Research field

l Nonlinear Mechanics

l Computational Fluid Dynamics (CFD)

l Applied Mathematics

l Ocean Engineering

Major Publications

BOOK

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.)

Ø BOOK REVIEWS:

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

Key articles about the HOMOTOPY ANALYSIS METHOD

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

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

3. Liao, S. J. (1997), “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. ( PDF download )

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

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

6. Allan, F.M. and Syam, M.I., “On the analytic solution of non-homogeneous Blasius problem”, J. Computational and Applied Mathematics, 182 (2005), pp. 362-371 ( PDF download )

7. S. Abbasbandy, “The application of the homotopy analysis method to nonlinear equations arising in heat transfer”, Physics Letters A, 360 (2006), pp. 109-113. ( PDF download )

8. S. Abbasbandy, “The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation”, Physics Letters A, 361 (2007), pp. 478-483. ( PDF download )

9. T. Hayat and M. Sajid, “On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder”, Physics Letter A, 361（2007）, pp. 316-322. ( PDF download )

10. K. Yabushita, M. Yamashita and K. Tsuboi, “An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method”, J. of Physics A: Mathematical and Theoretical, 40 (2007), pp. 8403-8416. ( PDF download )

11. M. Sajid, T. Hayat, and S. Asghar, “Comparison between the HAM and HPM solutions of tin film flows of non-Newtonian fluids on a moving belt”, Nonlinear Dynamics, 50 (2007), pp. 27-35. ( PDF download )

12. Allan, F.M., “Derivation of the Adomian decomposition method using the homotopy analysis method”, Applied Mathematics and Computation, 190 (2007), pp.6-14. ( PDF download )

13. 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. ( PDF download with electronic version of Mathematica Code )

14. Cheng, J. and Liao, S.J., Mohapatra , R.N. and Vajravelu, K., “Series solutions of Nano-boundary-layer flows by means of the homotopy analysis method”, Journal of Mathematical Analysis and Applications, 343(1):233-245, 2008. (http://dx.doi.org/10.1016/j.jmaa.2008.01.050) ( PDF download )

15. Liao, S.J., “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). ( PDF download )

16. M. Sajid and T. Hayat, “Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations”, Nonlinear Analysis: Real World Applications, 9 (2008), pp. 2296-2301. ( PDF download )

17. S.J. Liao , “Notes on the homotopy analysis method: Some definitions and theorems”, Commun Nonlinear Sci Numer Simulat, 14 (2009), pp. 983-997. ( PDF download )

18. S.J. Liao , “A general approach to get series solution of non-similarity boundary-layer flows”, Commun Nonlinear Sci Numer Simulat, 14 (2009), pp. 2144-2159. ( PDF download )

19. J. Cheng, J. Cang, and S.J. Liao, “On the interaction of deep water waves and exponential shear currents”, Z. angew. Math. Phys., online. ( PDF download )

20. M. Sajid and T. Hayat, “The application of homotopy analysis method to thin film flows of a third order fluid”, Chaos, Solitons and Fractals , 38 (2008), pp. 506-515. ( PDF download )

21. M. Sajid and T. Hayat, “Comparison of HAM and HPM solutions in heat radiation equations”, Int. Communications in Heat and Mass Transfer, 36 (2009), pp. 59-62. ( PDF download )

22. S.X. Liang and D. J. Jeffrey, “Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation”, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), pp. 4057-4064. ( PDF download )

23. Y.Y. Wu and K. F. Cheung, “Homotopy solution for nonlinear differential equations in wave propagation problems”, Wave Motion, 46 (2009), pp. 1-14. ( PDF download )

24. X.Y. Jiao, Y. Gao and S.Y. LOU, “Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation”, Science in China Series G: Physics, Mechanics & Astronomy, (2009, online). ( PDF download ) (New)

25. S.J. Liao, “On the relationship between the homotopy analysis method and Euler transform”, Communications in Nonlinear Science and Numerical Simulation, (2009, online). ( PDF download ) (New)

Some selected journal articles

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

2.        S. J. Liao,“An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate”, Communications in Nonlinear Science and Numerical Simulation, Vol. 11, No. 3, pp. 326-339, 2006. ( PDF download ).

3.         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 ( PDF )

4.        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. ( PDF ).

5.        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. ( PDF )

6.        S.J. Liao, A challenging nonlinear problem for numerical techniques, J. Computational and Applied Mathematics, vol. 181, pp.467-472, 2005. ( PDF download )

7.        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. ( PDF download ).

8.        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. ( PDF download )

9.        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. ( PDF )

10.    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. ( PDF )

11.    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. ( PDF )

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

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

14.     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. ( PDF download )

15.        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. ( PDF download )

16.        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. ( PDF download )

Full list of journal articles since 2001

Foreign visitors： Dr. S. Abbasbandy      Yann Bouremel

Group Members

One of Millennium Problems: Navier-Stokes Equation

Links

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