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Series solutions of non-linear Riccati differential equations with fractional order. (English) Zbl 1197.34006

Summary: Based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve non-linear Riccati differential equation with fractional order. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter \(\hbar/2 \pi\). Besides, it is proved that well-known Adomian’s decomposition method is a special case of the homotopy analysis method when \(\hbar/2 \pi - 1\). This work illustrates the validity and great potential of the homotopy analysis method for the non-linear differential equations with fractional order. The basic ideas of this approach can be widely employed to solve other strongly non-linear problems in fractional calculus.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
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[1] Podlubny, I., Fractional differential equations. an introduction to fractional derivatives, fractional differential equations, some methods of their solution and some of their applications, (1999), Academic Press SanDiego · Zbl 0924.34008
[2] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[3] Kemple S, Beyer H. Global and causal solutions of fractional differential equations. In: Transform methods and special functions: Varna96, Proceedings of 2nd international workshop (SCTP), Singapore; 1997. p. 210-6.
[4] Shawagfeh, N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl math comput, 131, 517-529, (2002) · Zbl 1029.34003
[5] Momani, S., Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, Appl math comput, 165, 459-472, (2005) · Zbl 1070.65105
[6] Al-Khaled, K.; Momani, S., An approximate solution for a fractional diffusion-wave equation using the decomposition method, Appl math comput, 165, 473-483, (2005) · Zbl 1071.65135
[7] Momani, S.; Shawagfeh, N.T., Decomposition method for solving fractional Riccati differential equations, Appl math comput, 182, 1083-1092, (2006) · Zbl 1107.65121
[8] Caputo, M., Linear models of dissipation whose q is almost frequency independent, Part II geophys JR astr soc, 13, 529-539, (1967)
[9] Luchko Y, Gorenflo R. The initial value problem for some fractional differential equations with the Caputo derivative. Fachbereich Mathematik und Informatick, Freie Universitat Berlin., Preprint Series A08-98. · Zbl 0931.44003
[10] Adomian, G., Nonlinear stochastic differential equations, J math anal appl, 55, 441-452, (1976) · Zbl 0351.60053
[11] Liao SJ. The proposed homotopy analysis techniques for the solution of nonlinear problems. Ph.D. dissertation. Shanghai Jiao Tong University; 1992 [in English].
[12] Liao, S.J., A kind of approximate solution technique which does not depend upon small parameters: a special example, Int J non-linear mech, 30, 371-380, (1995)
[13] Liao, S.J., An approximate solution technique which does not depend upon small parameters (part 2): an application in fluid mechanics, Int J non-linear mech, 32, 815-822, (1997) · Zbl 1031.76542
[14] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton
[15] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl math comput, 147, 499-513, (2004) · Zbl 1086.35005
[16] Liao SJ, Tan Y. A general approach to obtain series solutions of nonlinear differential equations. Stud Appl Math; in press.
[17] Lyapunov, A.M., (1892) general problem on stability of motion, (1992), Taylor & Francis London, [English translation] · Zbl 0786.70001
[18] Karmishin, A.V.; Zhukov, A.T.; Kolosov, V.G., Methods of dynamics calculation and testing for thin-walled structures, (1990), Mashinostroyenie Moscow, [in Russian]
[19] Hayat, T.; Sajid, M., On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder, Phys lett A, 361, 316-322, (2007) · Zbl 1170.76307
[20] Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of tin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dynam; in press. · Zbl 1181.76031
[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
[22] Abbasbandy, S., The application of homotopy analysis method to solve a generalized hirota – satsuma coupled KdV equation, Phys lett A, 361, 478-483, (2007) · Zbl 1273.65156
[23] He, J.H., Homotopy perturbation technique, Comput methods appl mech eng, 178, 257-262, (1999) · Zbl 0956.70017
[24] Liao, S.J., An analytic approximate approach for free oscillations of self-excited systems, Int J non-linear mech, 39, 2, 271-280, (2004) · Zbl 1348.34071
[25] Liao, S.J.; Cheung, K.F., Homotopy analysis of nonlinear progressive waves in deep water, J eng math, 45, 2, 105-116, (2003) · Zbl 1112.76316
[26] Liao, S.J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J fluid mech, 488, 189-212, (2003) · Zbl 1063.76671
[27] Xu, H.; Liao, S.J., Series solutions of unsteady magnetohydrodynamic flows of non-Newtonian fluids caused by an impulsively stretching plate, J non-Newtonian fluid mech, 129, 46-55, (2005) · Zbl 1195.76069
[28] Liao, S.J., Series solutions of unsteady boundary-layer flows over a stretching flat plate, Stud appl math, 117, 3, 2529-2539, (2006) · Zbl 1189.76142
[29] Liao, S.J., An explicit analytic solution to the thomas – fermi equation, Appl math comput, 144, 495-506, (2003) · Zbl 1034.34005
[30] Wang, C., On the explicit analytic solution of cheng-chang equation, Int J heat mass transfer, 46, 10, 1855-1860, (2003) · Zbl 1029.76050
[31] 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
[32] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int J eng sci, 42, 123-135, (2004) · Zbl 1211.76009
[33] Xu, H., An explicit analytic solution for convective heat transfer in an electrically conducting fluid at a stretching surface with uniform free stream, Int J eng sci, 43, 859-874, (2005) · Zbl 1211.76159
[34] Zhu, S.P., A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield, Anziam J, 47, 477-494, (2006) · Zbl 1147.91336
[35] Zhu, S.P., An exact and explicit solution for the valuation of American put options, Quantitative finance, 6, 229-242, (2006) · Zbl 1136.91468
[36] Baker, G.A., Essentials of PadĂ© approximants, (1975), Academic Press London · Zbl 0315.41014
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