How Can I Solve Complex PDEs Using Mathematica's NDSolve?

[crazybird]

I have a pde set as following:

parameters: γ, ω, α, β, c, η
variables: z,t; x,y
want: S = S(z,t;x,y)
A = A(z,t)

∂S/∂t = -γ*S - i ω*A*exp{-i*[(-θ-α*t)*x+β*t*y]}
[∂/∂t + (1/c)*∂/∂t] A = -i η*∫∫dxdy S*exp{i*[(-θ-α*t)*x+β*t*y]}

The integral range is angle:(0,2Pi), radius: (0,R)

How to solve this equation with NDSolve? I tried the following, which obviously does not work:

t1 = 500;(*ns, duration=5*10^-7 s*)
\[Mu] = -250;(*ns, central=-2.5*10^-7 s*)
\[Sigma] = 100;(*ns, width=10^-7 s*)
L = 1;
R = 0.2;
c = c = 29.979;
\[Gamma] = 1/100000;
\[Omega] = 1.329489268210057*10^-8;
\[Eta] = 2.0034565952485216*10^9;
\[Theta] = 1022.4;
\[Alpha] = 4.09;
\[Beta] = 0;

sol = NDSolve[{\!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\(sS[z, t, x]\)\) == -\[Gamma]*
      sS[z, t, x] - 
     I \[Omega]* E^(-I ((-\[Theta] - \[Alpha] t)*x))*aS[z, t, x], (\!\(
\*SubscriptBox[\(\[PartialD]\), \(z\)]\(aS[z, t, x]\)\) + 1/c  \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\(aS[z, t, 
         x]\)\)) == -I \[Eta]* 
     NIntegrate[
      E^(I ((-\[Theta] - \[Alpha] t)*x))*sS[z, t, x], {y, -R, 
       R}, {x, -Sqrt[R^2 - y^2], Sqrt[R^2 - y^2]}] , 
   sS[z, -t1, x] == 0, 
   aS[z, -t1, x] == 
    1/(Sqrt[2 Pi] \[Sigma]) E^(-((-t1 - \[Mu])^2/(2 \[Sigma]^2))), 
   aS[0, t, x] == 
    1/(Sqrt[2 Pi] \[Sigma]) E^(-((t - \[Mu])^2/(2 \[Sigma]^2)))}, {sS,
    aS，x}, {z, 0, L}, {t, -t1, 0}, {x, -R, R}, MaxSteps -> Infinity, 
  StartingStepSize -> 0.01, PrecisionGoal -> 1000, 
  MaxStepSize -> 0.01]

Anyone know how to do it easily?

 

[jackmell]

I don't think NDSolve can solve that. Strip it down into it's canonicalized form so that's it's easier to see what's going on. Looks like:

\frac{\partial S}{\partial t}=-yS-iwA g(t,x,y)

\frac{\partial A}{\partial t}=-ik\int_0^{2\pi}\int_0^{R} S(z,t,u,v) g(t,u,v)dudv

So since the derivative are only with respect to t, those are ordinary coupled integrodifferential equations. However you need appropriate initial conditions. For example, you need an initial region for S so that the integration can be performed for every time step starting at t=0. So the initial conditions would be:

A(z,0)= h(z)

S(z,0,x,y)=g(z,x,y),\quad 0\leq x\leq R,\quad 0\leq y\leq 2\pi

for some constant z. Then I think just start by coding a simple Euler method. Do just like you would do for two ordinary DEs, but at each time step, numerically compute the integral and add it into the calculations.

That's a start anyway. May need to tweek it.

 

[crazybird]

Hi, jackmell,

After many trials I also realize that it is not quite possible to simply use NDSolve to get it done. Thanks for the suggestion to go to a canonicalized form and I find that one can put the e^ factor into the variables to make a better looking form. I find that I made a mistake--the second equation involves a derivative of z: [∂/∂z + (1/c)*∂/∂t] A=... . It is not only ODEs. So things get complicated. I am trying to find a numerical way to solve it. Thanks for you answer~