Re: Partial Differential Equation

*To*: mathgroup at smc.vnet.net*Subject*: [mg26663] Re: [mg26623] Partial Differential Equation*From*: Reza Malek-Madani <research at usna.edu>*Date*: Wed, 17 Jan 2001 00:47:18 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Dear Phillipe: You need to write a shooting method or use the Galerkin scheme to solve a nonlinear BVP. Here is how you can use the shooting method in Mathematica for your equation: tfinal = 1; b1 = 0; F[b_] := Module[{sol}, sol = NDSolve[{T'[t] == S[t], S'[t] == -1/(t + 0.05) S[t] + 10^(-8) T[t]^4, T[0] == b, S[0] == -0.07}, {T, S}, {t, 0, tfinal}]; First[Evaluate[S[t]] /. sol /. t -> tfinal]] shoot=FindRoot[F[b] - b1, {b, 1, 0.9}]; sol=NDSolve[{T'[t] == S[t], S'[t]== -1/(t + 0.05) S[t] + 10^(-8) T[t]^4, T[0] == b/. shoot, S[0] == -0.07}, {T, S}, {t, 0, tfinal}]; Plot[Evaluate[S[t]/.sol], {t,0,tfinal}] Galerkin method is particularly useful if you have a nonlinear PDE or a system of ODEs in higher than two dimensions. Reza. ------------------------------------------------------------------------- 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 Sat, 13 Jan 2001, P. Poinas wrote: > Dear Group, > > I am trying to solve the following problem: > > \!\(\(\(NDSolve[{\(T''\)[R] + \((1/\((R + 0.05)\))\)\ \(T'\)[R] - > 10\^\(-8\)\ \ T[R]\^4 == 0, \(T'\)[0] == \(-0.07\), \ > \(T'\)[1] == > 0}, \ T, \ {R, \ 0, \ 1}]\)\(\ \)\)\) > > and I get the following message: > > NDSolve::"inrhs": "Differential equation does not evaluate to a number > or the \ > equation is not an nth order linear ordinary differential equation." > > I know that this error is because I am defining the derivative at 2 > different R values: > > at R=0, T'[0 ]= -0.07 > and R=1, T'[1] = 0 > > hence it is not an initial boundary condition. > > 1) How can I turn around the problem? > > > Actually, I found the Mathematica Book (V4) very weak on the subject. In > page 924, it seems possible in In[7] to find a solution to a linear > differential equation, even with 2 boundary conditions defined at 2 > different x values! My problem being not linear cannot therefore be > solved. But Mathematica does not mentioned the linearity as a show > stopper. > > 2) Does anybody know a better description of Mathematica's capacity? > > Thank you for helping me, > > > Philippe Poinas > > >