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I have the following code:

sys = {
    D[f[ϵ, x], {x, 2}] + Iee[ϵ, x] + Iph[ϵ, x] == 0,  
    f[ϵ, -L/2] == fF[ϵ - eV/2], 
    f[ϵ, L/2] == fF[ϵ + eV/2]
}
where:
Iee[ϵ_, x_] := a Integrate[
    Integrate[
        (f[ϵ, x] f[ϵ1 - ω, x] (1 - f[ϵ - ω, x]) (1 - f[ϵ1, x]) - 
         f[ϵ - ω, x] f[ϵ1, x] (1 - f[ϵ, x]) (1 - f[ϵ1 - ω, x])), 
        {ϵ1, -∞, ∞}, 
        {ω, -∞, ∞}
    ]
]

Iph[ϵ_, x_] := b Integrate[
    ω^2 ((1 - f[ϵ, x]) f[ϵ + ω, x] (1 + n[ω]) + 
         (1 - f[ϵ, x]) f[ϵ - ω] n[ω] - 
         f[ϵ, x] (1 - f[ϵ - ω]) (1 + n[ω]) - 
         f[ϵ, x] (1 - f[ϵ + ω]) n[ω]), 
    {ω, 0, ∞}
]
n[ω_] = 1/(Exp[ω/T] - 1);
fF[ϵ_] = 1/(1 + Exp[ϵ/T]);

where a,b,T,L,eV are constants.

I need to solve this equation numerically. As I understand, simple functions as well as NDSolve don't work here, and I need to use The Numerical Method of Lines (like here Integrating over $x$ in numerically solving a partial integrodifferential equation)

But anyway I still dont understand what to do with limits of integration and how to connect it with my boundary conditions.

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    $\begingroup$ For a numerical solution it would be helpful to find finite integration limits. I don't know the parameter values Do you think it's possible to find an appropriate transformation w,ϵ? $\endgroup$ Dec 13, 2023 at 20:03
  • $\begingroup$ As the first step towards a numerical solution, it would be necessary to determine the parameters a,b,T, L, eV. $\endgroup$ Dec 14, 2023 at 3:35
  • $\begingroup$ @UlrichNeumann Im not sure, but i gonna try to use something like 2 last functions for such transform. It might be naturally in this contecst. $\endgroup$ Dec 14, 2023 at 22:37
  • $\begingroup$ @AlexTrounev I thought that its possible to solve for all real. For the first solution it should be ok use any non zero constants. $\endgroup$ Dec 14, 2023 at 22:42
  • $\begingroup$ @PabloEscobar In Iph definition there are f[ϵ, x] and f[ϵ - ω]. Is it a typo? $\endgroup$ Dec 15, 2023 at 14:13

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