同伦分析方法与非线性微分方程(英文版)

.appendix 2.1 derivation of ~ in (2.57)
appendix 2.2 derivation of (2.55) by the 2nd approach
appendix 2.3 proof of theorem 2.3
appendix 2.4 mathematica code (without iteration) for example 2.2
appendix 2.5 mathematica code (with iteration) for example 2.2
problems
references
3 optimal homotopy analysis method
3.1 introduction
3.2 an illustrative description
3.2.1 basic ideas
3.2.2 different types of optimal methods
3.3 systematic description
3.4 concluding remarks and discussions
appendix 3.1 mathematica code for blasins flow
problems
references
systematic descriptions and related theorems
4.1 brief frame of the homotopy analysis method
4.2 properties of homotopy-derivative
4.3 deformation equations
4.3.1 a brief history
4.3.2 high-order deformation equations
4.3.3 examples
4.4 convergence theorems
4.5 solution expression
4.5.1 choice of initial approximation
4.5.2 choice of auxiliary linear operator
4.6 convergence control and acceleration
4.6.1 optimal convergence-control parameter
4.6.2 optimal initial approximation
4.6.3 homotopy-iteration technique
4.6.4 homotopy-pade technique
4.7 discussions and open questions
references
5 relationship to euler transform
5.1 introduction
5.2 generalized taylor series
5.3 homotopy transform
5.4 relation between homotopy analysis method and euler
transform
5.5 concluding remarks
references
6 some methods based on the ham
6.1 a brief history of the homotopy analysis method
6.2 homotopy perturbation method
6.3 optimal homotopy asymptotic method
6.4 spectral homotopy analysis method
6.5 generalized boundary element method
6.6 generalized scaled boundary finite element method
6.7 predictor homotopy analysis method
references
part ii mathematica package bvph and its applications
mathematica package bvph
7.1 introduction
7.1.1 scope
7.1.2 brief mathematical formulas
7.1.3 choice of base function and initial guess
7.1.4 choice of the auxiliary linear operator
7.1.5 choice of the auxiliary function
7.1.6 choice of the convergence-control parameter co
7.2 approximation and iteration of solutions
7.2.1 polynomials
7.2.2 trigonometric functions
7.2.3 hybrid-base functions
7.3 a simple users guide of the bvph 1.0
7.3.1 key modules
7.3.2 control parameters
7.3.3 input
7.3.4 output
7.3.5 global variables
appendix 7.1 mathematica package bvph (version 1.0)
references
nonlinear boundary-value problems with multiple solutions
8.1 introduction
8.2 brief mathematical formulas
8.3 examples
8.3.1 nonlinear diffusion-reaction model
8.3.2 a three-point nonlinear boundary-value problem
8.3.3 channel flows with multiple solutions
8.4 concluding remarks
appendix 8.1 input data of bvph for example 8.3.1
appendix 8.2 input data of bvph for example 8.3.2
appendix 8.3 input data of bvph for example 8.3.3
problems
references
nonlinear eigenvalne equations with varying coefficients
9.1 introduction
9.2 brief mathematical formulas
9.3 examples
9.3.1 non-uniform beam acted by axial load
9.3.2 gelfand equation
9.3.3 equation with singularity and varying coefficient
9.3.4 multipoint boundary-value problem with multiple solutions
9.3.5 orr-sommerfeld stability equation with complex coefficient
9.4 concluding remarks
appendix 9.1 input data of bvph for example 9.3.1
appendix 9.2 input data of bvph for example 9.3.2
appendix 9.3 input data of bvph for example 9.3.3
appendix 9.4 input data of bvph for example 9.3.4
appendix 9.5 input data of bvph for example 9.3.5
problems
references
10 a boundary-layer flow with an infinite number of solutions
10.1 introduction
10.2 exponentially decaying solutions
10.3 algebraically decaying solutions
10.4 concluding remarks
appendix 10.1 input data of bvph for exponentially decaying solution.
appendix 10.2 input data of bvph for algebraically decaying solution
references
11 non-similarity boundary-layer flows
11.1 introduction
11.2 brief mathematical formulas
11.3 homotopy-series solution
11.4 concluding remarks
appendix 11.1 input data of bvph
references
12 unsteady boundary-layer flows
12.1 introduction
12.2 perturbation approximation
12.3 homotopy-series solution
12.3.1 brief mathematical formulas
12.3.2 homotopy-approximation
12.4 concluding remarks
appendix 12.1 input data of bvph
references
part iii applications in nonlinear partial differential equations
13 applications in finance: american put options
13.1 mathematical modeling
13.2 brief mathematical formulas
13.3 validity of the explicit homotopy-approximations
13.4 a practical code for businessmen
13.5 concluding remarks
appendix 13.1 detailed derivation offn(z) and gn()
appendix 13.2 mathematica code for american put option
appendix 13.3 mathematica code apoe for businessmen
references
14 two and three dimensional gelfand equation
14.1 introduction
14.2 homotopy-approximations of 2d gelfand equation
14.2.1 brief mathematical formulas
14.2.2 homotopy-approximations
14.3 homotopy-approximations of 3d gelfand equation
14.4 concluding remarks
appendix 14.1 mathematica code of 2d gelfand equation
appendix 14.2 mathematica code of 3d gelfand equation
references
15 interaction of nonlinear water wave and nonuniform currents
15.1 introduction
15.2 mathematical modeling
15.2.1 original boundary-value equation
15.2.2 dubreil-jacotin transformation
15.3 brief mathematical formulas
15.3.1 solution expression
15.3.2 zeroth-order deformation equation
15.3.3 high-order deformation equation
15.3.4 successive solution procedure
15.4 homotopy approximations
15.5 concluding remarks
appendix 15.1 mathematica code of wave-current interaction
references
16 resonance of arbitrary number of periodic traveling water waves
16.1 introduction
16.2 resonance criterion of two small-amplitude primary waves
16.2.1 brief mathematical formulas
16.2.2 non-resonant waves
16.2.3 resonant waves
16.3 resonance criterion of arbitrary number of primary waves
16.3.1 resonance criterion of small-amplitude waves
16.3.2 resonance criterion of large-amplitude waves
16.4 concluding remark and discussions
appendix 16.1 detailed derivation of high-order equation
references
index