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Re: NDSolve with integral equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36150] Re: [mg36144] NDSolve with integral equation
  • From: Andrzej Kozlowski <andrzej at lineone.net>
  • Date: Fri, 23 Aug 2002 21:34:39 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

First of all, just look at your own posting below. You clearly have a 
*(Times) where a , (comma) should be. Presumably your input ought to be:

n = 5; NDSolve[{D[\[Sigma]norm[z, t], t] == 3*z*Integrate[\[Sigma]norm[z,
t]^n*z, {z, 0, 1}] - \[Sigma]norm[z, t]^n,
      \[Sigma]norm[z, 0] == 1.5*z, \[Sigma]norm[0, t] == 
0},\[Sigma]norm[z,
t], {z, 0.01, 1}, {t, 0.01, 2}]

However, even in the corrected version the equation can't be solved. 
First of all you will get the complaint:

NDSolve::"bcedge": "Boundary conditions must be specified at the edge of 
the \
spatial domain."

In other words Mathematica wants a boundary condition for  
\[Sigma]norm[z, 0.1] or alternatively you should use {z,0,1} in NDSolve. 
But actually I do not think this equation is solvable by any numerical 
scheme even if you could provide the initial conditions at the edge of 
the boundary that Mathematica requests. To evaluate the integral in your 
equation NDSolve needs to know the values of \[Sigma]norm[z, t] for all 
z between 0 and 1 and a given t, but this knowledge is not available at 
any stage of the evaluation. I am not really an expert, but this seems 
to me a clear example of an equation that is not solvable by any 
numerical means.

By the way, the fact that you know a solution to a differential 
equation, and even the fact that the solution is very simple does not 
imply that the equation can be solved by any known method, except of 
course guessing, which computer programs generally do not use.

  Andrzej Kozlowski
Toyama International University
JAPAN



On Friday, August 23, 2002, at 05:25  AM, Toshiyuki ((Toshi)) Meshii 
wrote:

> Hello,
>
> NDSolve seems to have difficulties with solving integral equation.
>
> n = 5; NDSolve[{D[\[Sigma]norm[z, t], t] == 
> 3*z*Integrate[\[Sigma]norm[z,
> t]^n*z, {z, 0, 1}] - \[Sigma]norm[z, t]^n,
>      \[Sigma]norm[z, 0] == 1.5*z, \[Sigma]norm[0, t] == 
> 0}*\[Sigma]norm[z,
> t], {z, 0.01, 1}, {t, 0.01, 2}]
>
> Mathematica returns a message
>
> NDSolve::"deql": "The first argument must have both an equation and an \
> initial condition."
>
> which I cannot understand.
> Can anybody tell what's wrong with my attempt?
>
> -Toshi
>
>
>
>
>



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