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Feb
13 |
comment |
Asymptotic expansions of solutions of nonlinear PDE's
Now I'm quite convinced that method you just presented can handle such nonlinearities. Also you encouraged me to learn more about that gradient expansion technique :) Thank you very much again, great answer! |
Feb
13 |
comment |
Asymptotic expansions of solutions of nonlinear PDE's
Thank you very much! Your answer is very helpful. May I just ask you to write about higher order expansion in just little more detail? (I'am unfamiliar with such techniques and don't see all things immidietly...) Also how does this expansion relates to actual solution? |
Feb
13 |
comment |
Asymptotic expansions of solutions of nonlinear PDE's
Thank you for references. I added my actual equation in main post. |
Feb
13 |
revised |
Asymptotic expansions of solutions of nonlinear PDE's
added 760 characters in body |
Feb
13 |
comment |
Asymptotic expansions of solutions of nonlinear PDE's
Also please look on second part of my post: is it obvious that we are getting series that is in any way related to actual true solution of our problem? Im quite sure that i even saw divergent " series solutions". Why noone cares if "solution" obtained in such a manner really defines a true solution? Or is it obvious and I just dont quite get all that black magic? |
Feb
13 |
comment |
Asymptotic expansions of solutions of nonlinear PDE's
Ok, I see it works in case of that example. The problem is that equations im currently working on are more complicated (they are in many dimensions, highly nonlinear: they involve more complicated rational functions of unknown and its variables). Could you reccomend me some sources where I can see those techniques explained? |
Feb
13 |
awarded | Editor |
Feb
13 |
revised |
Asymptotic expansions of solutions of nonlinear PDE's
added 1011 characters in body |
Feb
13 |
comment |
Asymptotic expansions of solutions of nonlinear PDE's
Does it really work in case of nonlinear equations? Even in case of equation like: $u_t=\frac{1}{u^2}u_{xx}+\frac{1}{u^4}(u_x)^2$ where functions $\alpha,\beta,\gamma,\phi$ are very simple rational functions inserting series into equation leads to terms that cant be ordered in polynomial fashion. (in case of rational functions i can of course multiply by denominators, but what in case of something like logarithm ?) |