I have a pde set as following:(adsbygoogle = window.adsbygoogle || []).push({});

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:

Anyone know how to do it easily?Code (Text):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]

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# Mathematica: Need help to solve this pde set

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