I am working on a mathematical model describing the crystallization process, represented by an integral equation involving the crystalline fraction $\xi_V(T, t)$, where $T$ is the temperature and $t$ is time. The equation is as follows:
$$ \xi_V(T(t)) = \frac{4 \pi}{3} \int_0^t I_V(T(z))\left[\int_z^t G(T(\tau)) \, \mathrm{d}\tau\right]^3[1-\xi_V(T(z))]^2 \, \mathrm{d}z, $$
where $\xi_V(T(t))$ represents the crystalline fraction, varying from 0 (initial condition, no crystallization) to 1 (fully crystallized). The functions $I_V$ and $G$ follow an Arrhenius form, indicating their dependence on temperature in a manner typical for chemical reaction rates.
Is this even a nonlinear volterra equation of the second kind?
In my attempts to solve this equation, I've employed numerical methods such as Picard iteration with and without relaxation (partial update) and a more direct approach with "brutal" discretization and forward time stepping. However, these methods tend to overshoot, producing values of $\xi_V$ greater than 1, which is physically unrealistic.
Given the complexities introduced by the double integrals and the self-referential nature of $\xi_V$, I'm considering a different approach. I wonder if it's feasible to derive a time derivative of this equation to transform it into an ordinary differential equation (ODE) that could be more straightforwardly solved using ODE solvers. However, the presence of the double integrals and the overall structure have left me uncertain about how to proceed with this transformation or if it's even a viable strategy.
Has anyone faced a similar challenge or could offer insights into whether converting this integral equation into an ODE is possible? And if so, what steps should be taken to achieve this transformation while ensuring the physical constraint of $0 \leq \xi_V(T, t) \leq 1$ is maintained throughout the solution process? Any suggestions or references to similar problems would be greatly appreciated.
Or are there any other approaches to solving this equation?