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)
Homotopy analysis method for fractional IVPs. (English) Zbl 1221.65277
Summary: The homotopy analysis method is applied to solve linear and nonlinear fractional initial-value problems (fIVPs). The fractional derivatives are described by Caputo’s sense. Exact and/or approximate analytical solutions of the fIVPs are obtained. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the approach.
MSC:
65M99Numerical methods for IVP of PDE
45A05Linear integral equations
References:
[1]Oldham, K. B.; Spanier, J.: The fractional calculus, (1974)
[2]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[3]Luchko Y, Gorenflo R. The initial-value problem for some fractional differential equations with Caputo derivative. Preprint Series A08-98. Fachbereich Mathematik und Informatic, Berlin, Freie Universitat; 1998.
[4]Podlubny, I.: Fractional differential equations, (1999)
[5]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
[6]Ray, S. S.; Bera, R. K.: Solution of an extraordinary differential equation by Adomian decomposition method, J appl math 4, 331-338 (2004) · Zbl 1080.65069 · doi:10.1155/S1110757X04311010
[7]Ray, S. S.; Bera, R. K.: An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl math comput 167, 561-571 (2005) · Zbl 1082.65562 · doi:10.1016/j.amc.2004.07.020
[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, No. 1, 95-112 (2007) · Zbl 1134.26300
[9]Abdulaziz O, Hashim I, Ismail ES. Approximate analytical solutions to fractional modified KdV equations. Far East J Appl Math, [in press]. · Zbl 1165.35441 · doi:10.1016/j.mcm.2008.01.005
[10]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
[11]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
[12]Odibat, Z.; Momani, S.: Application of variational iteration method to nonlinear differential equation of fractional order, Int J nonlinear sci numer simul 1, No. 7, 271-279 (2006)
[13]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
[14]Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D thesis, Shanghai Jiao Tong University; 1992.
[15]Liao, S. J.: An approximate solution technique which does not depend upon small parameters: a special example, Int J nonlinear mech 30, 371-380 (1995) · Zbl 0837.76073 · doi:10.1016/0020-7462(94)00054-E
[16]Liao, S. J.: An approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int J nonlinear mech 32, 815-822 (1997) · Zbl 1031.76542 · doi:10.1016/S0020-7462(96)00101-1
[17]Liao, S. J.: An explicit, totally analytic approximation of Blasius viscous flow problems, Int J nonlinear mech 34, No. 4, 759-778 (1999)
[18]Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[19]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
[20]Liao, S. J.; Campo, A.: Analytic solutions of the temperature distribution in Blasius viscous flow problems, J fluid mech 453, 411-425 (2002) · Zbl 1007.76014 · doi:10.1017/S0022112001007169
[21]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
[22]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
[23]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
[24]Hayat, T.; Khan, M.; Asghar, S.: Magnetohydrodynamic flow of an Oldroyd 6 – constant fluid, Appl math comput 155, 417-425 (2004) · Zbl 1126.76388 · doi:10.1016/S0096-3003(03)00787-2
[25]Abbasbandy, S.: Homotopy analysis method for heat radiation equations, Int comm heat mass transfer 34, 380-387 (2007)
[26]Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota – satsuma coupled KdV equation, Phys lett A 361, 478-483 (2007)
[27]Abbasbandy S. Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method. Chem Eng J 2007. doi:10.1016/j.cej.2007.03.022.
[28]Bataineh, A. S.; Noorani, M. S. M.; Hashim, I.: Solving systems of odes by homotopy analysis method, Commun nonlinear sci numer sim 13, No. 10, 2060-2070 (2008) · Zbl 1221.65194 · doi:10.1016/j.cnsns.2007.05.026
[29]Bataineh AS, Noorani MSM, Hashim I. Application of homotopy analysis method to nonlinear heat transfer equation [submitted for publication].
[30]Song L, Zhang H. Application of homotopy analysis method to fractional KdV – Burgers – Kuramoto equation. Phys Lett A, 2007. doi:10.1016/j.physleta.2007.02.083.
[31]Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order, , 223-276 (1997)
[32]Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor – corrector approach for the numerical solution of fractional differential equation, Nonlinear dyn 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341