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MathGroup Archive 2001

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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
> 
> 
> 



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