What numerical scheme could I use to solve this integral equation?

Question

I have the following equation which I would like to solve numerically (python) but I am not sure what scheme could be used. Any suggestions?

$$\phi_{2} = \int_{0}^{^{\phi_{1}}} \frac{r_{1}(\phi_{1})}{A-r_{1}(\phi_{1})}d\phi_{1}$$ where A is a constant = $$r_{1} + r_{2}$$

$$\phi_{1}$$ and $$\phi_{2}$$ are in polar coordinates.

$$r_{1}(\phi_{1})$$ will be a non-intersecting smooth and closed curve.

Answer 1

This answer is simply one possible approach that should work, although there may be more efficient means. The trapezoidal method is in general good with periodic functions. Let's say you know the function $\phi_2(t)$ and you want to solve this integral equation for $\phi_1$ on a set of $n+1$ grid points $0,h,\dots,nh$. Then, I would suggest the following:

0. Initialize $t = h$, $i = 0$.
1. Evaluate $\phi_2(t)$.
2. integrate $I(T) = \int_{0}^{T} \frac{r_1(t)}{A-r_1(t)}dt$ using trapezoidal rule.
3. update $T$ until you find a $T$ such that $I(T) = \phi_2(t)$ within precision. Then, save $\phi_1(t) = T$.
4. $i \leftarrow i+1$.
5. $t\leftarrow t+ih$
6. Go to step 1.

To speed things up a bit, replace the lower limit of integration each time with $(i-1)h$.