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I am modeling a cyclic voltammetry experiment. The mathematical model for it boils down to a Volterra integral equation of the first kind, with an unknown function $\chi(z)$ and known constants $\xi$ and $\theta$:

$$\int_0^{\sigma t}\frac{\chi(z)\mathrm{d}z}{(\sigma t-z)^{1/2}} = \frac{1}{1+\xi\theta e^{-\sigma t}}.$$

Is this a well-posed integral equation? I am troubled by the fact that the right-hand side does not vanish as $\sigma t$ tends to zero.

To solve this integral equation, I am differentiating both sides with respect to $\sigma t$ to obtain an ordinary differential equation:

$$\frac{\chi(z)}{\sqrt{z}}+2\sqrt{z}\chi^\prime(z) = \frac{-\xi\theta e^{-z}}{(1+\xi\theta e^{-z})^2}.$$

What boundary condition should I assume to solve this differential equation?

svavil
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  • If the integrand, $\chi(z)/\sqrt{z}$ diverges at $z=0$, then the integral cannot vanish at $\sigma t=0$; you should use Cauchy principal value integral. – MathematicalPhysicist May 13 '17 at 19:37
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    You complicate the problem by differentiating the equation, it does not require an additional boundary condition. This equation is of Abel type see EqWorld for example. – Paul Enta May 13 '17 at 19:39
  • @PaulEnta Wow. I didn't find Abel equations when I was reading on how to solve integral equations. I think this qualifies as an answer to my specific problem. – svavil May 13 '17 at 19:44

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