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Homotopy analysis method in nonlinear differential equations

Author: Shijun Liao
Publisher: Berlin ; London : Springer, 2012.
Series: SpringerLink : Bücher
Edition/Format:   eBook : Document : EnglishView all editions and formats
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Genre/Form: Electronic books
Additional Physical Format: Print version
Print version:
Liao, Shijun, 1963-
Homotopy analysis method in nonlinear differential equations.
Beijing : Higher Education Press ; Heidelberg ; New York, N.Y. : Springer, ©2012
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Shijun Liao
ISBN: 9783642251313 3642251315 9783642251320 3642251323
OCLC Number: 1058136519
Language Note: English.
Reproduction Notes: Electronic reproduction.
Description: xiv, 565 Seiten : Illustrationen ; 25 cm
Details: Mode of access: World Wide Web.
Contents: Note continued: 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 Blasius flow --
Problems --
References --
4. 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 Note continued: 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 --
pt. II Mathematica Package BVPh and Its Applications --
7. 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 Note continued: 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 --
8. 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 Note continued: Problems --
References --
9. Nonlinear Eigenvalue 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 Note continued: 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 --
pt. 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 Note continued: 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.
Series Title: SpringerLink : Bücher
Responsibility: by Shijun Liao.

Abstract:

This book presents the latest developments and applications of the analytic approximation method for highly nonlinear problems, namely the homotopy analysis method (HAM). Reviews basic concepts and  Read more...

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