Homotopy Analysis Method (or Homotopy Perturbation Method)?

[crackjack]

Homotopy Analysis Method (or Homotopy Perturbation Method)??

How effective is this Homotopy Analysis Method (HAM) in solving coupled non-linear PDE? I see some papers, but they seem to be cross-referencing a small group of people most of the time. This sounds strange for a method that is so generic and so powerful (and has been around since 1992), as those papers all say.

Homotopy Perturbation Method (HPM) is a recent variant of HAM. But this also seems to suffer from the same strange behaviors as HAM's papers.

 

[HallsofIvy]

"HAM" and "HPM" are methods for finding approximate solutions to non-linear equations. The WKB method used in Quantum Mechanics to solve Shrodinger's equation is an example of a "HPM".

 

[crackjack]

Thanks for the reply.

I understand that it is an approximation method. But the difference from WKB seems to be that, in HAM, there is no apriori need for a small-parameter (with which to perturb). HAM introduces an artificial parameter with which a perturbation solution is constructed and then the parameter is taken to unity. The convergence of the solution is controlled by another parameter.

I want to know if HAM really brings something powerful to the table and, if so, why do I not see it beyond a relatively small self-referencing group.

 

[hunt_mat]

I will ask my supervisor this, as he is an authority on this kind of stuff.

 

[blue_raver22]

I would say that the Homotopy Analysis Method (HAM) and its variant, the Homotopy Perturbation Method (HPM), have been shown to be effective in solving coupled non-linear partial differential equations (PDEs). However, there may be some limitations or challenges associated with these methods that may explain the lack of wider application and cross-referencing in the literature.

One possible limitation is that these methods require a certain level of mathematical expertise and understanding in order to be properly applied. This may explain why there are only a small group of researchers who are actively using and publishing papers on these methods.

Additionally, the success of these methods may also depend on the specific problem being solved and the choice of the homotopy parameter. This could lead to variations in results and make it difficult to compare and cross-reference different studies.

Furthermore, there may be other more established methods that are preferred by researchers for solving coupled non-linear PDEs, leading to a smaller pool of papers using HAM and HPM.

Overall, while HAM and HPM have shown promise in solving coupled non-linear PDEs, further research and exploration is needed to fully understand their capabilities and limitations. It is important for researchers to continue to investigate and improve upon these methods in order to expand their applicability and impact in the scientific community.