Abstract
In this paper, the application of the homotopy perturbation method (HPM) to the nonlinear Schrodinger equation with variable coefficient (NLSV) is presented to obtain approximate analytical solution. The procedure of the method is systematically illustrated. The result derived by this method is then compared with the progressive wave solution to verify the accuracy of the HPM solution. The solution obtained by the HPM is an infinite series for appropriate initial condition that can be expressed in a closed form to the exact solution. The absolute errors of the HPM solution of the NLSV equation with the progressive wave solution will later be carried out using the MAPLE program. The results of the HPM solution are of high accuracy, verifying that the method is indeed effective and promising. The HPM is found to be a powerful mathematical tool which can be used to solve nonlinear partial differential equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
He, J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)
He, J.H.: A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Non-Linear Mech. 35, 37–43 (2000)
He, J.H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73–79 (2003)
He, J.H.: Comparison of homotopy perturbation method and homotopy analysis method. Appl. Math. Comput. 156, 528–539 (2004)
Li, J.L.: Adomian’s decomposition method and homotopy perturbation method in solving nonlinear equations. J. Comput. Appl. Math. 228(1), 168–173 (2009)
Noor, M.A., Khan, W.A.: New iterative methods for solving nonlinear equation by using homotopy perturbation method. Appl. Math. Comput. 219(8), 3365–3374 (2012)
Ghasemi, M., Tavassoli, K.M., Davari, A.: Numerical solution of two dimensional nonlinear differential equation by homotopy perturbation method. Appl. Math. Comput. 189(1), 341–345 (2007)
Ravindran, R., Prasad, P.: A mathematical analysis of nonlinear waves in a fluid-filled viscoelastic tube. Acta Mech. 31(3–4), 253–280 (1979)
Antar, N., Demiray, H.: Nonlinear wave modulation in a prestressed fluid-filled thin elastic tube. Int. J. Nonlinear Mech. 34, 123–138 (1999)
Bakirtaş, İ., Demiray, H.: Amplitude modulation of nonlinear waves in a fluid-filled tapered elastic tube. Appl. Math. Comput. 154(3), 747–767 (2004)
Choy, Y.Y.: Nonlinear Wave Modulation in a Fluid-Filled Thin Elastic Stenosed Artery. Doctorial dissertation, UTM, Skudai (2014)
Ablowitz, M.A., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1992)
Taha, T.B., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation. J. Comput. Phys. 55(2), 203–209 (1984)
Wang, H.: Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Appl. Math. Comput. 170(1), 17–35 (2005)
Dehghan, M., Taleei, A.: A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. Comput. Phys. Commun. 181(1), 43–51 (2010)
El-Sayed, S.M., Kaya, D.: A numerical solution and an exact explicit solution of the NLS equation. Appl. Math. Comput. 172(2), 1315–1322 (2006)
Bratsos, A., Ehrhardt, M., Famelis, I.T.: A discrete Adomian decomposition method for discrete nonlinear Schrödinger equations. Appl. Math. Comput. 172(1), 190–205 (2008)
Mousa, M.M., Ragab, S.F.: Application of the homotopy perturbation method to linear and nonlinear Schrodinger equations. Verlag der Zeitschrift Fur Natueforschung 63(3–4), 140–144 (2008)
Acknowledgments
The authors wish to express appreciation to the Ministry of Education Malaysia for financial support and Universiti Tun Hussein Onn Malaysia for the RAGS’s Grant Vote R026.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Yazid, N.M., Tay, K.G., Choy, Y.Y., Sudin, A.M. (2017). Solving Nonlinear Schrodinger Equation with Variable Coefficient Using Homotopy Perturbation Method. In: Ahmad, AR., Kor, L., Ahmad, I., Idrus, Z. (eds) Proceedings of the International Conference on Computing, Mathematics and Statistics (iCMS 2015). Springer, Singapore. https://doi.org/10.1007/978-981-10-2772-7_26
Download citation
DOI: https://doi.org/10.1007/978-981-10-2772-7_26
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-2770-3
Online ISBN: 978-981-10-2772-7
eBook Packages: EducationEducation (R0)