Ph.D. Dissertation


Liao, S.J.: Proposed homotopy analysis techniques for the solution of nonlinear problems. Ph.D. dissertation, Shanghai Jiaotong University, 1992.




Liao, S.J.: Beyond Perturbation – Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC, Boca Raton, 2003. ( Booking )


Liao, S.J.: Homotopy Analysis Method in Nonlinear Differential Equations. Springer & Higher Education Press, Heidelberg, 2012. ( Booking ) [ PDF ]


Liao, S.J., (ed.): Advances in Homotopy Analysis Method. World Scientific Press, 2013. ( Booking ) [Chapter 1]


Selected Journal Articles


1. Liao, S.J.: An approximate solution technique which does not depend upon small parameters: a special example.  Int. J. Non-Linear Mechanics, 30: 371-380 (1995).  [ PDF ]

2. Liao, S.J.: An approximate solution technique which does not depend upon small parameters (Part 2): an application in fluid mechanics. Int. J. Non-Linear Mechanics, 32: 815-822 (1997).  [ PDF ]

3. Liao, S.J.: An explicit, totally analytic approximation of Blasius’ viscous flow problems.  Int. J. Non-Linear Mechanics, 34(4): 759-778 (1999).  [ PDF ]

4. Liao, S.J.: A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate. J. Fluid Mechanics, 385: 101-128 (1999).  [ PDF ]

5. Liao, S.J. and Campo, A.: Analytic solutions of the temperature distribution in Blasius viscous flow problems. J. Fluid Mechanics, 453: 411-425 (2002).  [ PDF ]

6. Liao, S.J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mechanics, 488: 189-212 (2003).  [ PDF ]

7. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147: 499-513 (2004). [ PDF ]

8. Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat and Mass Transfer, 48: 2529-2539 (2005). [ PDF

9. Liao, S.J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate. Studies in Applied Mathematics, 117: 239-263 (2006). [ PDF ]  

10. Liao, S.J. and Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Studies in Applied Mathematics, 119: 297-354 (2007). [ PDF ]

11. Liao, S.J.: “Notes on the homotopy analysis method: some definitions and theorems”, Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 983-997, 2009. [ PDF ]

12. Xu, H., Lin, Z.L., Liao, S.J., Wu, J.Z. and Majdalani, J.: Homotopy-based solutions of the Navier–Stokes equations for a porous channel with orthogonally moving walls. Physics of Fluids, 22(5): 053601 (2010). [ PDF ]

13. Li, Y.J., Nohara, B.T. and Liao, S.J.: Series solutions of coupled Van der Pol equation by means of homotopy analysis method. J. Mathematical Physics, 51: 063517 (2010). [ PDF ]

14. Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 15:2003-2016 (2010). [ PDF ]

15. 廖世俊,“超越摄动:同伦分析方法基本思想及其应用”,《力学进展》,2008年,第38卷,第1期,1-34页(中文综述论文)[ PDF ]

16. 廖世俊,“同伦分析方法:求解强非线性问题的一个新途径”,《科学观察》,2009年,第4卷,第5期,48-49页(热点研究论文特约稿)[ PDF ]

17. 廖世俊,“同伦分析方法:研究背景和现状”,《科学观察》,2011年,第6卷,第6期,55-58页(热点研究论文特约稿)[ PDF ]

18. Liao, S.J.: On the reliability of computed chaotic solutions of non-linear differential equations. Tellus, 61(A): 550-564 (2009). [ PDF ] (arXiv:0901.2986)

19. Liao, S.J.: Chaos – A bridge from micro-level uncertainty to macroscopic randomness. Communications in Nonlinear Science and Numerical Simulation, 17: 2564-2569 (2012) [PDF] (arXiv:1108.4472). 

20.  Liao, S.J.: On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems. Chaos, Solitons & Fractals, 47: 1-12 (2013). [PDF]

21.  Xu, D.L., Lin, Z.L., Liao, S.J. and Stiassnie, M.: On the steady-state fully resonant progressive waves in water of finite depth. Journal of Fluid Mechanics, 710:379-418 (2012). [ PDF ]

22.  Liao, S.J.: Physical limit of prediction for chaotic motion of three-body problem. Communications in Nonlinear Science and Numerical Simulation, 19:601-616 (2013). [PDF]

23.  Liao, S.J.: “Do peaked solitary water waves indeed exist?”. Communications in Nonlinear Science and Numerical Simulation, 19:1792-1821 (2014). [PDF]  [Elsevier AudioSlides]

24.  Liao, S.J., Wang, P. F.: On the mathematically reliable long-term simulation of chaotic solutions of Lorenz equation in the interval [0, 10000]. SCIENCE CHINA Physics, Mechanics & Astronomy, 57:330-335 (2014). [PDF]

25.  Liu, Z. and Liao, S.J.: Steady-state resonance of multiple wave interactions in deep water. Journal of Fluid Mechanics, 742:664-700 (2014). [PDF]

26.  Li, X.C., Xu, D.L., Liao, S.J.: Observations of highly localized oscillons with multiple crests and troughs. PHYSICAL REVIEW E, 90:031001(R) (2014). [PDF] [Movie 1][Movie 2]

27.  Liu, Z., Xu, D. L., Li, J., Peng T., Alsaedi, A., Liao, S. J.: On the existence of steady-state resonant waves in experiments. Journal of Fluid Mechanics, 763:1-23 (2015). [PDF]

28.  Li, X.C., Yu, Z.Y., Liao, S.J.: Observation of two-dimensional Faraday waves in extremely shallow depth. PHYSICAL REVIEW E, 92:033014 (2015). [PDF] [Movie 1][Movie 2][Movie 3]

29.  Liao, S.J., Zhao, Y. L.: On the method of directly defining inverse mapping for nonlinear differential equations. Numerical Algorithms, pp.1-32 (2015). [PDF]

30.  Qin, S.J., Liao, S. J.: Influence of numerical noises on computer-generated simulation of spatio-temporal chaos. Chaos, Solitons & Fractals, 136:109790 (2020). [PDF]

31.  Hu, T.L., Liao, S. J.: On the risks of using double precision in numerical simulations of spatio-temporal chaos. Journal of Computational Physics, 418:109629 (2020). [PDF]

32.  Liao, S. J., Qin, S.J.: Ultra-Chaos: an insurmountable objective obstacle of reproducibility and replication. Advances in Applied Mathematics and Mechanics, 14(4):799-815 (2022). [PDF]

33.  Qin, S.J., Liao, S. J.: Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence. Journal of Fluid Mechanics, 948:A7 (2022). [PDF]

34.  Liao, S. J.: Avoiding small denominator problems by means of the homotopy analysis method. Advances in Applied Mathematics and Mechanics, 15(2):267-299 (2023). [PDF]

35.  Yang, Y., Qin, S.J., Liao, S. J.: Ultra-chaos of a mobile robot: a higher disorder than normal-chaos. Chaos, Solitons & Fractals, 167:113037 (2023). [PDF]