流動非線性及其同倫分析

內(nèi)容概要

　　科學(xué)工程中的很多問題是非線性的，難以解決。傳統(tǒng)的解析近似方法只對弱非線性問題有效，但無法很好地解決強非線性問題。同倫分析方法是近20年發(fā)展起來的一種有效的求解強非線性問題的解析近似方法。《流動非線性及其同倫分析：流體力學(xué)和傳熱（英文版）》介紹了同倫分析方法的最新理論進展，但不局限于方法的理論架構(gòu)，也給出了大量的流體力學(xué)和傳熱中的非線性問題實例，來體現(xiàn)同倫分析方法的應(yīng)用性?！　　读鲃臃蔷€性及其同倫分析：流體力學(xué)和傳熱（英文版）》適合于物理、應(yīng)用數(shù)學(xué)、非線性力學(xué)、金融和工程等領(lǐng)域?qū)姺蔷€性問題解析近似解感興趣的科研人員和研究生。

作者簡介

作者：（美國）瓦捷拉維魯（ Kuppalapalle Vajravelu） （美國）隔德（Robert A.Van Gorder）  瓦捷拉維魯，為美國中佛羅里達(dá)大學(xué)數(shù)學(xué)系教授，機械、材料與航空和航天工程教授，Differential Equations and Nonlinear Mechanics的創(chuàng)刊主編。 隔德，任職于美國中佛羅里達(dá)大學(xué)。

書籍目錄

1 Introduction
References
2　　Principles of Homotopy Analysis
　2.1　Principles of homotopy and the homotopy analysis method
　2.2　Construction of the deformation equations
　2.3　Construction of the series solution
　2.4　Conditions for the convergence of the series solutions
　2.5　Existence and uniqueness of solutions obtained by
homotopyanalysis
　2.6　Relations between the homotopy analysis method and
otheranalytical methods
　2.7　Homotopy analysis method for the Swift-Hohenberg
equation
　　2.7.1　Application of the homotopy analysis.method
　　2.7.2　Convergence of the series solution and discussion of
results
　2.8　Incompressible viscous conducting fluid approaching a
permeable stretching surface
　　2.8.1　Exact solutions for some special cases
　　2.8.2　The case of G　0
　　2.8.3　The case of G = 0
　　2.8.4　Numerical solutions and discussion of the results
　2.9　Hydromagnetic stagnation point flow of a second grade fluid
over
　　a stretching sheet
　　2.9.1　Formulation of the mathematical problem
　　2.9.2　Exact solutions
　　2.9.3　Constructing analytical solutions via homotopy
analysis
References
……

章節(jié)摘錄

版權(quán)頁：   插圖：   Thus, while arbitrary functions H (x) which vanish over portions of the relevantdomain are not useful in the homotopy analysis method, one has the option to employ such functions provided they only vanish over a set of measure zero. One maylook at this in another way. In the homotopy given in (3.22), we introduce the newauxiliary operator (3.23) which depends on 1/H (x). If we do the same here, we seethat if H (x) vanishes over a set of measure zero, then the auxiliary linear operatorconstructed via (3.23) will have singularities at all members of this set of measurezero. Such singularities greatly complicate the problem of solving the linear operator to obtain the terms gm (x) in the mth order deformation equations. In practice,these vanishing auxiliary functions will modify the particular solutions obtainedwhen solving for the gm (X)'S, which may complicate the recursive solution process.As such, it is usually best to avoid auxiliary functions H (x) which vanish at anypoint over the domain of the problem, unless one has a good reason to use them. Yet, if we are to avoid all such H (x) which vanish over any portion of the domain, we can just as well elect to solve the modified homotopy (3.22) using themodified auxiliary linear operator (3.23). This is why, in many cases, one simplytakes H (x) = 1 and then attempts to obtain the appropriate initial guess and auxiliary linear operator. In those cases where a different, yet nonvanishing auxiliaryfunction is used, one may simply modify the auxiliary linear operator to arrive atthe same results (i.e., the same series solutions). However, one should point out that the solution expression is determined by thechoice of auxiliary linear operator, L, the initial approximation and the functionH (x). When one does not know, a priori, the expression of solution, then one cansimply choose H (x) = 1. However, we should point out that simple and elegant solutions may be obtained in many cases by properly choosing an appropriate functionalform for H (x) = 1. 3.3 Selection of the convergence control parameter The convergence control parameter, h ≠ 0, was introduced by Liao in order to control the manner of convergence in the series solutions obtained via homotopy analysis. As a consequence, once the initial approximation, auxiliary linear operator,and auxiliary function are selected, the homotopy analysis method still provides onewith a family of solutions, dependent upon the convergence control parameter. Sincewe are free to select a member of this family as the approximate solution to a nonlinear equation, we find that the convergence region and the convergence rate of theseries solutions obtained via the homotopy analysis method depend on the convergence control parameter. As a consequence, we are free to enhance the convergenceregion and the convergence rate of a series solution via an appropriate choice of theconvergence control parameter h even for fixed choices of the initial approximation,auxiliary linear operator, and auxiliary function. Such a property makes the homotopy analysis method unique among analytical techniques and provides us with avery powerful tool to study nonlinear differential equations.

編輯推薦

《流動非線性及其同倫分析:流體力學(xué)和傳熱(英文版)》適合于物理、應(yīng)用數(shù)學(xué)、非線性力學(xué)、金融和工程等領(lǐng)域?qū)姺蔷€性問題解析近似解感興趣的科研人員和研究生。

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