On a Generalized Taylor Theorem:a Rational Proof of the Validity of the So-Called Homotopy Analysis Method

LIAO Shi-jun

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LIAO Shi-jun. On a Generalized Taylor Theorem:a Rational Proof of the Validity of the So-Called Homotopy Analysis Method[J]. Applied Mathematics and Mechanics, 2003, 24(1): 47-54.

Citation:

LIAO Shi-jun. On a Generalized Taylor Theorem:a Rational Proof of the Validity of the So-Called Homotopy Analysis Method[J]. Applied Mathematics and Mechanics, 2003, 24(1): 47-54.

Citation:

LIAO Shi-jun. On a Generalized Taylor Theorem:a Rational Proof of the Validity of the So-Called Homotopy Analysis Method[J]. Applied Mathematics and Mechanics, 2003, 24(1): 47-54.

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On a Generalized Taylor Theorem:a Rational Proof of the Validity of the So-Called Homotopy Analysis Method

LIAO Shi-jun

School of Naval Architecture & Ocean Engineering, Shanghai Jiaotong University, Shanghai 200030, China

Received Date: 2002-01-28
Rev Recd Date: 2002-10-15
Publish Date: 2003-01-15

Abstract

Abstract

A generalized Taylor series of a complex function was derived and some related theorems about its convergence region were given. The generalized Taylor theorem can be applied to greatly enlarge convergence regions of approximation series given by other traditional techniques. The rigorous proof of the generalized Taylor theorem also provides us with a rational base of the validity of a new kind of powerful analytic technique for nonlinear problems, namely the homotopy analysis method.
- Taylor series,
- convergence and summability of series,
- homotopy analysis method,
- perturbation

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References(17)

References

[1]	Newton I.On the binomial theorem for fractional and negative exponents[A].In:G D Walcott Ed.A Source Book in Mathematics[C].New York:McGraw Hill Book Company,1929,224-228.
[2]	Dienes P.The Taylor Series[M].Oxford:Dover,1931.
[3]	廖世俊.The proposed homotopy analysis method for nonlinear problems[D].博士论文.上海:上海交通大学,1992.
[4]	LIAO Shi-jun,An approximate solution technique not depending on small parameters:a special example[J].Internat J Non-Linear Mech,1995,30:371-380.
[5]	LIAO Shi-jun.A kind of approximate solution technique which does not depend upon small parameters (Part 2):an application in fluid mechanics[J].Internat J Non-Linear Mech,1997,32:815-822.
[6]	LIAO Shi-jun.A simple way to enlarge the convergence region of perturbation approximations[J].Nonlinear Dynamics,1999,19:93-110.
[7]	LIAO Shi-jun.An explicit,totally analytic approximation of Blasius' viscous flow problems[J].Internat J Non-Linear Mech,1999,34:759-778.
[8]	LIAO Shi-jun.A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate[J].J Fluid Mech,1999,385:101-128.
[9]	LIAO Shi-jun,Campo A.Analytic solutions of the temperature distribution in Blasius viscous flow problems[J].J Fluid Mech,2002,453:411-425.
[10]	LIAO Shi-jun,An analytic approximation of the drag coefficient for the viscous flow past a sphere[J].Internat J Non-Linear Mech,2002,37:1-18.
[11]	Nayfeh A H.Perturbation Methods[M].New York:John Wiley & Sons,Inc,2000.
[12]	Lyapunov A M.General Problem on Stability of Motion[M].London:Taylor & Francis,1992.(English version).
[13]	Karmishin A V,Zhukov A I,Kolosov V G.Methods of Dynamics Calculation and Testing for Thin-Walled Structures[M].Moscow:Mashinostroyenie,1990.
[14]	Adomian G.Nonlinear stochastic differential equations[J].J Math Anal Appl,1976,55:441-452.
[15]	Adomian,G.Solving Frontier Problems of Physics:the Decomposition Method[M].Boston:Kluwer Academic Publishers,1994.
[16]	Wazwaz A M.The decomposition method applied to systems of partial differential equations and to the reactioncdiffusion Brusselator model[J].Applied Mathematics and Computation,2000,110:251-264.
[17]	Shawagfeh N T.Analytical approximate solutions for nonlinear fractional differential equations[J].Applied Mathematics and Computation,2002,131:517-529.