powerful Computational Technique, Biology, 9, (2020), 107, doi:10.3390/biology9050107.
[13] W. Gao, H. F. Ismael, H. Bulut, and H. M. Baskonus, Instability modulation for the (2 + 1)− dimension paraxial wave equation and its new optical soliton solutions in Kerr media, Phys. Scr., 95, (2020), 035207.
[14] W. Gao, H. F. Ismael, A. M. Husien, H. Bulut, and H. M. Baskonus, Optical Soliton Solutions of the Cubic-Quartic Nonlinear Schrdinger and Resonant Nonlinear Schrdinger Equation with the Parabolic Law, Appl. Sci., 10(219) (2020), doi:10.3390/app10010219.
[15] W. Gao, H. Rezazadeh, Z. Pinar, H. M. Baskonus, S. Sarwar, and G. Yel, Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Optical and Quantum Electronics, 52 (2020), https://doi.org/10.1007/s11082-019-2162-8.
[16] W. Gao, M. Senel, G. Yel, H. M. Baskonus, and B. Senel, New complex wave patterns to the electrical transmission line model arising in network system, AIMS Mathematics, 5, (2020), 18811892.
[17] W. Gao, G. Yel, H. M. Baskonus, and C. Cattani, Complex solitons in the conformable (2+ 1)−dimensional Ablowitz Kaup-Newell-Segur equation, AIMS Mathematics, 5 (2019), 507521.
[18] S. Gowrisankar and N. Srinivasan, The parameter uniform numerical method for singularly perturbed parabolic reaction diffusion problems on equidistributed grids, Applied Mathematics Letters, 26 (2013), 10531060.
[19] S. Gowrisankar and N. Srinivasan, Robust numerical scheme for singularly perturbed convectiondiffusion parabolic initialboundary-value problems on equidistributed grids, Computer Physics Communications, 185 (2014), 2008-2019.
[20] J. L. Gracia and E. ORiordan, Numerical approximation of solution derivatives in the case of singularly perturbed time dependent reactiondiffusion problems, Journal of Computational and Applied Mathematics, 273 (2015), 1324.
[21] 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, International Journal of Computer Mathematics, (2018), DOI:10.1080/00207160.2018.1432856.
[22] J. J. H. Miller, E. ORiordan, and G.I. Shishkin, Fitted numerical methods for singular perturbation problems, Error estimate in the maximum norm for linear problems in one and two dimensions, Revised Edition, World Scientific, 2012.
[23] K. W. Morton, Numerical solution of convection-diffusion problems, CRC Press, Taylor and Francis Group, 1996.
[24] J. B. Munyakazi and K. C. Patidar, A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems, Computational and Applied Mathematics,32 (2013), 509 519.
[25] M. P. Rajan and G. D. Reddy , An iterative technique for solving singularly perturbed parabolic PDE, J. Appl. Math. Comput., (2015), DOI:10.1007/s12190-015-0866-x.
[26] H. G. Roos, M. Stynes, and L. Tobiska, Robust numerical methods for singularly perturbed differential equations, Convection-Diffusion-Reaction and Flow Problems, Second Edition, Springer Series in Computational Mathematics ISSN0179-3632, 2008.
[27] J. Singh, D. Kumar, and Z. Hammouch, Abdon Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316 (2018), 504-515.
[28] G. D. Smith, Numerical solution of partial differential equations, Finite difference methods, Third edition, Clarendon press, Oxford, 1984.
[29] L. Zhilin, Z. Qiao, and T. Tang, Numerical solution of differential equations, Introduction to finite difference and finite element methods, printed in the United Kingdom by Clays, 2018
Computational Methods for Differential Equations
Volume 9, Issue 3
July 2021
Pages 886-898

Files

History

  • Receive Date: 06 May 2020
  • Revise Date: 23 July 2020
  • Accept Date: 23 August 2020

Share

How to cite

Statistics