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I'm trying to solve this form of Fredholm equation:

$$ g(v)=f_1(v)+\int\limits_{0}^{v_\mathrm{th}} g(v_s)\frac{e^{-\tfrac{\big[(v-v_\mathrm{init})-(1-v_\mathrm{leak})(v_s-v_\mathrm{init})\big]^2}{2v_\mathrm{step}^2}}+e^{-\tfrac{\big[(v+v_\mathrm{init})+(1-v_\mathrm{leak})(v_s-v_\mathrm{init})\big]^2}{2v_\mathrm{step}^2}}}{\sqrt{2\pi}v_\mathrm{step}}\mathrm{d} v_s, $$ where,

  • $v_\mathrm{init}$, $v_\mathit{th}$, $v_\mathrm{step}$, $v_\mathrm{leak}$ are constant and
  • $f_1$ has the following form: $$ f_1(v)=\frac{e^{-\tfrac{(v-v_\mathrm{init})^2}{2v_\mathrm{step}^2}}+e^{-\tfrac{(v+v_\mathrm{init})^2}{2v_\mathrm{step}^2}}}{\sqrt{2\pi}v_\mathrm{step}} $$

Are there any methods or theorems that can solve this problem?

Since I'm freshman in university, it's very challenging for me. But I really want to untangle this problem.

Can you help me to solve this kind of equation or let me know what theory I should study?

I would be grateful if you could just give me directions, so I can know where to start studying. Sorry for my bad English.

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  • $\begingroup$ Where is the equation? $\endgroup$
    – Pietro Majer
    Commented Jul 1, 2020 at 13:58
  • $\begingroup$ I signed up yesterday, and I think I was restricted to upload figures. Thank you for editing my question. $\endgroup$
    – ground
    Commented Jul 1, 2020 at 19:59
  • $\begingroup$ Are you sure the second exponential term is correct in the kernel? Looks there is no connection between kernel and your f1 function $\endgroup$
    – DuFong
    Commented Jul 2, 2020 at 0:08
  • $\begingroup$ Oh, I think the editor made a mistake by changing the picture to an expression. I edited second term in kernel, so that K(v,vinit)=f1(v). $\endgroup$
    – ground
    Commented Jul 2, 2020 at 4:02

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