A Comparison Between Fourier Transform Adomian Decomposition Method and Homotopy Perturbation ethod for Linear and Non-Linear Newell-Whitehead-Segel Equations

Document Type : Research Article

Authors

Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

Abstract

In this paper, a comparison among the hybrid of Fourier Transform and Adomian
Decomposition Method (FTADM) and Homotopy Perturbation Method (HPM) is investigated.
The linear and non-linear Newell-Whitehead-Segel (NWS) equations are solved and the results are
compared with the exact solution. The comparison reveals that for the same number of components
of recursive sequences, the error of FTADM is much smaller than that of HPM. For the non-linear
NWS equation, the accuracy of FTADM is more pronounced than HPM. Moreover, it is shown that
as time increases, the results of FTADM, for the linear NWS equation, converges to zero. And for the
non-linear NWS equation, the results of FTADM converges to 1 with only six recursive components.
This is in agreement with the basic physical concept of NWS diffusion equation which is in turn in
agreement with the exact solution.

Highlights

[1] J.-H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons & Fractals, 26(3) (2005) 695-700.

[2] J.-H. He, Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation, 6(2) (2005) 207-208.

[3] J.-H. He, Homotopy perturbation method for solving boundary value problems, Physics letters A, 350(1) (2006) 87-88.

[4] A.-M. Wazwaz, Partial differential equations and solitary waves theory, Springer Science & Business Media, 2010.

[5] A. Yildirim, Homotopy perturbation method for the mixed Volterra–Fredholm integral equations, Chaos, Solitons & Fractals, 42(5) (2009) 2760-2764.

[6] S.S. Nourazar, M. Soori, A. Nazari-Golshan, On the exact solution of Newell-Whitehead-Segel equation using the homotopy perturbation method, arXiv preprint arXiv:1502.08016,  (2015).

[7] M.M. Rashidi, H. Shahmohamadi, Analytical solution of three-dimensional Navier–Stokes equations for the flow near an infinite rotating disk, Communications in Nonlinear Science and Numerical Simulation, 14(7) (2009) 2999-3006.

[8] O.A. Bég, M. Rashidi, T.A. Bég, M. Asadi, Homotopy analysis of transient magneto-bio-fluid dynamics of micropolar squeeze film in a porous medium: a model for magneto-bio-rheological lubrication, Journal of Mechanics in Medicine and Biology, 12(03) (2012) 1250051.

[9] M.H. Abolbashari, N. Freidoonimehr, F. Nazari, M.M. Rashidi, Entropy analysis for an unsteady MHD flow past a stretching permeable surface in nano-fluid, Powder Technology, 267 (2014) 256-267.

[10] S. Nourazar, A. Nazari-Golshan, A. Yıldırım, M. Nourazar, On the hybrid of Fourier transform and Adomian decomposition method for the solution of nonlinear Cauchy problems of the reaction-diffusion equation, Zeitschrift für Naturforschung A, 67(6-7) (2012) 355-362.

[11] A. Nazari-Golshan, S. Nourazar, H. Ghafoori-Fard, A. Yildirim, A. Campo, A modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane–Emden equations, Applied Mathematics Letters, 26(10) (2013) 1018-1025.

[12] A. Saravanan, N. Magesh, A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell–Whitehead–Segel equation, Journal of the Egyptian Mathematical Society, 21(3) (2013) 259-265.

[13] G. Adomian, Solving Frontier Problems of Physics: The Decomposition MethodKluwer, Boston, MA,  (1994).

[14] A.-M. Wazwaz, M.S. Mehanna, The combined Laplace-Adomian method for handling singular integral equation of heat transfer, International Journal of Nonlinear Science, 10(2) (2010) 248-252.

[15] R.G. Pratt, C. Shin, G. Hick, Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion, Geophysical Journal International, 133(2) (1998) 341-362.

[16] R.G. Pratt, M. Worthington, Inverse theory applied to multi-source cross-hole tomography. Part 1: Acoustic wave-equation method, Geophysical prospecting, 38(3) (1990) 287-310.

[17] T. Wu, Z. Chen, A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML, Journal of Computational and Applied Mathematics, 267 (2014) 82-95.

Keywords


[1] J.-H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons & Fractals, 26(3) (2005) 695-700.
[2] J.-H. He, Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation, 6(2) (2005) 207-208.
[3] J.-H. He, Homotopy perturbation method for solving boundary value problems, Physics letters A, 350(1) (2006) 87-88.
[4] A.-M. Wazwaz, Partial differential equations and solitary waves theory, Springer Science & Business Media, 2010.
[5] A. Yildirim, Homotopy perturbation method for the mixed Volterra–Fredholm integral equations, Chaos, Solitons & Fractals, 42(5) (2009) 2760-2764.
[6] S.S. Nourazar, M. Soori, A. Nazari-Golshan, On the exact solution of Newell-Whitehead-Segel equation using the homotopy perturbation method, arXiv preprint arXiv:1502.08016,  (2015).
[7] M.M. Rashidi, H. Shahmohamadi, Analytical solution of three-dimensional Navier–Stokes equations for the flow near an infinite rotating disk, Communications in Nonlinear Science and Numerical Simulation, 14(7) (2009) 2999-3006.
[8] O.A. Bég, M. Rashidi, T.A. Bég, M. Asadi, Homotopy analysis of transient magneto-bio-fluid dynamics of micropolar squeeze film in a porous medium: a model for magneto-bio-rheological lubrication, Journal of Mechanics in Medicine and Biology, 12(03) (2012) 1250051.
[9] M.H. Abolbashari, N. Freidoonimehr, F. Nazari, M.M. Rashidi, Entropy analysis for an unsteady MHD flow past a stretching permeable surface in nano-fluid, Powder Technology, 267 (2014) 256-267.
[10] S. Nourazar, A. Nazari-Golshan, A. Yıldırım, M. Nourazar, On the hybrid of Fourier transform and Adomian decomposition method for the solution of nonlinear Cauchy problems of the reaction-diffusion equation, Zeitschrift für Naturforschung A, 67(6-7) (2012) 355-362.
[11] A. Nazari-Golshan, S. Nourazar, H. Ghafoori-Fard, A. Yildirim, A. Campo, A modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane–Emden equations, Applied Mathematics Letters, 26(10) (2013) 1018-1025.
[12] A. Saravanan, N. Magesh, A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell–Whitehead–Segel equation, Journal of the Egyptian Mathematical Society, 21(3) (2013) 259-265.
[13] G. Adomian, Solving Frontier Problems of Physics: The Decomposition MethodKluwer, Boston, MA,  (1994).
[14] A.-M. Wazwaz, M.S. Mehanna, The combined Laplace-Adomian method for handling singular integral equation of heat transfer, International Journal of Nonlinear Science, 10(2) (2010) 248-252.
[15] R.G. Pratt, C. Shin, G. Hick, Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion, Geophysical Journal International, 133(2) (1998) 341-362.
[16] R.G. Pratt, M. Worthington, Inverse theory applied to multi-source cross-hole tomography. Part 1: Acoustic wave-equation method, Geophysical prospecting, 38(3) (1990) 287-310.
[17] T. Wu, Z. Chen, A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML, Journal of Computational and Applied Mathematics, 267 (2014) 82-95.