Adomian Decomposition and Successive Approximation Methods for Solving Kaup-Boussinesq System
DOI:
https://doi.org/10.25271/sjuoz.2019.7.3.582Keywords:
Adomian Decomposition Method (ADM), Successive Approximation Method (SAM), Kaup-Boussinesq System (KB)Abstract
The Kaup-Boussinesq system has been solved numerically by using two methods, Successive approximation method (SAM) and Adomian decomposition method (ADM). Comparison between the two methods has been made and both can solve this kind of problems, also both methods are accurate and has faster convergence. The comparison showed that the Adomian decomposition method much more accurate than Successive approximation method.
References
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Adomian, G. (1986). A new approaches to the heat equation an application of the decomposition method. Journal of Mathematical Analysis and Applications, 202-209.
Adomian, G. (1986). Nonlinear Stochastic operator equations.San Digo: Academic Press.
Adomian, G. (1994). Solving frontier problems of physics: the decomposition method.
Al Awawdah, E. (2016). The Adomian decomposition method for solving partial differential equations: D.Alaeddin the-sis, BirzeitUnivercity..
Aminikhan, H., Sheikhani, A. H. R., &Rezazadeh, H. (2016.). Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method.Boletim da Society Paranaense de Mathematica, 34(2), 213-229.
Biazar, J., &Ghazvini, H. (2009).Heshomotopy perturbation method for solving systems of volterra integral equa-tions of the second kind. Chaos, Solitons& Fractals, 39(2), 770-777.
Broer, L. (1974). On the Hamiltonian theory of surface waves.Applied Scientific Research, 29(1), 430-446.
Chen, W., & Lu, Z (2004).An algorithm for Adomian decomposi-tion method. Applied Mathematics and Computation, 159 (1), 221-235.
Debnath, L. (2011). Nonlinear partial differential equations for scientists and engineers.Springer Science & Business Media.
Hashem, H. (2015). On successive approximation method for coupled system of Chandrasekhar quadratic integral equations.Journal of the Egyptian Mathematical Socie-ty, 23(1), 108-112.
Hosseini, K., Ansari, R., &Gholamin, P. (2012).Exact solutions of some nonlinear systems of partial differential equations by using the first integral method. Journal of Mathe-matical Analysis and Applications, 387(2), 807-814. doi: 10.1016/j.jmaa.2011.09.044.
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Juliussen, B.-S.H. (2014). Investigation of the Kaup-Boussinesqmodel equations for water waves: P.Henrik thesis, The University of Bergen.
Kaup, D. (1975). A higher-order water-waves equation and the method for solving it.Progress of Theoretical Physics, 54(2), 396-408.
Koparan, M., Kaplan, M., Bekir, A., &Guner, O. (2017). A novel generalized Kudryashov method for exact solutions of nonlinear evolution equations. Paper represented at the AIP Conference Proceedings, 1798, doi:10.1063/1.4972674.
Manna, S. A., Easif, F. H., & Ali, B. Y. (2017).Successive ap-proximation method for solving (1+1)-dimentional dis-persive long wave equations.International Journal of Advanced and Applied Science, 4(8), 98-103.
Nwogu, O. (1993). Alternative form of Boussinesq equations for nearshore wave propagation.Journal of waterway, Port, Coastal, and Ocean engineering, 119(6), 618-638.
Peregrine, D. H. (1967).Long waves on a beach.Journal of Fluid Mechanics, 27(4), 815-827.
Ruan, J., & Lu, Z. (2007). A modified algorithm for the adomian decomposition method with applications to lotka-volterra systems. Mathematical and Computer Model-ing, 46(9-10), 1214-1224.
Somali, S., &Gokmen, G. (2007).Adomian decomposition method for nonlinear sturm-liouville problems.Surveys in mathematica and its Applications, 2, 11-20.
Yang, A.-M., Zhang, C., Jafari, H., Cattani, C., & Jiao, Y. (2014).Picard successive approximation method for solving differential equations arising in fractal heat transfer with local fractional derivative.In Abstract and Applied analysis.
Yassien, S. M. (2014). Modification adomian decomposition method for solving seventh order integro-differential equations.IOSR Journal of Mathematics (IOSR-JM), 72-77
Zhou, J., Tian, L., & Fan, X. (2009). Solitary-wave solutions to a dual equations of the Kaup-boussinesq system. Nonlin-ear Analysis: Real Word Applications 11(4) 3229-3235
Adomian, G. (1986). A new approaches to the heat equation an application of the decomposition method. Journal of Mathematical Analysis and Applications, 202-209.
Adomian, G. (1986). Nonlinear Stochastic operator equations.San Digo: Academic Press.
Adomian, G. (1994). Solving frontier problems of physics: the decomposition method.
Al Awawdah, E. (2016). The Adomian decomposition method for solving partial differential equations: D.Alaeddin the-sis, BirzeitUnivercity..
Aminikhan, H., Sheikhani, A. H. R., &Rezazadeh, H. (2016.). Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method.Boletim da Society Paranaense de Mathematica, 34(2), 213-229.
Biazar, J., &Ghazvini, H. (2009).Heshomotopy perturbation method for solving systems of volterra integral equa-tions of the second kind. Chaos, Solitons& Fractals, 39(2), 770-777.
Broer, L. (1974). On the Hamiltonian theory of surface waves.Applied Scientific Research, 29(1), 430-446.
Chen, W., & Lu, Z (2004).An algorithm for Adomian decomposi-tion method. Applied Mathematics and Computation, 159 (1), 221-235.
Debnath, L. (2011). Nonlinear partial differential equations for scientists and engineers.Springer Science & Business Media.
Hashem, H. (2015). On successive approximation method for coupled system of Chandrasekhar quadratic integral equations.Journal of the Egyptian Mathematical Socie-ty, 23(1), 108-112.
Hosseini, K., Ansari, R., &Gholamin, P. (2012).Exact solutions of some nonlinear systems of partial differential equations by using the first integral method. Journal of Mathe-matical Analysis and Applications, 387(2), 807-814. doi: 10.1016/j.jmaa.2011.09.044.
Javadi, S. (2014).A modification in successive approximation method for solving nonlinear volttera Hammerstein in-tegral equations of the second kind.Journal of mathe-matical Extension.
Juliussen, B.-S.H. (2014). Investigation of the Kaup-Boussinesqmodel equations for water waves: P.Henrik thesis, The University of Bergen.
Kaup, D. (1975). A higher-order water-waves equation and the method for solving it.Progress of Theoretical Physics, 54(2), 396-408.
Koparan, M., Kaplan, M., Bekir, A., &Guner, O. (2017). A novel generalized Kudryashov method for exact solutions of nonlinear evolution equations. Paper represented at the AIP Conference Proceedings, 1798, doi:10.1063/1.4972674.
Manna, S. A., Easif, F. H., & Ali, B. Y. (2017).Successive ap-proximation method for solving (1+1)-dimentional dis-persive long wave equations.International Journal of Advanced and Applied Science, 4(8), 98-103.
Nwogu, O. (1993). Alternative form of Boussinesq equations for nearshore wave propagation.Journal of waterway, Port, Coastal, and Ocean engineering, 119(6), 618-638.
Peregrine, D. H. (1967).Long waves on a beach.Journal of Fluid Mechanics, 27(4), 815-827.
Ruan, J., & Lu, Z. (2007). A modified algorithm for the adomian decomposition method with applications to lotka-volterra systems. Mathematical and Computer Model-ing, 46(9-10), 1214-1224.
Somali, S., &Gokmen, G. (2007).Adomian decomposition method for nonlinear sturm-liouville problems.Surveys in mathematica and its Applications, 2, 11-20.
Yang, A.-M., Zhang, C., Jafari, H., Cattani, C., & Jiao, Y. (2014).Picard successive approximation method for solving differential equations arising in fractal heat transfer with local fractional derivative.In Abstract and Applied analysis.
Yassien, S. M. (2014). Modification adomian decomposition method for solving seventh order integro-differential equations.IOSR Journal of Mathematics (IOSR-JM), 72-77
Zhou, J., Tian, L., & Fan, X. (2009). Solitary-wave solutions to a dual equations of the Kaup-boussinesq system. Nonlin-ear Analysis: Real Word Applications 11(4) 3229-3235
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Published
2019-09-30
How to Cite
Manaa, S. A., & Mosa, N. M. (2019). Adomian Decomposition and Successive Approximation Methods for Solving Kaup-Boussinesq System. Science Journal of University of Zakho, 7(3), 101–107. https://doi.org/10.25271/sjuoz.2019.7.3.582
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Science Journal of University of Zakho
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