Is there a method that would allow one to find a common solution to a differential equation and an integral equation?
Currently I am trying to solve the following Integral Equation:
$$kT\ln\left |g(x) \right |=-\Phi(x)+\pi n\int_{x}^{\infty}\frac{1}{{x}^2}\Bigg(\int_{0}^{\infty}\frac{\mathrm{d} \Phi(y) }{\mathrm{d} x}g(y)\Bigg(\int_{\left | x-y \right |}^{x+y}[(x^2+y^2)-z^2]zg(z)dz\Bigg)dy\Bigg)dx$$
where
1) $k,T,$ and $n$ are known or given parameters or constants, and
2) $\Phi(x)$ is a given function, such as
$$\Phi(x)=4\varepsilon\Bigg(\bigg(\frac{\sigma }{x} \bigg)^{q}-\bigg(\frac{\sigma }{x} \bigg)^{p}\Bigg)$$
where $\sigma$ and $\varepsilon$ are known parameters.
$g(x)$ is the function that is to be determined, and should oscillate at regular intervals based on $\sigma$ while decaying to zero at infinity. Currently the idea is to use orthogonal polynomials combined with some function related to the system as a model for $g(x)$. For example:
$$g(x) \sim\exp{\bigg[\frac{-\Phi(x)}{kT}\bigg]}\Bigg(\sum_{i=0}^n w_{i}L_{i}\Bigg)\Bigg(\frac{cos(x)}{x^m}\Bigg)$$
where $L_{i}$'s are orthogonal polynomials to each other, and $w_{i}$'s are weights that are unique to each system.
This doesn't work, at least as simply as it is presented there, as I have tried it, but I started wondering if there was a general way to do it. Low and behold, there is a whole field of functional analysis (I'm a chemical engineer), as well as the fact that all "classical orthogonal polynomials" derive from solutions to the Sturm-Liouville Differential Equation:
$$Q(x)g''(x)+P(x)g'(x)+\lambda g(x)=0$$
I'm very, very far out of my mathematical knowledge zone, as I've only taken through multivariate calc., but right now I have the idea that there is some sort of space that represents family wherein each point represents a single equation. These two equations, the integral equation and the differential equation, each span a space of functions that are solutions to the equations, and the intersection of those spaces is the unique solution or set of solutions common to both equations.
Does this make sense, and if so, is there a way to find this solution or sets of solutions?
Again, so sorry if this is unintelligible. I feel like a five year old with a tank with the words and concepts I'm trying to use.
Thanks so much for the help in advance.
Note: for those of you that have seen these before, $g(x)$ is the two particle radial distribution function, the Integral equation is used to determine $g(x)$ by extracting it directly from the three body distribution function, and $\Phi(x)$ is any inter-molecular potential function, though here specifically I am using the L-J 6-12 potential function.