Abstract
In this paper, the time-fractional Newell–Whitehead–Segel equation has been solved numerically using the Kansa-radial basis function collocation method. In the numerical scheme, the finite difference approach and the Kansa method have been utilized for the temporal and spatial discretization, respectively. The unconditional stability and convergence of the time-discretized scheme are also proven in this paper. In addition, the Kudryashov technique has been employed to acquire the soliton solutions for comparison with the numerical results. Numerical experiments are performed to establish the good accuracy of the proposed scheme.
Similar content being viewed by others
References
Bakkyaraj T, Sahadevan R (2016) Approximate analytical solution of two coupled time fractional nonlinear Schrödinger equations. Int J Appl Comput Math 2:113–135
Chen Y (2003) Generalized extended Tanh-function method to construct new explicit exact solutions for the approximate equations for long water waves. Int J Mod Phys C 14(5):601–611
Golbabai A, Nikan O, Nikazad T (2019) Numerical investigation of the time fractional mobile-immobile advection–dispersion model arising from solute transport in porous media. Int J Appl Comput Math 5:50
Gupta AK, Saha Ray S (2017) On the solitary wave solution of fractional Kudryashov–Sinelshchikov equation describing nonlinear wave processes in a liquid containing gas bubbles. Appl Math Comput 298:1–12
Hosseini VR, Chen W, Avazzadeh Z (2014) Numerical solution of fractional telegraph equation by using radial basis functions. Eng Anal Bound Elem 38:31–39
Kansa EJ (1990) Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 19(8–9):147–161
Korkmaz A (2018) Complex wave solutions to mathematical biology models I: Newell–Whitehead–Segel and Zeldovich equations. J Comput Nonlinear Dyn 13(8):081004
Kumar D, Sharma RP (2016) Numerical approximation of Newell–Whitehead–Segel equation of fractional order. Nonlinear Eng 5(2):81–86
Nourazar SS, Soori M, Nazari-Golshan A (2011) On the exact solution of Newell–Whitehead–Segel equation using the Homotopy perturbation method. Aust J Basic Appl Sci 5(8):1400–1411
Prakash A, Kumar M (2016) He’s variational iteration method for the solution of nonlinear Newell–Whitehead–Segel equation. J Appl Anal Comput 6(3):738–748
Saha Ray S (2016) New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods. Chin. Phys. B 25(4):040204
Saha Ray S (2020) Nonlinear differential equations in physics. Springer, Singapore
Saha Ray S, Sahoo S (2015) New exact solutions of fractional Zakharov–Kuznetsov and modified Zakharov–Kuznetsov equations using fractional sub-equation method. Commun. Theor. Phys. 63(1):25–30
Saha Ray S, Chaudhuri KS, Bera RK (2006) Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method. Appl Math Comput 182:544–552
Saha Ray S (2013) Soliton solutions for time fractional coupled modified KdV equations using new coupled fractional reduced differential transform method. J Math Chem 51:2214–2229
Shah FA, Abass R, Debnath L (2017) Numerical solution of fractional differential equations using Haar wavelet operational matrix method. Int J Appl Comput Math 3:2423–2445
Sun Z, Wu X (2006) A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math 56(2):193–209
Syam MI (2018) Analytical solution of the fractional initial Emden–Fowler equation using the fractional residual power series method. Int J Appl Comput Math 4:106
Tarasov VE (2010) Fractional dynamics. Higher Education Press, Springer, Berlin
Zafar A, Khalid B, Fahand A, Rezazadeh H, Bekir A (2020) Analytical behaviour of travelling wave solutions to the Van der Waals model. Int J Appl Comput Math 6:131
Zheng B (2012) \((G^{\prime }/G)\)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun Theor Phys 58(5):623–630
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Vasily E. Tarasov.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sagar, B., Ray, S.S. Numerical soliton solutions of fractional Newell–Whitehead–Segel equation in binary fluid mixtures. Comp. Appl. Math. 40, 290 (2021). https://doi.org/10.1007/s40314-021-01676-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-021-01676-3
Keywords
- Fractional Newell–Whitehead–Segel equation
- Caputo fractional derivative
- Kudryashov method
- Kansa method
- Radial basis function