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)
A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations. (English) Zbl 1185.65139
Summary: Based on the homotopy analysis method (HAM), a powerful algorithm is developed for the solution of nonlinear ordinary differential equations of fractional order. The proposed algorithm presents the procedure of constructing the set of base functions and gives the high-order deformation equation in a simple form. 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 . The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.
MSC:
65L99Numerical methods for ODE
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
References:
[1]Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations, Appl. numer. Math. 56, 80-90 (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[2]Tadjeran, C.; Meerschaert, M. M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. comput. Phys. 220, 813-823 (2007) · Zbl 1113.65124 · doi:10.1016/j.jcp.2006.05.030
[3]Lynch, V. E.; Carreras, B. A.; Del-Castillo-Negrete, D.; Ferriera-Mejias, K. M.; Hicks, H. R.: Numerical methods for the solution of partial differential equations of fractional order, J. comput. Phys. 192, 406-421 (2003) · Zbl 1047.76075 · doi:10.1016/j.jcp.2003.07.008
[4]Momani, S.; Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. math. Comput. 162, No. 3, 1351-1365 (2005) · Zbl 1063.65055 · doi:10.1016/j.amc.2004.03.014
[5]Momani, S.: An explicit and numerical solutions of the fractional KdV equation, Math. comput. Simul. 70, No. 2, 110-118 (2005) · Zbl 1119.65394 · doi:10.1016/j.matcom.2005.05.001
[6]Momani, S.: Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos soliton. Fract. 28, No. 4, 930-937 (2006) · Zbl 1099.35118 · doi:10.1016/j.chaos.2005.09.002
[7]Momani, S.; Odibat, Z.: Analytical solution of a time-fractional Navier – Stokes equation by Adomian decomposition method, Appl. math. Comput. 177, 488-494 (2006) · Zbl 1096.65131 · doi:10.1016/j.amc.2005.11.025
[8]Odibat, Z.; Momani, S.: Approximate solutions for boundary value problems of time-fractional wave equation, Appl. math. Comput. 181, 1351-1358 (2006)
[9]Odibat, Z.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlin. sci. Numer. simul. 7, No. 1, 27-34 (2006)
[10]Momani, S.; Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order, Chaos soliton. Fract. 31, No. 5, 1248-1255 (2007) · Zbl 1137.65450 · doi:10.1016/j.chaos.2005.10.068
[11]Momani, S.; Odibat, Z.: Numerical approach to differential equations of fractional order, J. comput. Appl. math. 207, No. 1, 96-110 (2007) · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015
[12]Odibat, Z.; Momani, S.: Numerical methods for solving nonlinear partial differential equations of fractional order, Appl. math. Model. 32, No. 1, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025
[13]Odibat, Z.; Momani, S.: Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos soliton. Fract. 36, No. 1, 167-174 (2008) · Zbl 1152.34311 · doi:10.1016/j.chaos.2006.06.041
[14]Momani, S.; Odibat, Z.: Comparison between homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. math. Appl. 54, No. 7-8, 910-919 (2007) · Zbl 1141.65398 · doi:10.1016/j.camwa.2006.12.037
[15]Momani, S.; Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. lett. A 365, No. 5-6, 345-350 (2007) · Zbl 1203.65212 · doi:10.1016/j.physleta.2007.01.046
[16]Chowdhury, M. S.; Hashim, I.; Momani, S.: The multistage homotopy perturbation method: a powerful scheme for handling the Lorenz system, Chaos soliton. Fract. 40, No. 4, 1929-1937 (2009) · Zbl 1198.65135 · doi:10.1016/j.chaos.2007.09.073
[17]Erturk, V.; Momani, S.; Odibat, Z.: Application of generalized differential transform method to multi-order fractional differential equations, Comm. nonlin. Sci. numer. Simul. 13, No. 8, 1642-1654 (2008) · Zbl 1221.34022 · doi:10.1016/j.cnsns.2007.02.006
[18]Odibat, Z.; Momani, S.: Generalized differential transform method for linear partial differential equations of fractional order, Appl. math. Lett. 21, No. 2, 194-199 (2008) · Zbl 1132.35302 · doi:10.1016/j.aml.2007.02.022
[19]Momani, S.; Odibat, Z.; Erturk, V.: Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Phys. lett. A 370, No. 5-6, 379-387 (2007) · Zbl 1209.35066 · doi:10.1016/j.physleta.2007.05.083
[20]Momani, S.; Odibat, Z.: A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula, J. comput. Appl. math. 220, No. 1-2, 85-95 (2008) · Zbl 1148.65099 · doi:10.1016/j.cam.2007.07.033
[21]Cang, J.; Tan, Y.; Xu, H.; Liao, S. J.: Series solutions of non-linear Riccati differential equations with fractional order, Chaos soliton. Fract. 40, No. 1, 1-9 (2009) · Zbl 1197.34006 · doi:10.1016/j.chaos.2007.04.018
[22]Hashim, I.; Abdulaziz, O.; Momani, S.: Homotopy analysis method for fractional ivps, Comm. nonlin. Sci. numer. Simul. 14, No. 3, 674-684 (2009) · Zbl 1221.65277 · doi:10.1016/j.cnsns.2007.09.014
[23]S.J. Liao, The Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems, Ph.D. dissertation, Shanghai Jiao Tong University, 1992 [in English].
[24]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
[25]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
[26]Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[27]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
[28]Liao, S. J.; Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations, Stud. appl. Math. 119, 297-354 (2007)
[29]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. 136, No. 2-3, 144-150 (2008)
[30]Abbasbandy, S.: Solitary wave solutions to the Kuramoto – Sivashinsky equation by means of the homotopy analysis method, Nonlinear dynam. 52, 35-40 (2008) · Zbl 1173.35646 · doi:10.1007/s11071-007-9255-9
[31]Abbasbandy, S.; Parkes, E. J.: Solitary smooth hump solutions of the Camassa – Holm equation by means of the homotopy analysis method, Chaos soliton. Fract. 36, No. 3, 581-591 (2008) · Zbl 1139.76013 · doi:10.1016/j.chaos.2007.10.034
[32]Podlubny, I.: Fractional differential equations, (1999)
[33]Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II, J. roy. Astral. soc. 13, 529-539 (1967)
[34]Cheng, J.; Liao, S.; Mohapatra, R. N.; Vajravelub, K.: Series solutions of nano boundary layer flows by means of the homotopy analysis method, J. math. Anal. appl. 343, No. 1, 233-245 (2008) · Zbl 1135.76016 · doi:10.1016/j.jmaa.2008.01.050