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~