Re: NDSolve with NIntegrate for a PDE where the unknown function is integrated wrt its variables
- To: mathgroup at smc.vnet.net
- Subject: [mg119072] Re: NDSolve with NIntegrate for a PDE where the unknown function is integrated wrt its variables
- From: Dan Dubin <ddubin at ucsd.edu>
- Date: Sat, 21 May 2011 06:46:51 -0400 (EDT)
>Hello, i know that this issue has been raised many times, but i cannot
>find an answer in the previous posts.
>
>I tried to reformulate my problem in a simple way:
>
>NDSolve[{Derivative[0, 1][Yg][r, t]==NIntegrate[A[r1] Yg[r1, t] , {r1,
>1, r}] + B[r]/Yg[r, t], Yg[r, 0] == r}, {Yg}, {r, 1, 10}, {t, 0, 0.3}]
>
>where A and B are some functions, for example:
>
>A[r_] = r^-1; B[r_] = r + 10;
>
>Please help!
I think you need to do a little math to get this integro-differential
equation into shape. One simple approach -- take the derivative of
your equation with respect to r. That turns it into a PDE, first
order in both r and t. Then I would solve it by finite-differencing
the PDE on a grid in r. An extra boundary condition needs to be
specified then, Derivative[0, 1][Yg][0, t] == B[0]/Yg[0, t] . Second
approach -- simply finite-difference the original equation in r --
write the integral as a sum over the grid using Simpson's rule, for
instance. In both cases, note that you are solving many coupled ODEs
in time for Yg at different grid points. NDSolve should be able to
handle that even though the ODEs are nonlinear.
--
---------------
| Professor Dan Dubin
| Dept of Physics , Mayer Hall Rm 3126,
| UC San Diego La Jolla CA 92093-0319
| phone (858) - 534-4174 fax: (858)-534-0173
| ddubin at ucsd.edu