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
>
>
>
>
>