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The homotopy perturbation method applied to the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations. (English) Zbl 1219.35347
Summary: The fractional derivatives in the sense of Caputo, and the homotopy perturbation method are used to construct approximate solutions for nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equations with respect to time and space fractional derivatives. Also, we apply complex transformation to convert a time and space fractional nonlinear KPP equation to an ordinary differential equation and use the homotopy perturbation method to calculate the approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.

35R11Fractional partial differential equations
26A33Fractional derivatives and integrals (real functions)
35A35Theoretical approximation to solutions of PDE
35A22Transform methods (PDE)
35A24Methods of ordinary differential equations for PDE
Full Text: DOI
[1] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[2] He, J. H.: Some applications of nonlinear fractional differential equations and their applications, Bull. sci. Technol. 15, No. 2, 86-90 (1999)
[3] Diethelm, K.; Luchko, Y.: Numerical solution of linear multi-term differential equations of fractional order, J. comput. Anal. appl. 6, 243-263 (2004) · Zbl 1083.65064
[4] Erturk, V. S.; Momani, Sh.; Odibat, Z.: Application of generalized differential transform method to multi-order fractional differential equations, Commun. nonlinear sci. Numer. simul. 13, 1642-1654 (2008) · Zbl 1221.34022 · doi:10.1016/j.cnsns.2007.02.006
[5] Daftardar-Gejji, V.; Bhalekar, S.: Solving multi-term linear and non-linear diffusion wave equations of fractional order by Adomian decomposition method, Appl. math. Comput. 202, 113-120 (2008) · Zbl 1147.65106 · doi:10.1016/j.amc.2008.01.027
[6] Daftardar-Gejji, V.; Jafari, H.: Solving a multi-order fractional differential equation using Adomian decomposition, Appl. math. Comput. 189, 541-548 (2007) · Zbl 1122.65411 · doi:10.1016/j.amc.2006.11.129
[7] Sweilam, N. H.; Khader, M. M.; Al-Bar, R. F.: Numerical studies for a multi-order fractional differential equation, Phys. lett. A 371, 26-33 (2007) · Zbl 1209.65116 · doi:10.1016/j.physleta.2007.06.016
[8] A. Golbabai, K. Sayevand, Fractional calculus -- a new approach to the analysis of generalized fourth-order diffusion--wave equations, Comput. Math. Appl. (2011) (in press). · Zbl 1219.65117
[9] Golbabai, A.; Sayevand, K.: The homotopy perturbation method for multi-order time fractional differential equations, Nonlinear sci. Lett. A 1, 147-154 (2010) · Zbl 1197.35014
[10] Nikitin, A. G.; Barannyk, T. A.: Solitary waves and other solutions for nonlinear heat equations, Cent. eur. J. math. 2, 840-858 (2005) · Zbl 1116.35035 · doi:10.2478/BF02475981
[11] He, J. H.: New interpretation of homotopy perturbation method, Int. J. Modern phys. B 20, 1-7 (2006)
[12] Li, Z. B.; He, J. H.: Fractional complex transformation for fractional differential equations, Math. comput. Appl. 15, 970-973 (2010) · Zbl 1215.35164
[13] Rafiq, A.; Ahmed, M.; Hussain, S.: A general approach to specific second order ordinary differential equations using homotopy perturbation method, Phys. lett. A 372, 4973-4976 (2008) · Zbl 1221.34030 · doi:10.1016/j.physleta.2008.05.070