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 homotopy analysis 2.6 Relations between the homotopy analysis method and other analytical 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 3 Methods for the Control of Convergence in Obtained Solutions 3.1 Selection of the auxiliary linear operator and base function representation 3.1.1 Method oflinear partition matching 3.1.2 Method of highest order differential matching 3.1.3 Method of complete differential matching 3.1.4 Initial versus boundary value problems 3.1.5 Additional options for the selection of an auxiliary linear operator 3.1.6 Remarks on the solution expression 3.2 The role of the auxiliary function 3.3 Selection of the convergence control parameter 3.4 0ptimal convergence control parameter value and the Lane-Emden equation of the first kind 3.4.1 Physical background 3.4.2 Analytic solutions via Taylor series 3.4.3 Analytic solutions via homotopy analysis References 4 Additional Techniques 4.1 Construction of multiple homotopies for coupled equations 4.2 Selection of an auxiliary nonlinear operator 4.3 Validation of the convergence control parameter 4.3.1 Convergence controlparameter plots （"h-plots"） 4.3.2 Minimized residual errors 4.3.3 Minimized approximate residual errors 4.4 Multiple homotopies and the construction of solutions to the Foppl-von Karman equations governing defiections of a thin flat plate 4.4.1 Physical background 4.4.2 Linearization and construction of perturbation solutions 4.4.3 Recursive solutions for the clamped edge boundary data 4.4.4 Special case: The thin plate limit h→0,v2→1 4.4.5 Control of error and selection of the convergence control parameters 4.5 Nonlinear auxiliary operators and local solutions to the Drinfel'd-Sokolov equations 4.6 Recent work on advanced techniques in HAM 4.6.1 Mathematical properties of h-curve in the frame work of the homotopy analysis method 4.6.2 Predictor homotopy analysis method andits application to some nonlinear problems 4.6.3 An optimal homotopy-analysis approach for strongly nonlinear differential equations 4.6.4 On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity References 5 Application of the Homotopy Analysis Method to Fluid Flow Problems 6 Further Applications of the Homotopy Analysis Method Subject Index Author Index
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