Abstract
In this paper, the Homotopy analysis method (HAM) is employed to obtain the analytical/numerical solutions for linear and nonlinear Kolmogorov Petrovskii-Piskunov (KPP) and fractional KPP equations. The proposed method is a powerful and easy-to-use analytical tool for linear and nonlinear problems. This method contains the auxiliary parameter \(h\), which provides us with a simple way to adjust and control the convergence region of solution series. Some illustrative examples are presented. Moreover the use of HAM is found to be accurate, simple, convenient, flexible and computationally attractive.
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References
S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006)
M.D. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Am. Math. Soc. 285, 1–190 (1983)
V. Daftardar-Gejji, H. Jafari, Solving a multi-order fractional differential equation using adomian decomposition. Appl. Math. Comput. 189, 541–548 (2007)
K. Diethelm, Y. Luchko, Numerical solution of linear multi-term differential equations of fractional order. J. Comput. Anal. Appl. 6, 243–263 (2004)
K.A. Gepreel, The homotopy perturbation method applied to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations. Appl. Math. Lett. 24, 1428–1434 (2011)
A. Golbabai, K. Sayevand, The homotopy perturbation method for multi-order time fractional differential equations. Nonlinear Sci. Lett. A 1, 147–154 (2010)
J.H. He, Some applications of nonlinear fractional differential equations and their applications. Bull. Sci. Technol. 15(2), 86–90 (1999)
H. Jafari, S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Commun. Nonlinear Sci. Numer. Simulat. 14, 2006–2012 (2009)
A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, A study of the diffusion equation with increase in the quantity of matter, and its application to a biological problem. Bull. Moscow Univ. Math. ser. A 1, 1–25 (1937)
S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University; (1992)
S.J. Liao, An approximate solution technique which does not depend upon small parameters: a special example. Int. J. Nonlinear. Mech. 30, 37180 (1995)
S.J. Liao, On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)
S.J. Liao, Comparison between the homotopy analysis method and homotopy perturbation method. Appl. Math. Comput. 169, 118–164 (2005)
S.J. Liao, Notes on homotopy analysis method: some definitions and theorems. Commun Nonlinear Sci. Numer. Simul. 14, 983–997 (2009)
W.X. Ma, B. Fuchssteiner, Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation. Int. J. Non-Linear Mech. 31(3), 329–338 (1996)
A.G. Nikitin, T.A. Barannyk, Solitary waves and other solutions for nonlinear heat equations. Cent. Eur. J. Math. 2, 840–858 (2005)
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
N.H. Sweilam, M.M. Khader, R.F. Al-Bar, Numerical studies for a multi-order fractional differential equation. Phys. Lett. A 371, 26–33 (2007)
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I am very grateful to the reviewers for their useful comments that led to improvement of my manuscript.
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Hariharan, G. The homotopy analysis method applied to the Kolmogorov–Petrovskii–Piskunov (KPP) and fractional KPP equations. J Math Chem 51, 992–1000 (2013). https://doi.org/10.1007/s10910-012-0132-5
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DOI: https://doi.org/10.1007/s10910-012-0132-5