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  • 流动非线性及其同伦分析:流体力学和传热(英文版)[精装]
  • 共3个商家     51.80元~58.60
  • 作者:瓦捷拉维鲁(KuppalapalleVajravelu)(作者),隔德(RobertA.VanGorder)(作者)
  • 出版社:高等教育出版社;第1版(2012年8月1日)
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  • ISBN:9787040354492

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    《流动非线性及其同伦分析:流体力学和传热(英文版)》适合于物理、应用数学、非线性力学、金融和工程等领域对强非线性问题解析近似解感兴趣的科研人员和研究生。

    作者简介

    作者:(美国)瓦捷拉维鲁( Kuppalapalle Vajravelu) (美国)隔德(Robert A.Van Gorder)

    瓦捷拉维鲁,为美国中佛罗里达大学数学系教授,机械、材料与航空和航天工程教授,Differential Equations and Nonlinear Mechanics的创刊主编。
    隔德,任职于美国中佛罗里达大学。

    目录

    Introduction 1
    References 3
    2 Principles of Homotopy Analysis 7
    2.1 Principles of homotopy and the homotopy analysis method 7
    2.2 Construction of the deformation equations 9
    2.3 Construction of the series solution 11
    2.4 Conditions for the convergence of the series solutions 12
    2.5 Existence and uniqueness of solutions obtained by homotopy analysis 14
    2.6 Relations between the homotopy analysis method and other analytical methods 14
    2.7 Homotopy analysis method for the Swift-Hohenberg equation 14
    2.7.1 Application of the homotopy analysismethod 16
    2.7.2 Convergence of the series solution and discussion of results 17
    2.8 Incompressible viscous conducting fluid approaching a permeable stretching surface 21
    2.8.1 Exact solutions for some special cases 23
    2.8.2 The case of G ≠ 0 25
    2.8.3 The case of G = 0 29
    2.8.4 Numerical solutions and discussion of the results 32
    2.9 Hydromagnetic stagnation point flow of a second grade fluid over a stretching sheet 34
    2.9.1 Formulation of the mathematical problem 38
    2.9.2 Exact solutions 38
    2.9.3 Constructing analytical solutions via homotopy analysis 40
    References 46
    3 Methods for the Control of Convergence in Obtained Solutions 53
    3.1 Selection of the auxiliary linear operator and base function representation 53
    3.1.1 Method of linear partition matching 56
    3.1.2 Method of highest order differential matching 57
    3.1.3 Method of complete differential matching 58
    3.1.4 Initial versus boundary value problems 59
    3.1.5 Additional options for the selection of an auxiliary linear operator 60
    3.1.6 Remarks on the solution expression 60
    3.2 The role of the auxiliary function 61
    3.3 Selection of the convergence control parameter 63
    3.4 Optimal convergence control parameter value and the Lane-Emden equation of the first kind 65
    3.4.1 Physical background 65
    3.4.2 Analytic solutions via Taylor series 66
    3.4.3 Analytic solutions via homotopy analysis 69
    References 75
    Additional Techniques 77
    4.1 Construction of multiple homotopies for coupled equations 77
    4.2 Selection of an auxiliary nonlinear operator 79
    4.3 Validation of the convergence control parameter 80
    4.3.1 Convergence control parameter plots ("h-plots") 80
    4.3.2 Minimized residual errors 80
    4.3.3 Minimized approximate residual errors 82
    4.4 Multiple homotopies and the construction of solutions to the F(o)ppl-von Kármán equations governing deflections of a thin flat plate 82
    4.4.1 Physical background 82
    4.4.2 Linearization and construction of perturbation solutions 84
    4.4.3 Recursive solutions for the clamped edge boundary data 85
    4.4.4 Special case: The thin plate limit h → 0,v2 → 1 87
    4.4.5 Control of error and selection of the convergence control parameters 88
    4.4.6 Results 90
    4.5 Nonlinear auxiliary operators and local solutions to the Drinfel'd-Sokolov equations 91
    4.6 Recent work on advanced techniques in HAM 97
    4.6.1 Mathematical properties of h-curve in the frame work of the homotopy analysis method 97
    4.6.2 Predictor homotopy analysis method and its application to some nonlinear problems 98
    4.6.3 An optimal homotopy-analysis approach for strongly nonlinear differential equations 98
    4.6.4 On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves 98
    References 99
    Application of the Homotopy Analysis Method to Fluid Flow Problems 101
    5.1 Thin film flow of a Sisko fluid on a moving belt 102
    5.1.1 Mathematical analysis of the problem 103
    5.1.2 Application of the homotopy analysis method 106
    5.1.3 Numerical results and discussion 108
    5.2 Nano boundary layers over stretching surfaces 112
    5.2.1 Formulation of the problem 113
    5.2.2 Application of the homotopy analysis method 115
    5.2.3 Analytical solutions via the homotopy analysis method 116
    5.2.4 Numerical solutions 118
    5.2.5 Discussion of the results 121
    5.3 Rotating flow of a third grade fluid by homotopy analysis method 123
    5.3.1 Mathematical formulation 124
    5.3.2 Solution of the problem 126
    5.3.3 Results and discussions 129
    5.4 Homotopy analysis for boundary layer flow of a micropolar fluid through a porous channel 133
    5.4.1 Description of the problem 134
    5.4.2 HAM solutions for velocity and micro-rotation fields 136
    5.4.3 Convergence of the solutions 139
    5.4.4 Results and discussion 140
    5.5 Solution of unsteady boundary-layer flow caused by an impulsively stretching plate 142
    5.5.1 Mathematical description 143
    5.5.2 Homotopy analytic solution 145
    5.5.3 Results and discussion 147
    References 151
    Further Applications of the Homotopy Analysis Method 157
    6.1 Series solutions of a nonlinear model of combined convective and radiative cooling of a spherical 157
    6.1.1 Basic equations 157
    6.1.2 Series solutions given by the HAM 159
    6.1.3 Result analysis 164
    6.1.4 Conclusions and discussions 169
    6.2 An ill-posed problem related to the flow of a thin fluid film over a sheet 171
    6.2.1 Introduction and physical motivation 171
    Contents
    6.2.2 Formulation of the three-parameterproblem 173
    6.2.3 A related four-parameter ill-posed problem 175
    6.2.4 Analytical solution for f (η) via the homotopy analysis method 178
    6.2.5 Results and discussion 181
    References 184
    Subject Index 185
    Author Index 187

    文摘

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    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.