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Re: Functions in partial differential equations with different number

  • To: mathgroup at smc.vnet.net
  • Subject: [mg99522] Re: Functions in partial differential equations with different number
  • From: Blue Fly <blueflyspin at gmail.com>
  • Date: Thu, 7 May 2009 06:34:56 -0400 (EDT)
  • References: <887b51cc0905051139t2a53cfbeydc85817e54684c03@mail.gmail.com>

Thanks for those who replied to my message and provided the suggestions. I
would like to clarify my question a bit more:
Consider the following set of equations of motion:

{-I*v_g*D[R[x,t],x] + V*DiracDelta[x]*F[t] ==I*D[R[x,t],t], \Omega*F[t] +
V*R[0,t]== I*D[F[t],t]},

which describes the one-dimensional wave in a one-way waveguide (i.e., no
reflection) incident upon an excitable localized target at x=0. v_g is the
wave velocity; V is the coupling constant, while \Omega is the resonance
frequency of the localized target (in the following, v_g = 1, V=1, and
\Omega = 2). Also, R[0, t] is taken as (R[0^+, t] + R[0^-, t])/2, for all t.

This set of equations has analytic solution for constant frequency e (take
v_g=1):

R_E =( UnitStep[-x] + UnitStep [x] t_e ) Exp[I* e*x - I*e*t] , where t_e =
(e-\Omega - I* V^2/2)/(e-\Omega +I* V^2/2),

F_E = V/(e-\Omega+ I*V^2/2) Exp[-I*e*t].

I was wondering if it's possible to simulate the original set of equations
of motions numerically (with DiracDelta approximated by some regularized
functions, of course).

Thanks.

Dave

On Tue, May 5, 2009 at 11:39 AM, Blue Fly <blueflyspin at gmail.com> wrote:

> Hi, I was trying to solve the following set of PDE:
> s=0.01;
> NDSolve[{-I*D[R[x,t],x] + (s/(Pi (x^2+ s^2)))*F[t] ==I*D[R[x,t],t], 2 F[t]
> + R[0,t]== I*D[F[t],t], R[-20, t]==R[20,t], R[x,0]==Exp[-(x+5)^2],
> F[0]==0},{R, F}, {x,-20, 20}, {t, 0,10}]
>
> where I is the imaginary number Sqrt[-1].
>
> However, Mathematica gave an error message saying that R and F have
> different number of dependent variables:
> "NDSolve::"dvlen" :  "The function F[t] does not have the same number of
> arguments as independent variables (2)."
>
> This set of equations simulate a one-dimensional wave hits a localized
> target at x=0 (approximates using a Lorentzian). Initial wave form at t=0 is
> given by R[x, 0]. The computation domain in x is assumed to be periodic.
>
> Thank you for any help and suggestions.
>
> Dave
>



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