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)
On the homotopy analysis method for nonlinear problems. (English) Zbl 1086.35005
Summary: A powerful, easy-to-use analytic tool for nonlinear problems in general, namely the homotopy analysis method, is further improved and systematically described through a typical nonlinear problem, i.e., the algebraically decaying viscous boundary layer flow due to a moving sheet. Two rules, the rule of solution expression and the rule of coefficient ergodicity, are proposed, which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering. An explicit analytic solution is given for the first time, with recursive formulas for coefficients. This analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem.

35A25Other special methods (PDE)
76M25Other numerical methods (fluid mechanics)
76D10Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids)
Full Text: DOI
[1] Cole, J. D.: Perturbation methods in applied mathematics. (1968) · Zbl 0162.12602
[2] Nayfeh, Ali Hasan: Perturbation methods. (2000) · Zbl 0995.35001
[3] Lyapunov, A. M.: General problem on stability of motion (English translation). (1992) · Zbl 0786.70001
[4] Karmishin, A. V.; Zhukov, A. I.; Kolosov, V. G.: Methods of dynamics calculation and testing for thin-walled structures. (1990)
[5] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[6] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992
[7] Liao, S. J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Non-linear mech. 34, No. 4, 759-778 (1999) · Zbl 05137896
[8] Liao, S. J.: A simple way to enlarge the convergence region of perturbation approximations. Int. J. Non-linear dynam. 19, No. 2, 93-110 (1999) · Zbl 0949.70003
[9] Liao, S. J.: A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate. J. fluid mech. 385, 101-128 (1999) · Zbl 0931.76017
[10] 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
[11] Liao, S. J.: An analytic approximation of the drag coefficient for the viscous flow past a sphere. Int. J. Non-linear mech. 37, 1-18 (2002) · Zbl 1116.76335
[12] Liao, S. J.: An explict analytic solution to the Thomas--Fermi equation. Appl. math. Comput. 144, 433-444 (2003)
[13] S.J. Liao, K.F. Cheung, Analytic solution for nonlinear progressive waves in deep water, J. Engrg. Math., in press · Zbl 1112.76316
[14] Kuiken, H. K.: On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small. IMA J. Appl. math. 27, 387-405 (1981) · Zbl 0472.76045