zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 new spectral-homotopy analysis method for solving a nonlinear second order BVP. (English) Zbl 1222.65090
Summary: A modification of the homotopy analysis method (HAM) for solving nonlinear second-order boundary value problems (BVPs) is proposed. The implementation of the new approach is demonstrated by solving the Darcy–Brinkman–Forchheimer equation for steady fully developed fluid flow in a horizontal channel filled with a porous medium. The model equation is solved concurrently using the standard HAM approach and numerically using a shooting method based on the fourth order Runge–Kutta scheme. The results demonstrate that the new spectral homotopy analysis method is more efficient and converges faster than the standard homotopy analysis method.
65L99Numerical methods for ODE
[1]Abbasbandy, S.: The application of the homotopy analysis method to nonlinear equations arising in heat transfer, Phys lett A 360, 10913 (2006)
[2]Abbasbandy, S.: Homotopy analysis method for heat radiation equations, Int commun heat mass transfer 34, 3807 (2007)
[3]Adomian, G.: Nonlinear stochastic differential equations, J math anal appl 55, 44152 (1976) · Zbl 0351.60053 · doi:10.1016/0022-247X(76)90174-8
[4]Adomian, G.: A review of the decomposition method and some recent results for nonlinear equations, Comp math appl 21, 10127 (1991) · Zbl 0732.35003 · doi:10.1016/0898-1221(91)90220-X
[5]Awartani, M. M.; Hamdan, M. H.: Fully developed flow through a porous channel bounded by flat plates, Appl math comput 169, 749-757 (2005) · Zbl 1151.76592 · doi:10.1016/j.amc.2004.09.087
[6]Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A.: Spectral methods in fluid dynamics, (1988) · Zbl 0658.76001
[7]Dinarvand, S.; Rashidi, M. M.: A reliable treatment of a homotopy analysis method for two-dimensional viscous flow in a rectangular domain bounded by two moving porous walls, Nonlinear anal: real world appl (2009)
[8]Don, W. S.; Solomonoff, A.: Accuracy and speed in computing the Chebyshev collocation derivative, SIAM J sci comput 16, No. 6, 1253-1268 (1995) · Zbl 0840.65010 · doi:10.1137/0916073
[9]Ellahi, R.: Effects of the slip boundary condition on non-Newtonian flows in a channel, Commun nonlinear sci numer simul 14, 1377-1384 (2009)
[10]Hayat, T.; Sajid, M.: Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet, Int J heat mass transfer 50, 7584 (2007)
[11]Hayat, T.; Abbas, Z.; Sajid, M.; Asghar, S.: The influence of thermal radiation on MHD flow of a second grade fluid, Int J heat mass transfer 50, 93141 (2007) · Zbl 1124.80325 · doi:10.1016/j.ijheatmasstransfer.2006.08.014
[12]Hayat, T.; Sajid, M.: Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid, Int J eng sci 45, 393401 (2007) · Zbl 1213.76137 · doi:10.1016/j.ijengsci.2007.04.009
[13]Hayat, T.; Ahmed, N.; Sajid, M.; Asghar, S.: On the MHD flow of a second grade fluid in a porous channel, Comp math appl 54, 40714 (2007) · Zbl 1123.76072 · doi:10.1016/j.camwa.2006.12.036
[14]Hayat, T.; Khan, M.; Ayub, M.: The effect of the slip condition on flows of an Oldroyd 6-constant fluid, J comput appl 202, 40213 (2007) · Zbl 1147.76550 · doi:10.1016/j.cam.2005.10.042
[15]Hayat, T.; Ahmed, N.; Sajid, M.: Analytic solution for MHD flow of a third order fluid in a porous channel, J comp appl math 214, 572-582 (2008) · Zbl 1144.76059 · doi:10.1016/j.cam.2007.03.013
[16]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
[17]Hayat, T.; Naz, R.; Sajid, M.: On the homotopy solution for poiseulle flow of a fourth grade fluid, Commun nonlinear sci numer simul 15, No. 3, 581-589 (2010) · Zbl 1221.76033 · doi:10.1016/j.cnsns.2009.04.024
[18]Hayat, T.; Ellahi, R.; Ariel, P. D.; Asghar, S.: Homotopy solutions for the channel flow of a third grade fluid, Nonlinear dyn 45, 55-64 (2006) · Zbl 1100.76005 · doi:10.1007/s11071-005-9015-7
[19]He, J. H.: Homotopy perturbation technique, Comput methods appl mech eng 178, 257-262 (1999)
[20]He, J. H.: A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int J non-linear mech 35, 37-43 (2000) · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[21]Hooman, K.: A perturbation solution for forced convection in a porous-saturated duct, J comp appl math 211, 57-66 (2008) · Zbl 1132.76051 · doi:10.1016/j.cam.2006.11.005
[22]Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University; 1992.
[23]Liao, S. J.: Beyond perturbation: introduction to homotopy analysis method, (2003)
[24]Liao, S. J.: Comparison between the homotopy analysis method and the homotopy perturbation method, Appl math comput 169, 1186-1194 (2005) · Zbl 1082.65534 · doi:10.1016/j.amc.2004.10.058
[25]Liao, S. J.: Notes on the homotopy analysis method: some definitions and theories, Commun nonlinear sci numer simul 14, 983-997 (2009) · Zbl 1221.65126 · doi:10.1016/j.cnsns.2008.04.013
[26]Lyapunov, A. M.: General problem on stability of motion, (1992)
[27]Mehmood, A.; Ali, A.: Heat transfer analysis of three dimensional flow in a channel of lower stretching wall, J Taiwan inst chem eng (2009)
[28]Rach, R.: On the Adomian method and comparisons with picards method, J math anal appl 10, 13959 (1984)
[29]Rashidi, M. M.; Dinarvand, S.: Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by homotopy analysis method, Nonlinear anal: real world appl 10, 2346-2356 (2009) · Zbl 1163.34307 · doi:10.1016/j.nonrwa.2008.04.018
[30]Sajid, M.; Abbas, Z.; Hayat, T.: Homotopy analysis for boundary layer flow of micropolar fluid through a porous channel, Appl math model 33, 4120-4125 (2009) · Zbl 1205.76201 · doi:10.1016/j.apm.2009.02.006
[31]Trefethen LN. Spectral methods in MATLAB, SIAM; 2000.
[32]Van Gorder, R. A.; Vajravelu, K.: On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach, Commun nonlinear sci numer simul 14, 4078-4089 (2009) · Zbl 1221.65208 · doi:10.1016/j.cnsns.2009.03.008