zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
WorldCat.org
Full Text: DOI
References:
[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) · Zbl 0924.34008
[2] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · 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 · doi:10.1016/S0096-3003(01)00167-9
[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 · doi:10.1016/j.amc.2004.06.025
[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 · doi:10.1016/j.amc.2004.06.026
[7] Momani, S.; Shawagfeh, N. T.: Decomposition method for solving fractional Riccati differential equations, Appl math comput 182, 1083-1092 (2006) · Zbl 1107.65121 · doi:10.1016/j.amc.2006.05.008
[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 · doi:10.1016/0022-247X(76)90174-8
[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) · Zbl 0837.76073 · doi:10.1016/0020-7462(94)00054-E
[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 · doi:10.1016/S0020-7462(96)00101-1
[14] Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[15] 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
[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) · Zbl 0786.70001
[18] Karmishin, A. V.; Zhukov, A. T.; Kolosov, V. G.: Methods of dynamics calculation and testing for thin-walled structures, (1990)
[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 · doi:10.1016/j.physleta.2006.09.060
[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 · doi:10.1007/s11071-006-9140-y
[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, No. 2, 271-280 (2004) · Zbl 05138450
[25] Liao, S. J.; Cheung, K. F.: Homotopy analysis of nonlinear progressive waves in deep water, J eng math 45, No. 2, 105-116 (2003) · Zbl 1112.76316 · doi:10.1023/A:1022189509293
[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 · doi:10.1017/S0022112003004865
[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 · doi:10.1016/j.jnnfm.2005.05.005
[28] Liao, S. J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate, Stud appl math 117, No. 3, 2529-2539 (2006) · Zbl 1145.76352 · doi:10.1111/j.1467-9590.2006.00354.x
[29] Liao, S. J.: An explicit analytic solution to the Thomas -- Fermi equation, Appl math comput 144, 495-506 (2003) · Zbl 1034.34005 · doi:10.1016/S0096-3003(02)00423-X
[30] Wang, C.: On the explicit analytic solution of cheng-chang equation, Int J heat mass transfer 46, No. 10, 1855-1860 (2003) · Zbl 1029.76050 · doi:10.1016/S0017-9310(02)00470-2
[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 · doi:10.1016/S0020-7225(03)00207-6
[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 · doi:10.1016/S0020-7225(03)00281-7
[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 · doi:10.1016/j.ijengsci.2005.01.005
[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 · doi:10.1017/S1446181100010087 · http://www.austms.org.au/Publ/ANZIAM/V47P4/2378.html
[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 · doi:10.1080/14697680600699811
[36] Baker, G. A.: Essentials of Padé approximants, (1975) · Zbl 0315.41014