I need to find a solution to this unconventional integral equation: I need to find some nonnegative function $f$ that satisfies the following for any $\phi \in[0,\pi]$:
$\int^\phi_0f(\theta)f(\phi-\theta)d\theta +\int^\pi_\phi f(\theta)f(\theta-\phi)d\theta +\int^\pi_{\pi-\phi}f(\theta)f(2\pi-\phi-\theta)d\theta +\int^{\pi-\phi}_0f(\theta)f(\theta+\phi)d\theta = \cos^2\frac{\phi}{2}$
I would be very much in your debt if you help me solve it. Thank you in advance.
P. S. A Greek physicist, who is a friend of mine, has made the following transformation, maybe it can help:
Let $\theta\rightarrow g_i(x):[a_i, b_i]\rightarrow [0,1]$ such that all boundary terms become identical. The choice of $g$ is arbitrary and only a linear interpolant is necessary like g(x) = a + bx in which case you get $g(x)=g(0)+(g(1)-g(0))x$ where $g(0) = a$ and $g(1) = b$ each integral's bounds . We can then drive every term under the same integral sign to write
$\int_0^1dx[I_1(g_1(x),\phi)g_1'(x)+I_2(g_2(x),\phi)g_2'(x)+I_3(g_3(x),\phi)g_3'(x)+I_4(g_4(x),\phi)g_4'(x) - g_5(x)cos^2(\phi)] = 0$
where $g_5$ any function satisfying $G(1)-G(0)=1: G=\int dx g(x)$