Numerical Solution of Newell-Whitehead-Segel Equation

Document Type : Research Paper

Authors

1 School of Mathematics and Computer Science, Damghan University, Damghan, Iran

2 Department of Mathematics, Semnan University, Semnan, Iran.

3 Department of Biology, Payam Noor University of Isfahan, Isfahan, Iran.

4 Faculty of Psychology, Islamic Azad University, Karaj Branch.

Abstract

The Newell-Whitehead-Segel (NWS) equation is an important model arising in fluid mechanics. Various researchers worked on approximate solutions to this model by using different methods. In this paper, the Sine-Cosine wavelets method is applied for solving numerically the NWS equation. The Sine-Cosine wavelet operational matrix of integration is obtained and used to transform the equations into a system of algebraic equations. To demonstrate the effectiveness and applicability of this method, two numerical examples are included.

Keywords

References

  1. I. Aziz, F. Khan, et al. A new method based on haar wavelet for the numerical solution of two-dimensional nonlinear integral equations, Journal of Computational and Applied Mathematics, 272, 70–80 (2014).
  2. N. Azizi, R. Pourgholi, Applications of Sine-Cosine wavelets method for solving Drinfel’d–Sokolov–Wilson system, Advances in Systems Science and Applications, 21, 75–90 (2021).
  3. C. Chen, C. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proceedings-Control Theory and Applications, 144, 87–94 (1997).
  4. C. K. Chui, An introduction to wavelets, Elsevier (2016).
  5. I. Daubechies, Ten lectures on wavelets, SIAM (1992).
  6. I. Daubechies, J. C. Lagarias, Two-scale difference equations ii. local regularity, infinite products of matrices and fractals, SIAM Journal on Mathematical Analysis, 23, 1031– 1079 (1992).
  7. S. Foadian, R. Pourgholi, S. H. Tabasi, J. Damirchi, The inverse solution of the coupled nonlinear reaction–diffusion equations by the haar wavelets, International Journal of Computer Mathematics, 96, 105–125 (2019).
  8. C. Heil, Ten lectures on wavelets (ingrid daubechies), SIAM Review, 35, 666–669 (1993).
  9. M. Hosseininia, M. Heydari, Z. Avazzadeh, Numerical study of the variable-order fractional version of the nonlinear fourth-order 2d diffusion-wave equation via 2d chebyshev wavelets, Engineering with Computers, 1–10 (2020).
  10. N. Irfan, A. Siddiqi, Sine-Cosine wavelets approach in numerical evaluation of Hankel transform for seismology, Applied Mathematical Modelling, 40, 4900–4907 (2016).
  11. H. K. Jassim, Homotopy perturbation algorithm using Laplace transform for NewellWhitehead-Segel equation, International Journal of Advances in Applied Mathematics and Mechanics, 2, 8–12 (2015).
  12. M. T. Kajani, M. Ghasemi, E. Babolian, Numerical solution of linear integro-differential equation by using Sine–Cosine wavelets, Applied Mathematics and Computation, 180, 569–574 (2006).
  13. H. Li, Z. Song, F. Zhang, A reduced-order modified finite difference method preserving unconditional energy-stability for the Allen–Cahn equation, Numerical Methods for Partial Differential Equations, 37, 1869–1885 (2021).
  14. S. G. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, Fundamental Papers in Wavelet Theory, 494–513 (2009).
  15. S. S. Nourazar, M. Soori, A. Nazari-Golshan, On the exact solution of NewellWhitehead-Segel equation using the homotopy perturbation method, arXiv preprint arXiv:1502.08016, (2015).
  16. R. Pourgholi, A. Esfahani, S. Foadian, S. Parehkar, Resolution of an inverse problem by haar basis and legendre wavelet methods, International Journal of Wavelets, Multiresolution and Information Processing, 11, 1350034 (2013).
  17. R. Pourgholi, N. Tavallaie, S. Foadian, Applications of haar basis method for solving some ill-posed inverse problems, Journal of Mathematical Chemistry, 50, 2317–2337 (2012).
  18. A. Prakash, M. Kumar, He’s variational iteration method for the solution of nonlinear Newell Whitehead–Segel equation, J. Appl. Anal. Comput., 6, 738–748 (2016).
  19. M. Razzaghi, S. Yousefi, Sine-cosine wavelets operational matrix of integration and its applications in the calculus of variations, International Journal of Systems Science, 33, 805–810 (2002).
  20. S. G. Rubin, R. A. Graves Jr, A cubic spline approximation for problems in fluid mechanics, NASA STI/Recon Technical Report N, 75, 33345 (1975).
  21. A. Saeed, U. Saeed, Sine-cosine wavelet method for fractional oscillator equations, Mathematical Methods in the Applied Sciences, 42, 6960–6971 (2019).
  22. Y. Yang, M. Heydari, Z. Avazzadeh, A. Atangana, Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations, Advances in Difference Equations, 2020, 1–24 (2020).
  23. B. Yuttanan, M. Razzaghi, Legendre wavelets approach for numerical solutions of distributed order fractional differential equations, Applied Mathematical Modelling, 70, 350–364 (2019).
  24. H. Zhang, J. Yan, X. Qian, S. Song, Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation, Applied Numerical Mathematics, 161, 372–390 (2021).
Analytical and Numerical Solutions for Nonlinear Equations
Volume 6, Issue 2
November 2021
Pages 321-330

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History

  • Receive Date: 16 February 2022
  • Revise Date: 12 March 2022
  • Accept Date: 12 March 2022

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