Is there any program capable of simulating partial differential equations formulated in curvilinear coordinates, which uses these coordinates internally for the computations (i.e., no transform to cartesian coordinates)? And is there any program doing bifurcation analyses for such equations? And, finally, is there any program which can do both?
Dear Mario, I use Mathematica and for solving pde in curvilinear coordinates (e.g. spherical or cylindrical) first I transform equation from cartesian to curvilinear coordinates, then I (hopefully) solve it using Mathematica internal built-in functions like NDSolve; I think that these internal functions apply numerical methods directly to curvilinear coordinates. See link below for an example from Quantum Mechanics. Gianluca
https://mathematica.stackexchange.com/questions/72391/boundary-sphere-partial-differential-equation
Hello. In the Maple you can set any curvilinear coordinate system at first. And after this you can work in this coordinate system without any restrictions.
As conserning bifurcation analysis, I do not have exact information, because I did not do this. But in my mind the Maple is the best program for working with PDE symbolically. For example, you can do the separation of variables for given PDE. As I know, this build-in function is not exist in any other mathematical package.
If you need some help, you can write me a message.
This is an area that insterests me as well. I have been informed about a Matlab based package called pde2path (http://www.staff.uni-oldenburg.de/hannes.uecker/pde2path/). It is limited to elliptic systems of PDEs.
I haven't had a chance to use it; perhaps if you do, you could share your findings and experiences in using this package.
Мне такие программы неизвестны. Но я думаю, что предварительный перевод в криволинейные координаты даст возможность что-нибудь скомбинировать в пакете Matlab.
To make ODE bifurcation analysis, you can use "matcont" in Matlab (lookfor in web) and in linux you can use auto. In any case you must transform the PDEs to a set of ODEs
Try:
PDEcont
Bifurcation analysis of periodic solutions of PDEs and large-scale ODE systems
By Kurt Lust
https://dsweb.siam.org/Software/pdecont
Freely available for download on: http://www.kurtlust.net/CODE/r_PDEcont.html
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