The problem I have is a system of three non-linear equations with three unknowns. Each equation has a integration term. But the integration term has all the three unknowns in there. Some suggested to discretize. So I need to discretize all three unknowns. And solving the system become looping over the three variables and find when the equations hold at certain combination of the values. I am not sure if this is the right way to go or there are better options The three equations are \begin{equation} L=N_a \int_S l(s)dF(s) \end{equation} where $l(s)=(\gamma(1-\theta)Ak\frac{P_a}{q})^{\frac{1}{1-\gamma}}[\theta((\frac{\theta}{1-\theta})\frac{q}{r})^{\frac{\rho}{1-\rho}}+(1-\theta)s^{\frac{\theta}{1-\theta}}]^{\frac{\gamma-\rho}{\rho(1-\gamma)}}s^{\frac{\rho}{1-\rho}}$ \begin{equation} K_a+K_n=K \end{equation} , where $K_a$ is determined by a similar integral as above $K_a=N_a \int_S k(s)dF(s)$, and $k(s)$ has a similar function as $l(s)$ There a similar third equation.The three unknowns are $q$, $N_a$ and $K_n$. But the problem is the integrand $l(s)$ has $q$ in there.
Numeric integration with unknowns
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numerical-methods
integro-differential-equations