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Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. (English) Zbl 1165.65375
Summary: Convergence and stability are main issues when an asymptotical method like the Homotopy Perturbation Method (HPM) has been used to solve differential equations. In this paper, convergence of the solution of fractional differential equations is maintained. Meanwhile, an effective method is suggested to select the linear part in the HPM to keep the inherent stability of fractional equations. Riccati fractional differential equations as a case study are then solved, using the Enhanced Homotopy Perturbation Method (EHPM). Current results are compared with those derived from the established Adams-Bashforth-Moulton method, in order to verify the accuracy of the EHPM. It is shown that there is excellent agreement between the two sets of results. This finding confirms that the EHPM is powerful and efficient tool for solving nonlinear fractional differential equations.

65L99Numerical methods for ODE
34A45Theoretical approximation of solutions of ODE
Full Text: DOI
[1] J.H. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering’98, Dalian, China, 1998, pp. 288--291
[2] Odibat, Z.; Momani, S.: Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos solitons fractals 36, No. 1, 167-174 (2008) · Zbl 1152.34311 · doi:10.1016/j.chaos.2006.06.041
[3] J. Cang, Y. Tan, H. Xu, S.J. Liao, Series solutions of non-linear Riccati differential equations with fractional order, Chaos Solitons Fractals (2007) (in press) · Zbl 1197.34006
[4] Jafari, H.; Momani, S.: Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys. lett. A 370, No. 5--6, 388-396 (2007) · Zbl 1209.65111 · doi:10.1016/j.physleta.2007.05.118
[5] Ge, Z. -M.; Ou, C. -Y.: Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal, Chaos solitons fractals 35, No. 4, 705 (2008)
[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] Abdulaziz, O.; Hashima, I.; Momani, S.: Solving systems of fractional differential equations by homotopy-perturbation method, Phys. lett. A 372, No. 4, 451-459 (2008) · Zbl 1217.81080 · doi:10.1016/j.physleta.2007.07.059
[8] Ganji, D. D.; Sadighi, A.: Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, J. comput. Appl. math. 207, No. 1, 24-34 (2007) · Zbl 1120.65108 · doi:10.1016/j.cam.2006.07.030
[9] Ganji, D. D.: The application of he’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys. lett. A 355, No. 4--5, 337-341 (2006) · Zbl 1255.80026
[10] He, J. H.: Homotopy perturbation technique, Comp. methods appl. Mech. eng. 178, No. 3--4, 257-262 (1999) · Zbl 0956.70017
[11] Nia, S. H. Hosein; Ranjbar, A. N.; Ganji, D. D.; Soltani, H.; Ghasemi, J.: Maintaining the stability of nonlinear differential equations by the enhancement of HPM, Phys. lett. A 372, No. 16, 2855-2861 (2008) · Zbl 1220.70018 · doi:10.1016/j.physleta.2007.12.054
[12] He, J. -H.: Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern phys. B 20, No. 10, 1141-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[13] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[14] Tavazoei, Mohammad Saleh; Haeri, Mohammad: A necessary condition for double scroll attractor existence in fractional-order systems, Phys. lett. A 367, 102-113 (2007) · Zbl 1209.37037 · doi:10.1016/j.physleta.2007.05.081
[15] Diethelm, K.; Ford, N. J.; Freed, A. D.: Detailed error analysis for a fractional Adams method, Numer. algorithms 36, 31-52 (2004) · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be
[16] Li, C.; Peng, G.: Chaos in Chen’s system with a fractional order, Chaos solitons fractals 22, 443-450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[17] Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor--corrector approach for the numerical solution of fractional differential equations, Nonlinear dynam. 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[18] Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order, Electron. trans. Numer. anal. 5, 1-6 (1997) · Zbl 0890.65071 · emis:journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.html
[19] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[20] Ahmed, E.; El-Sayed, A. M. A.; El-Saka, H. A. A.: Equilibrium points, stability and numerical solutions of fractional order predator--prey and rabies models, J. math. Anal. appl. 325, No. 1, 542-553 (2007) · Zbl 1105.65122 · doi:10.1016/j.jmaa.2006.01.087
[21] Matignon, D.: Stability results for fractional differential equations with applications to control processing, Computational engineering in systems applications 2, 963-968 (1996)
[22] Ranjbar, A.; Hosseinnia, S. H.; Soltani, H.; Ghasemi, J.: A solution of Riccati nonlinear differential equation using enhanced homotopy perturbation method (EHPM), Ije 21, No. 1, 27-38 (2008)