# How can I perform a scan and plot of a parameter space?

I have three functions dependent on three independent variables, like so:

H[u,m,s]
S[u,m,s]
SH[u,m,s]


However, I don't actually know these functions--they are solutions to three partial differential equations much like this (very simplified) example:

DEQ1 = u*D[H[u,m,s],u]-12*S[u,m,s]+1/4*SH[u,m,s]==0;
DEQ2 = u*D[S[u,m,s],u]-2*S[u,m,s]+6*SH[u,m,s]==0;
DEQ3 = u*D[SH[u,m,s],u]+3*S[u,m,s]+4*SH[u,m,s]==0;


I want to impose the following conditions:

H[u,m,s] > 0
0 < S[u,m,s] < 4*Pi
-4*Pi < SH[u,m,s] < 4*Pi


and plot the allowed parameter space that satisfies all of these conditions ('s' as a function of 'm', for a specified range of 'u'). How would I do this? I have absolutely no idea. I need to do a 'scan' of a parameter space and plot the points (the functions are such that I can only solve numerically), but I don't know how to do this. Any insight would be much appreciated.

• The ODEs as specified can be solved with DSolve up to three constants, which must be specified. With this done, the resulting expressions together with the constrains in the question probably can be used to define an ImplicitRegion. – bbgodfrey Sep 7 '16 at 19:15
• As I mentioned, those are overly simplified examples and the actual differential equations must be solved numerically. – Ash Arsenault Sep 7 '16 at 19:17
• Then, the equations can be solved with ParametricNDSolve, with the upper boundaries of the integration used as parameters, although constants of integration still must be specified. I am uncertain whether the resulting numerical functions can be used with ImplicitRegion. I imagine that the total computation will be very slow. It is difficult to say more without specifics of the PDEs. – bbgodfrey Sep 7 '16 at 19:28
• Thank you! ParametricNDSolve worked in terms of the solution. Just trying to work around the plotting issue now. :) – Ash Arsenault Sep 7 '16 at 20:03
• Perhaps, ContourPlot3D or ListContourPlot3D to display the boundary surfaces. – bbgodfrey Sep 7 '16 at 20:21