Homotopy perturbation method

The homotopy perturbation method (HPM) is a series expansion method used in the solution of nonlinear partial differential equations. The method employs a homotopy transform to generate a convergent series solution of differential equations. This gives flexibility in the choice of basis functions for the solution and the linear inversion operators (as compared to the Adomian decomposition method), while still retaining a simplicity that makes the method easily understandable from the standpoint of general perturbation methods. The HPM was introduced by Ji-Huan He of Shanghai University in 1998. The HPM is a special case of the homotopy analysis method (HAM) developed by Liao Shijun in 1992. The HAM uses a so-called convergence-control parameter to guarantee the convergence of approximation series over a given interval of physical parameters.

References

• He, J.H. (1999), "Homotopy perturbation technique", Computer Methods in Applied Mechanics and Engineering 178 (3-4): 257–262, Bibcode:1999CMAME.178..257H, doi:10.1016/S0045-7825(99)00018-3 ;
• Liao, S.J. (1992), The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University 
• Liao, S.J. (2003), Beyond Perturbation: Introduction to the Homotopy Analysis Method, Boca Raton: Chapman & Hall/ CRC Press, ISBN 1-58488-407-X 
• Liao, S.J. (2004), "On the homotopy analysis method for nonlinear problems", Applied Mathematics and Computation 147: 499–513, doi:10.1016/S0096-3003(02)00790-7 ;
• Liao, S.J. (2012), Homotopy Analysis Method in Nonlinear Differential Equation, Berlin & Beijing: Springer & Higher Education Press, ISBN 978-3642251313