So I was doing (my own style of) thermodynamics and ended up with this kind of self-referential integral equation.
\begin{equation} \label{self ref 2} \tilde P( \lambda) = A \int_{- \infty}^{\infty} (\tilde P (\lambda-x) \tilde P ( x) )^2d( x).\tag{1} \end{equation}
Is there a standard method of solving these integral equations?
Note the answer for this distribution is:
$$ \tilde P(p_x) = A \exp(-p_x^2/2), \tag{2}$$
where $A$ is a normalization constant for the distribution. While, later on I deal with relativity and vectors in $3$ dimensions and things get much more complicated.
Is Mathematica capable of solving this? Is there a standard algorithm to solve this? The more generic algorithm the better?