Re: Using NDSolve for 2-variables function ?

*To*: mathgroup at smc.vnet.net*Subject*: [mg31131] Re: Using NDSolve for 2-variables function ?*From*: Reza Malek-Madani <research at usna.edu>*Date*: Fri, 12 Oct 2001 03:36:50 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Dear Florent: NDSolve does not solve PDEs directly, but it is very effective if you combine it with the spectral method or the method of line. NDSolve is primarily an ODE initial value solver. You may want to look at Chapter 13 of my book "Advanced Engineering Mathematics with Mathematica and MATLAB", 1998. It is an introduction on how to use the characteristics method in Mathematica. In particular there is a treatment of the Burgers equation there. If I understood your code correctly, your f'[u] refers to the derivative of f[u] along characteristics so your equation seems to be Burgers-like. I would be happy to look at your quasilinear problem if you send me the exact initial-boundary value problem. Sincerely, Reza Malek-Madani ------------------------------------------------------------------------- Reza Malek-Madani Director of Research Research Office, MS 10m Phone: 410-293-2504 (FAX -2507) 589 McNair Road DSN: 281-2504 U.S. Naval Academy Nimitz Room 17 in ERC Annapolis MD 21402-5031 Email: research at usna.edu -------------------------------------------------------------------------- On Wed, 10 Oct 2001, Florent Saulnier wrote: > Dear MathGroup, > I'm trying to solve a quasi-linear PDE using the method of characteristics. > For this, I need to calculate a function - for instance f[r_,t_] - by > NDSolve (I simplified the equation for clarity) and then use it again in > another differential equation : > > Input[1] > f[r_,t_]=f[r]/.NDSolve[{f'[u]+u*f[u]==0,f[Sqrt[t]]==t^2},f,{u,Sqrt[t],10^9}] > [[1]][[1]] > > ... gives the following error messages : > NDSolve::ndnl : Endpoint Sqrt[t] in {u,Sqrt[t],1000000000} is not > a real number > ReplaceAll::reps : {uf[u]+f'[u]==0} is neither a list of > replacement rules nor a valid dispatch table, and so cannot be used for > replacing. > Output[1] Null (f[r]/.uf[u]+f'[u]==0) > > If I give the definition of f[r,t] with the sign :=, it gives me the > correct result at any given point, with the correct boundary conditions : > > Input[1] > f[r_,t_]:=f[r]/.NDSolve[{f'[u]+u*f[u]==0,f[Sqrt[t]]==t^2},f,{u,Sqrt[t],10^9} > ][[1]][[1]] > Out[1] Null^2 > Input[2] f[3,5] > Out[2] 3.3834 > Input[3] f[2,4] > Out[3] 16 > > The main problem is that I need f[r,t] for a second equation, and of course > its resolution cannot be achieved : > > Input[1] g[t_]=h[t]/.NDSolve[{h'[u]-f[h[u],u]==0,h[1]==1},h,{u,1,10}][[1]][[1]] > > ...which gives the same error messages : > NDSolve::ndnl : Endpoint Sqrt[t] in {u,Sqrt[t],1000000000} is not > a real number > ReplaceAll::reps : {uf[u]+f'[u]==0} is neither a list of > replacement rules nor a valid dispatch table, and so cannot be used for > replacing. > > Could you please help me about these problems ? > Is there any other instructions or objects I could use for it ? > Thanks a lot ! > Florent Saulnier > >