Fredholm integral equation of second kind in statistical physics of fluids

The article analyzes the applicability of algorithms for solving various approximations for nonlinear equations of statistical physics of fluids to the solution of the linear integral Fredholm equation of second kind, which has been proposed earlier for description of surface phenomena in liquids. We have considered a molecular system of solid spheres bordering a solid surface. In the Percus-Yevick approximation for the kernel and right-hand side, the solution is sought in the class of piecewise continuous functions. We formulate the method of analytical calculation for each interval in the domain of function. For other approximations, the core of the equation and the right-hand side are calculated numerically. Fredholm equation should be also solved numerically. To solve it, we propose the Labik-Malijevsky algorithm, which is a standard of precision in modern physics of fluids. It is proposed to use this algorithm to calculate the two-particle function of meta-stable states distribution in the theory of the first kind chaotic phase transition of supercooled liquid - ideal glass, which will allow describing surface phenomena in amorphous films.