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.