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Abstract
In this paper, the homotopy perturbation method is adopted to find explicit and numerical solutions for systems of non-linear fractional shallow water equations. The fractional derivatives are described in the Caputo sense. We apply both the homotopy perturbation method and the homotopy analysis method, to solve certain shallow water equations with time-fractional derivatives, and explicitly construct convergent power series solutions. The results obtained reveal that these methods are both very effective and simple for finding approximate solutions. Some numerical examples and plots are presented to illustrate the efficiency and reliability of these methods.
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References
- S. Abbasbandy, The application of homotopy analysis method to solve a genaralized Hiroto-Satsuma coupled KdV equations. Phys. Lett. A , Vol. 361 (2007) Pages 478-483.
- F.Allan, Derivation of the Adomian decomposition method using the homotopy analysis method. Appl. Math. Comput., Vol. 190 (2007) Pages 6-14.
- Kamel Al-Khaled; F. Allan, Construction of solutions for the shallow water equations by the decomposition method. Comput. Comput. Simul., Vol. 66 (2004) Pages 479-486.
- M.K. Al-Saafeen, Homotopy Perturbation Method for Fractional Hirota-Satsuma Coupled KdV Equations, Master Thesis, Jordan University of Science and Technology, May (2008).
- A.Bermudez, M. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, Vol. 23 (1994) Pages 1049-1071.
- M.Caputo, Linear models of dissipation whose is almost frequency independent II, Geophys. J. R. Astron. Soc. 13 (1967) Pages 529-539.
- J.H.He, Homotopy-Perturbation Technique. Comput. Math. Appl. Mech. Eng., Vol. 178 (1999).
- J.H. He,Homotopy-Perturbation Method: a new nonlinear analytical technique. Appl. Math. Comput., Vol. 135 (2003), Pages 73-79.
- Tomas Kisela, Power functions and essentials of fractional calculus on isolated time scales. Advance Difference Equations, 2013, 2013: 259.
- S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992.
- S.J. Liao, Comparison between the homotopy analysis method and homotopy perturbation method. Appl. Math. Comput., Vol. 169 (2005), Pages 1186-1194.
- S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and hall/CRC Press, Boca Raton, 2003.
- M. Matinafar, M. Saeidy, M. Eslami Solving a system of linear and nonlinear fractional partial differential equations using homotopy perturbation method, Intern. J. of Nonlinear Sci. Numer. Simul., Vol. 14, Issue 7-8, Pages 471-478.
- K.S. Miller, B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons Inc. New York, 1993.
- Shaher Momani, Non-perturbative analytical solutions of the space and time fractional Burgers' equations, Chaos Soliton Fractals, Volume 28, (2006), Pages 930-937.
- K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
- I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
- I. Podlubny, Gemmetric and physical interpretation of fractional integration and fractional differentiation, Frac. Calc. Appl. Anal. 5 (2002) Pages 367-386.
- I.Podlubny, What Euler could further write, or the unnoticed "big bang" of the fractional calculus, Frac. Calc. Appl. Anal. 16 (2013) Pages 501-506.
- M. Sahid; T. Hayat; S. Asghar, Comparison between the HAM and HPM solutions of tin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dynam, Vol. 50 (2007) pages 27-35.
- L.N. Song, Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation, Phys. Lett. A, 367, No. 1-2 (2007), Pages 88-94.
References
S. Abbasbandy, The application of homotopy analysis method to solve a genaralized Hiroto-Satsuma coupled KdV equations. Phys. Lett. A , Vol. 361 (2007) Pages 478-483.
F.Allan, Derivation of the Adomian decomposition method using the homotopy analysis method. Appl. Math. Comput., Vol. 190 (2007) Pages 6-14.
Kamel Al-Khaled; F. Allan, Construction of solutions for the shallow water equations by the decomposition method. Comput. Comput. Simul., Vol. 66 (2004) Pages 479-486.
M.K. Al-Saafeen, Homotopy Perturbation Method for Fractional Hirota-Satsuma Coupled KdV Equations, Master Thesis, Jordan University of Science and Technology, May (2008).
A.Bermudez, M. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, Vol. 23 (1994) Pages 1049-1071.
M.Caputo, Linear models of dissipation whose is almost frequency independent II, Geophys. J. R. Astron. Soc. 13 (1967) Pages 529-539.
J.H.He, Homotopy-Perturbation Technique. Comput. Math. Appl. Mech. Eng., Vol. 178 (1999).
J.H. He,Homotopy-Perturbation Method: a new nonlinear analytical technique. Appl. Math. Comput., Vol. 135 (2003), Pages 73-79.
Tomas Kisela, Power functions and essentials of fractional calculus on isolated time scales. Advance Difference Equations, 2013, 2013: 259.
S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992.
S.J. Liao, Comparison between the homotopy analysis method and homotopy perturbation method. Appl. Math. Comput., Vol. 169 (2005), Pages 1186-1194.
S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and hall/CRC Press, Boca Raton, 2003.
M. Matinafar, M. Saeidy, M. Eslami Solving a system of linear and nonlinear fractional partial differential equations using homotopy perturbation method, Intern. J. of Nonlinear Sci. Numer. Simul., Vol. 14, Issue 7-8, Pages 471-478.
K.S. Miller, B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons Inc. New York, 1993.
Shaher Momani, Non-perturbative analytical solutions of the space and time fractional Burgers' equations, Chaos Soliton Fractals, Volume 28, (2006), Pages 930-937.
K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
I. Podlubny, Gemmetric and physical interpretation of fractional integration and fractional differentiation, Frac. Calc. Appl. Anal. 5 (2002) Pages 367-386.
I.Podlubny, What Euler could further write, or the unnoticed "big bang" of the fractional calculus, Frac. Calc. Appl. Anal. 16 (2013) Pages 501-506.
M. Sahid; T. Hayat; S. Asghar, Comparison between the HAM and HPM solutions of tin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dynam, Vol. 50 (2007) pages 27-35.
L.N. Song, Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation, Phys. Lett. A, 367, No. 1-2 (2007), Pages 88-94.