Series solutions of systems of nonlinear fractional differential equations. (English) Zbl 1187.34007

Summary: Differential equations of fractional order (FDE) appear in many applications in physics, chemistry and engineering. An effective and easy-to-use method for solving such equations is needed. In this paper, series solutions of the FDEs are presented using the homotopy analysis method (HAM). The HAM provides a convenient way of controlling the convergence region and rate of the series solution. It is confirmed that the HAM series solutions contain the Adomian decomposition method solution as special cases.


34A08 Fractional ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A45 Theoretical approximation of solutions to ordinary differential equations
Full Text: DOI


[1] Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974) · Zbl 0292.26011
[2] Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) · Zbl 0789.26002
[3] Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) · Zbl 0924.34008
[4] Li, C.P., Peng, G.J.: Chaos in Chen’s system with fractional order. Chaos Solitons Fractals 22(2), 443–450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[5] Wang, Y.H., Li, C.P.: Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle? Phys. Lett. A 363(5–6), 414–419 (2007) · doi:10.1016/j.physleta.2006.11.038
[6] Jafari, H., Daftardar-Gejji, V.: Solving a system of nonlinear fractional differential equations using Adomian decomposition. J. Comput. Appl. Math. 196, 644–651 (2006) · Zbl 1099.65137 · doi:10.1016/j.cam.2005.10.017
[7] Shawagfeh, N.T.: Analytical approximate solutions for nonlinear fractional differential equations. Appl. Math. Comput. 131, 517–29 (2002) · Zbl 1029.34003 · doi:10.1016/S0096-3003(01)00167-9
[8] Abdulaziz, O., Hashim, I., Chowdhury, M.S.H., Zulkifle, A.K.: Assessment of decomposition method for linear and nonlinear fractional differential equations. Far East J. Appl. Math. 28, 95–112 (2007) · Zbl 1134.26300
[9] Abdulaziz, O., Hashim, I., Ismail, E.S.: Approximate analytical solutions to fractional modified KdV equations. Math. Comput. Model. (2008). doi: 10.1016/j.mcm.2008.01.005 · Zbl 1165.35441
[10] Abdulaziz, O., Hashim, I., Momani, S.: Solving systems of fractional differential equations by homotopy-perturbation method. Phys. Lett. A 372, 451–459 (2008) · Zbl 1217.81080 · doi:10.1016/j.physleta.2007.07.059
[11] Abdulaziz, O., Hashim, I., Momani, S.: Application of homotopy-perturbation method to fractional IVPs. J. Comput. Appl. Math. 216, 574–584 (2008) · Zbl 1142.65104 · doi:10.1016/j.cam.2007.06.010
[12] Momani, S., Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365, 345–350 (2007) · Zbl 1203.65212 · doi:10.1016/j.physleta.2007.01.046
[13] Odibat, Z., Momani, S.: Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order. Chaos Solitons Fractals 36, 167–174 (2008) · Zbl 1152.34311 · doi:10.1016/j.chaos.2006.06.041
[14] Odibat, Z., Momani, S.: Application of variational iteration method to nonlinear differential equation of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 1, 271–279 (2006) · Zbl 1378.76084
[15] Momani, S., Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 31, 1248–1255 (2007) · Zbl 1137.65450 · doi:10.1016/j.chaos.2005.10.068
[16] Liao, S.J.: The proposed homotopy analysis techniques for the solution of nonlinear problems. Ph.D. dissertation, Shanghai Jiao Tong University, Shanghai (1992)
[17] Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC/Chapman and Hall, Boca Raton (2003)
[18] Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. (2008). doi: 10.1016/j.cnsns.2008.04.013
[19] He, J.H.: An approximate solution technique depending upon an artificial parameter. Commun. Nonlinear Sci. Numer. Simul. 3(2), 92–97 (1998) · Zbl 0921.35009 · doi:10.1016/S1007-5704(98)90070-3
[20] He, J.H.: Newton-like iteration method for solving algebraic equations. Commun. Nonlinear Sci. Numer. Simul. 3, 106–109 (1998) · Zbl 0918.65034 · doi:10.1016/S1007-5704(98)90073-9
[21] Abbasbandy, S.: The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006) · Zbl 1236.80010 · doi:10.1016/j.physleta.2006.07.065
[22] Sajid, M., Hayat, T.: Comparison of HAM and HPM methods for nonlinear heat conduction and convection equations. Nonlinear Anal. Real World Appl. (2007). doi: 10.1016/j.nonrwa.2007.08.007 · Zbl 1156.76436
[23] Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 361, 316–322 (2007) · Zbl 1170.76307 · doi:10.1016/j.physleta.2006.09.060
[24] Liao, S.J.: An approximate solution technique which does not depend upon small parameters (Part 2): an application in fluid mechanics. Int. J. Nonlinear Mech. 32, 815–822 (1997) · Zbl 1031.76542 · doi:10.1016/S0020-7462(96)00101-1
[25] Liao, S.J.: An explicit totally analytic approximation of Blasius viscous flow problems. Int. J. Nonlinear Mech. 34, 759–778 (1999) · Zbl 1342.74180 · doi:10.1016/S0020-7462(98)00056-0
[26] Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004) · Zbl 1086.35005 · doi:10.1016/S0096-3003(02)00790-7
[27] Liao, S.J., Pop, I.: Explicit analytic solution for similarity boundary layer equations. Int. J. Heat Mass Transfer 47, 75–78 (2004) · Zbl 1045.76008 · doi:10.1016/S0017-9310(03)00405-8
[28] Liao, S.J.: Comparison between the homotopy analysis method and homotopy perturbation method. Appl. Math. Comput. 169, 1186–1194 (2005) · Zbl 1082.65534 · doi:10.1016/j.amc.2004.10.058
[29] Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Transfer 48, 2529–3259 (2005) · Zbl 1189.76142 · doi:10.1016/j.ijheatmasstransfer.2005.01.005
[30] Ayub, M., Rasheed, A., Hayat, T.: Exact flow of a third grade fluid past a porous plate using homotopy analysis method. Int. J. Eng. Sci. 41, 2091–2103 (2003) · Zbl 1211.76076 · doi:10.1016/S0020-7225(03)00207-6
[31] Hayat, T., Khan, M., Asghar, S.: Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid. Acta Mech. 168, 213–232 (2004) · Zbl 1063.76108 · doi:10.1007/s00707-004-0085-2
[32] Hayat, T., Khan, M.: Homotopy solutions for a generalized second-grade fluid past a porous plate. Nonlinear Dyn. 42, 395–405 (2005) · Zbl 1094.76005 · doi:10.1007/s11071-005-7346-z
[33] Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation. Phys. Lett. A 15, 1–6 (2006) · Zbl 1273.65156
[34] Xu, H., Liao, S.J.: A series solution of the unsteady Von Karman swirling viscous flows. Acta Appl. Math. 94, 215–231 (2006) · Zbl 1122.35102 · doi:10.1007/s10440-006-9076-8
[35] Bataineh, A.S., Noorani, M.S.M., Hashim, I.: The homotopy analysis method for Cauchy reaction-diffusion problems. Phys. Lett. A 372, 613–618 (2008) · Zbl 1217.35101 · doi:10.1016/j.physleta.2007.07.069
[36] Bataineh, A.S., Noorani, M.S.M., Hashim, I.: Solving systems of ODEs by homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 13, 2060–2070 (2008) · Zbl 1221.65194 · doi:10.1016/j.cnsns.2007.05.026
[37] Song, L., Zhang, H.: Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation. Phys. Lett. A 367, 88–94 (2007) · Zbl 1209.65115 · doi:10.1016/j.physleta.2007.02.083
[38] Hashim, I., Abdulaziz, O., Momani, S.: Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Numer. Simul. (2007). doi: 10.1016/j.cnsns.2007.09.014 · Zbl 1221.65277
[39] Cang, J., Tan, Y., Xu, H., Liao, S.J.: Series solutions of non-linear Riccati differential equations with fractional order. Chaos Solitons Fractals (2007). doi: 10.1016/j.chaos.2007.04.018 · Zbl 1197.34006
[40] Xu, H., Cang, J.: Analysis of a time fractional wave-like equation with the homotopy analysis method. Phys. Lett. A 372, 1250–1255 (2008) · Zbl 1217.35111 · doi:10.1016/j.physleta.2007.09.039
[41] Gorenflo, R., Mainardi, F.: Fractional calculus: Integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997) · Zbl 0917.73004
[42] Daftardar-Gejji, V., Babakhani, A.: Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 293, 511–522 (2004) · Zbl 1058.34002 · doi:10.1016/j.jmaa.2004.01.013
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