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You are the visitor since March 1st, 2009

In Chinese

CIP 2010978



Dr. Shijun LIAO

Cheung Kong Professor

School of Naval Architecture, Ocean and Civil Engineering

Shanghai Jiao Tong University

800 Dongchuan Road, Shanghai 200240, China

Email:      sjliao@sjtu.edu.cn

Phone:      0086-21-3420 4445 (O)

FAX:      0086-21-3420 4445 (O)


Research field


l        Nonlinear Mechanics

l        Computational Fluid Dynamics (CFD)

l        Applied Mathematics

l        Ocean Engineering


Major Publications


Shijun Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, 2003.


Ø         MATHEMATICA Codes for Chapters 2, 6, 7, 8, and 9.      (free download)

Ø         Maple Codes for Chapters 2, 6, 7, 8, 9, 10, 11, 12 and 13.    (All the Maple codes are provided by Miss Pei Yang from East China Normal University. We express our grateful thanks to Miss Yang.)



²       This book deals with a very interesting mathematical technique that is rather powerful. An excellent reference to researchers, engineers, and interested individuals in helping them tackle nonlinear problems in an analytical fashion. A good subject index and an outstanding list of bibliography with 136 references cited. Very well written and is relatively easy to follow to the mathematically literate person. I highly recommend that it be acquired by interested individuals and libraries throughout.” -Applied Mathematics Review, Vol. 57, No. 5, September 2004 ( in details )

²       This monograph offers the opportunity to explore the details of the valuable new approach both in the theory and on many interesting examples. It will be useful to specialists working in applied nonlinear analysis.” -Zentralblatt MATH 1051



Some selected journal articles

1.        S.J. Liao, “Proposed homotopy analysis techniques for the solution of nonlinear problems”, Ph.D. dissertation, Shanghai Jiao Tong University,1992.

2.        S. J. Liao, “An approximate solution technique which does not depend upon small parameters: a special example”, International Journal of Nonlinear Mechanics, 30, 371-380, 1995.

3.        S. J. Liao, “An approximate solution technique which does not depend upon small parameters (Part 2): an application in fluid mechanics”,International Journal of Nonlinear Mechanics, Vol. 32, No. 5, pp. 815-822, 1997.

4.        S. J. Liao, “An explicit, totally analytic approximation of Blasius’ viscous flow problems”,  International Journal of Non-Linear Mechanics, Vol. 34, No. 4, pp. 759-778, 1999.

5.        S. J. Liao, “A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate”, Journal of Fluid Mechanics, Vol. 385, pp. 101-128, 1999.

6.        S. J. Liao and A. Campo, “Analytic solutions of the temperature distribution in Blasius viscous flow problems”, Journal of Fluid Mechanics, Vol. 453, pp. 411-425, 2002.

7.        S. J. Liao, “On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet”, Journal of Fluid Mechanics, vol. 488, pp. 189-212, 2003.

8.        S. J. Liao and K. F. Cheung, “Homotopy analysis of nonlinear progressive waves in deep water”, J. Engineering Mathematics, vol. 45, No. 2, pp. 105-116, 2003.

9.        S. J. Liao, “An explicit analytic solution to the Thomas-Fermi equation”, Applied Mathematics and Computation, vol. 144, pp. 495-506, 2003.

10.     S. J. Liao, “A new analytic algorithm of Lane-Emden type equations”, Applied Mathematics and Computation, vol. 142, No.1, pp. 1-16, 2003.

11.     S. J. Liao, “On the homotopy analysis method for nonlinear problems”, Applied Mathematics and Computation, vol. 147/2, pp. 499-513, 2004.

12.     S. J. Liao, “An analytic approximate approach for free oscillations of self-excited systems”, Int. J. Non-Linear Mech., Vol. 39, No.2, pp. 271-280, 2004.

13.     S. J. Liao, “A challenging nonlinear problem for numerical techniques”, J. Computational and Applied Mathematics, vol. 181, pp.467-472, 2005.

14.     S. J. Liao, “A new branch of solutions of boundary-layer flows over an impermeable stretched plate”, Int. J. Heat and Mass Transfer, vol. 48, pp. 2529-2539, 2005. 

15.     S. J. Liao, “Comparison between the homotopy analysis method and homotopy perturbation method”, Applied Mathematics and Computation, Vol. 169, Issue 2, Pages 1186-1194, 2005.

16.     H. Xu and S. J. Liao, “Analytic solutions of magnetohydrodynamic flows of non-Newtonian fluids caused by an impulsively stretching plate”, Journal of Non-Newtonian Fluid Mechanics, vol. 129, pp. 46-55, 2005.

17.     S. J. Liao, “Series solutions of unsteady boundary-layer flows over a stretching flat plate”, Studies in Applied Mathematics, Vol. 117, Issue 3, 2006.

18.     S. J. Liao, “An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate”, Communications in Nonlinear Science and Numerical Simulation, Volume 11, Issue 3, pp. 326-339, 2006.

19.     S. J. Liao, J. Su and A.T. Chwang, “Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body”, Int. J. Heat and Mass Transfer, vol. 49, pp. 2437-2445, 2006.

20.     S. J. Liao and E. Magyari, “Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones”, Z. angew. Math. Phys. (ZAMP), Volume 57, Number 5, 777 – 792, 2006.

21.     H. Xu, S. J. Liao and I. Pop, “Series solution of unsteady boundary layer flow of a micropolar fluid near the forward stagnation point of a plane surface”, Acta Mechanica, 2006.

22.     S.J. Liao and Y. Tan, “A general approach to obtain series solutions of nonlinear differential equations”, Studies in Applied Mathematics, Vol. 119, pp. 297-354, 2007.

23.     J. Cheng, S.J. Liao, R.N. Mohapatra and K. Vajravelu, “Series solutions of Nano-boundary-layer flows by means of the homotopy analysis method”, Journal of Mathematical Analysis and Applications, 343(1), pp. 233-245, 2008.

24.     S.J. Liao, “Beyond perturbation: a review on the homotopy analysis method and its applications”, Advance in Mechanics, Vol. 153, No.1, pp. 1-34, 2008.(in Chinese)

25.     J. Cheng, J. Cang and S.J. Liao, “On the interaction of deep water waves and exponential shear currents”, Z. angew. Math. Phys., 60, pp. 450-478, 2009.

26.     X. Hang, Z.L. Lin, S.J. Liao, J.Z. Wu and J. Majdalani, “Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls”,Physics of Fluids, 22 (2010, online).(new)

27.     S.J. Liao, “An optimal homotopy-analysis approach for strongly nonlinear differential equations”, Communications in Nonlinear Science and Numerical Simulation , Volume 15, Issue 8, pp. 2003-2016, 2010.




Full list of journal articles since 2001

Foreign visitors Dr. S. Abbasbandy      Yann Bouremel

Group Members


One of Millennium Problems: Navier-Stokes Equation



l        Universities in China

l        Chinese Academy of Science

l        Chinese Mathematical Society

l        ScienceWorld

l        MathWorld

l        International Mathematical Union (IMU)

l        International Association of Mathematical Physics (IAMP)

l        European Mathematical Society (EMS)

l        American Mathematical Society (AMS)

l        Society for Industrial and Applied Mathematics (SIAM)

l        American Society Of Mechanical Engineers (ASME)

l        American Institute of Mathematical Sciences

l        Isaac Newton Institute for Mathematical Sciences

l        Department of Applied Mathematics and Theoretical Physics (DAMTP)

l        The Nonlinear Centre, UNIVERSITY of CAMBRIDGE

l        Centre for Nonlinear Dynamics and its Applications (at UCL)

l        Bristol Centre For Applied Nonlinear Mathematics

l        Clay Mathematics Institute

l        Mathematics Institutes, Centers and Laboratories

l        Science

l        Nature