NUMERICAL TREATMENT OF NEWELL-WHITEHEAD-SEGEL EQUATION

Abstract

In this paper, a comparative study of recently developed numerical methods in solving nonlinear Newell-–Whitehead—Segel equations is carried out. Results computed using Variational iteration method VIM are compared with Adomian decomposition method ADM , iterative method and the exact results. The numerical results obtained by VIM are discussed with the help of different figures and tables. The plotted graph and numerical results show accuracy and efficiency of this method in solving nonlinear equations.

Keywords

• Variational iteration method,Newell-–Whitehead-–Segel equation,He’s polynomials,Lagrange’s multiplier

References

1. Soliman, A. A. and Abdou, M. A., (2007), Numerical solutions of nonlinear evolution equations using variational iteration method, J. Comput. Appl. Math., 207, pp. 111-120.
2. Ablowitz, M. J. and Clarkson, P. A., (1991), Solitons, Nonlinear evolution equations and inverse scat- tering, Cambridge University Press, Cambridge.
3. Wazwaz, A. M., (2000), A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111, pp. 53-69.
4. Fadaei, J., (2011), Application of Laplace-Adomian decomposition method on linear and nonlinear system of PDEs, Appl. Math. Sci., 5, pp. 1307-1315.
5. He, J. H., (2000), A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlin. Mech., 35, pp. 37–43.
6. Saravanan, A. and Magesh, N., (2013), A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell—Whitehead—Segel equation, J. Egyp- tian Math. Soc., 21, pp. 259-265.
7. Bataineh, A. S., Noorani, M. S. M. and Hashim, I., (2008), Approximate analytical solutions of systems of PDEs by homotopy analysis method, Comput. Math. Appl., 55, pp. 2913-2923.
8. Parand, K. and Rad, J. A., (2012), Exp-function method for some nonlinear PDE’s and a nonlinear ODE’s, J. King Saud Univ. Sci., 24, pp. 1-10.

9. He, J. H., (1999), Variational iteration method-a kind of non–linear analytical technique: Some exam- ples, Int. J. Nonlin. Mech., 34, pp. 699-708.
10. Prakash, A. and Kumar, M., (2016), He’s Variational Iteration Method for solution of non-linear Newell-–Whitehead-–Segel Equation, J. Appl. Anal. Comput., 6, pp. 738-748.
11. He, J. H. and Wu, X. H., (2007), Variational iteration method: new development and applications, Comput. Math. Appl., 54, pp. 881-894.
12. Sulaiman, T. A., Yokus, A. and Gulluoglu, N., (2018), H.M.Baskonus, Regarding the Numerical and Stability Analysis of the Sharma-Tosso-Olver Equation, 3rd International Conference on Computational Mathematics and Engineering Sciences (CMES-2018), Girne/Cyprus
13. D¨u¸s¨unceli, F. and C¸ elik, E., (2015), An Effective Tool: Numerical Solutions by Legendre Polynomials for High-Order Linear Complex Differential Equations, Br. J. Appl. Sci. Tech., 8(4), pp. 348-355.
14. Yokus, A., Baskonus, H. M., Sulaiman, T. A. and Bulut, H., (2018), Numerical simulation and solutions of the two-component second order KdV evolutionary system, Numer Methods Partial Differ Equ., 34(1), pp. 211-227.
15. Belgacem, F. B. M., Baskonus, H. M. and Bulut, H., (2015), Variational Iteration Method for Hyper- chaotic Nonlinear Fractional Differential Equations Systems, Advances in Mathematics and Statistical Sciences, pp. 445-453.
16. D¨u¸s¨unceli, F. and C¸ elik, E., (2017), Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials, I˘gdır University Journal of the Institute of Science and Technol- ogy,7(4), pp. 189-201.
17. Baskonus, H. M., Belgacem, F. B. M. and Bulut, H., (2015), Solutions of Nonlinear Fractional Dif- ferential Equations Systems through an Implementation of the Variational Iteration Method, Frac- tional Dynamics, De Gruyter, ISBN: 333-342, 2015, ISBN(Online): 9783110472097; DOI(Chapter): 10.1515/9783110472097-020; DOI(Book): 10.1515/9783110472097.
18. Kumar, D., Singh, J., Baleanu, D. and Rathore, S., (2018), Analysis of a fractional model of Ambart- sumian equation, Eur. Phys. J. Plus, 133, pp. 259.
19. Singh, J., Kumar, D., Baleanu, D. and Rathore, S., (2018), An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation, Appl. Math. Comput., 335, pp. 12-24.
20. Kumar, D., Agarwal, R. P. and Singh, J., (2018), A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation, J. Comput. Appl. Math. 339, pp. 405-413.
21. Pue-on, P., (2013), Laplace Adomian Decomposition Method for Solving Newell–Whitehead–Segel Equation, Appl. Math. Sci., 7, pp. 6593-6600.
22. Aasaraai, A., (2011), Analytic solution for Newell-–Whitehead—Segel equation by differential trans- form method, Middle–East J. Sci. Res., 10, pp. 270-273.
23. Manaa, S. A., (2011), An Approximate solution to the Newell-Whitehead-Segel equation by the Ado- mian decomposition method, Raf. J. Comp. Math. 8, pp. 171-180.
24. Ezzati, C. R. and Shakibi, K., (2011), Using adomian’s decomposition and multi–quadric quasi- interpolation methods for solving Newell-–Whitehead—Segel equation, Procedia Comput. Sci., 3, pp. 1043-1048.
25. Mahgoub, M. A., (2016), Homotopy Perturbation Method for Solving Newell—Whitehead-–Segel Equation, Adv. Theor. Appl. Math., 11, pp. 399-406.
26. Patade, J. and Bhalekar, S., (2015), Approximate analytical solution of Newell–Whitehead–Segel Equation using a new iterative method, World J. Model Simul., 11, pp. 94-103.