International Journal of Nonlinear Analysis and Applications
Solving Benny-Lin Equation by Adomain Decomposition Method with Genetic Algorithm
Document Type : Research Paper
Authors
College of Computer Sciences and Mathematics, University of Mosul, Mosul, Republic of Iraq
Abstract
The nonlinear Benny-Lin equation has been solved In this paper using Adomian decomposition technique (ADM ) with different initial conditions and the results shown in Figures (1-8), also by using modified Adomian decomposition technique with genetic algorithm to expound the optimal equation parameters. The proposed method (GA-ADM) guarantees that the optimal parameters will be achieved precisely regardless of the complexity and multiple values of the equation. The proposed method gives more accurate results than the ADM.
Keywords
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analysis method with genetic algorithm,” Appl. Math, vol. 11, pp. 1-10, 2017.
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parameters by using PSO algorithm,” Indonesian Journal of Electrical Engineering and Computer Science, vol.
19, pp. 709-714, 2020.[14] K. A. Abed, ”Controlling of jerk chaotic system via linear feedback control strategies,” Indonesian Journal of
Electrical Engineering and Computer Science, vol. 20, pp. 370-378, 2020.
[15] Z. S. Al-Talib and S. F. AL-Azzawi, ”Projective synchronization for 4D hyperchaotic system based on adaptive
nonlinear control strategy,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 19, pp.
715-722, 2020.
[16] M. M. Aziz and S. F. Al-Azzawi, ”A modification of nonlinear feedback controller,” International Journal of
Computing Science and Mathematics, vol. 13, pp. 64-79, 2021.
[17] S. H. Mohammad and E. S. Al-Rawi, ”Solving Fractional Coupled EW and Coupled MEW Equations Using
Bernstein Collocation Method,” in Journal of Physics: Conference Series, 2021, p. 012021.
[18] A. F. Qasim and E. S. Al-Rawi, ”Adomian decomposition method with modified Bernstein polynomials for solving
ordinary and partial differential equations,” Journal of Applied Mathematics, vol. 2018, 2018.
[19] A. F. Qasim and A. A. Hamed, ”Treating Transcendental Functions in Partial Differential Equations Using the
Variational Iteration Method with Bernstein Polynomials,” International Journal of Mathematics and Mathematical Sciences, vol. 2019, 2019.
[20] Y. Liu, ”Adomian decomposition method with orthogonal polynomials: Legendre polynomials,” Mathematical
and Computer Modelling, vol. 49, pp. 1268-1273, 2009.
[21] Y. Mahmoudi, M. Abdollahi, N. Karimian, and H. Khalili, ”Adomian decomposition method with Laguerre
polynomials for solving ordinary differential equation,” Journal of Basic and Applied Scientific Research, vol. 2,
pp. 12236-12241, 2012.
[22] O. S. Qasim, N. A. Al-Thanoon, and Z. Y. Algamal, ”Feature selection based on chaotic binary black hole
algorithm for data classification,” Chemometrics and Intelligent Laboratory Systems, vol. 204, p. 104104, 2020.
[23] A. F. Qasim, K. Adel Abed, and O. S. Qasim, ”Optimal Parameters for Nonlinear Hirota-Satsuma Coupled KdV
System by Using Hybrid Firefly Algorithm with Modified Adomian Decomposition,” Journal of Mathematical &
Fundamental Sciences, vol. 52, 2020.
[24] X.-S. Yang and X. He, ”Firefly algorithm: recent advances and applications,” International journal of swarm
intelligence, vol. 1, pp. 36-50, 2013.
[25] G. Adomian, Nonlinear stochastic operator equations: Academic press, 2014.
[2] S. Lin, ”Finite amplitude side-band stability of a viscous film,” Journal of Fluid Mechanics, vol. 63, pp. 417-429,
1974.
[3] S. Rashidi and S. R. Hejazi, ”Analyzing Lie symmetry and constructing conservation laws for time-fractional
Benny–Lin equation,” International Journal of Geometric Methods in Modern Physics, vol. 14, p. 1750170, 2017.
[4] O. F. G¨oz¨uk ¨ ızıl and S¸. Ak¸ca˘gıl, ”Travelling wave solutions to the Benney-Luke and the higher-order improved
Boussinesq equations of Sobolev type,” in Abstract and applied analysis, 2012.
[5] P. K. Gupta, ”Approximate analytical solutions of fractional Benney–Lin equation by reduced differential transform method and the homotopy perturbation method,” Computers & Mathematics with Applications, vol. 61,
pp. 2829-2842, 2011.
[6] M. Safari, D. D. Ganji, and E. Sadeghi, ”Application of He’s homotopy perturbation and He’s variational iteration
methods for solution of Benney–Lin equation,” International Journal of Computer Mathematics, vol. 87, pp. 1872-
1884, 2010.
[7] A. M. Al-Rozbayani and M. O. Al-Amr, ”Discrete Adomian decomposition method for solving Burger’s-Huxley
Equation,” International Journal of Contemporary Mathematical Sciences, vol. 8, pp. 623-631, 2013.
[8] M. Saqib and M. Iqbal, ”Some multi-step iterative methods for solving nonlinear equations,” Open Journal of
Mathematical Sciences, vol. 1, pp. 25-33, 12/31 2017.
[9] N. Bildik and A. Konuralp, ”The use of variational iteration method, differential transform method and Adomian
decomposition method for solving different types of nonlinear partial differential equations,” International Journal
of Nonlinear Sciences and Numerical Simulation, vol. 7, pp. 65-70, 2006.
[10] Y.-J. Yang and L.-Q. Hua, ”Variational iteration transform method for fractional differential equations with local
fractional derivative,” in Abstract and Applied Analysis, 2014.
[11] W. Al-Hayani, L. Alzubaidy, and A. Entesar, ”Solutions of singular IVP’s of Lane-Emden type by homotopy
analysis method with genetic algorithm,” Appl. Math, vol. 11, pp. 1-10, 2017.
[12] H. Rehman, M. S. Saleem, and A. Ahmad, ”Combination of homotopy perturbation method (HPM) and double
sumudu transform to solve fractional KDV equations,” Open Journal of Mathematical Sciences, vol. 2, pp. 29-38,
2018.
[13] K. A. Abed, ”Solving Kuramoto-Sivashinsky equation by the new iterative method and estimate the optimal
parameters by using PSO algorithm,” Indonesian Journal of Electrical Engineering and Computer Science, vol.
19, pp. 709-714, 2020.[14] K. A. Abed, ”Controlling of jerk chaotic system via linear feedback control strategies,” Indonesian Journal of
Electrical Engineering and Computer Science, vol. 20, pp. 370-378, 2020.
[15] Z. S. Al-Talib and S. F. AL-Azzawi, ”Projective synchronization for 4D hyperchaotic system based on adaptive
nonlinear control strategy,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 19, pp.
715-722, 2020.
[16] M. M. Aziz and S. F. Al-Azzawi, ”A modification of nonlinear feedback controller,” International Journal of
Computing Science and Mathematics, vol. 13, pp. 64-79, 2021.
[17] S. H. Mohammad and E. S. Al-Rawi, ”Solving Fractional Coupled EW and Coupled MEW Equations Using
Bernstein Collocation Method,” in Journal of Physics: Conference Series, 2021, p. 012021.
[18] A. F. Qasim and E. S. Al-Rawi, ”Adomian decomposition method with modified Bernstein polynomials for solving
ordinary and partial differential equations,” Journal of Applied Mathematics, vol. 2018, 2018.
[19] A. F. Qasim and A. A. Hamed, ”Treating Transcendental Functions in Partial Differential Equations Using the
Variational Iteration Method with Bernstein Polynomials,” International Journal of Mathematics and Mathematical Sciences, vol. 2019, 2019.
[20] Y. Liu, ”Adomian decomposition method with orthogonal polynomials: Legendre polynomials,” Mathematical
and Computer Modelling, vol. 49, pp. 1268-1273, 2009.
[21] Y. Mahmoudi, M. Abdollahi, N. Karimian, and H. Khalili, ”Adomian decomposition method with Laguerre
polynomials for solving ordinary differential equation,” Journal of Basic and Applied Scientific Research, vol. 2,
pp. 12236-12241, 2012.
[22] O. S. Qasim, N. A. Al-Thanoon, and Z. Y. Algamal, ”Feature selection based on chaotic binary black hole
algorithm for data classification,” Chemometrics and Intelligent Laboratory Systems, vol. 204, p. 104104, 2020.
[23] A. F. Qasim, K. Adel Abed, and O. S. Qasim, ”Optimal Parameters for Nonlinear Hirota-Satsuma Coupled KdV
System by Using Hybrid Firefly Algorithm with Modified Adomian Decomposition,” Journal of Mathematical &
Fundamental Sciences, vol. 52, 2020.
[24] X.-S. Yang and X. He, ”Firefly algorithm: recent advances and applications,” International journal of swarm
intelligence, vol. 1, pp. 36-50, 2013.
[25] G. Adomian, Nonlinear stochastic operator equations: Academic press, 2014.
)
Volume 12, Special Issue
December 2021Pages 861-871
- Receive Date: 10 January 2021
- Revise Date: 25 June 2021
- Accept Date: 13 July 2021