There's no general analytical method to solve for a material with specific properties, including superconductivity. The properties of materials are governed by the quantum mechanical behavior of electrons and their interactions with the crystal lattice. These interactions are typically described by the laws of solid-state physics, which involve solving complex quantum mechanical equations for the electrons in a crystalline structure.
For simple systems, such as some low-temperature superconductors and idealized models, researchers can sometimes make analytical approximations or derive simplified equations that describe the behavior of the material. But for most materials, the equations are too complex to solve analytically, and researchers rely on numerical methods, computational simulations, and empirical data to understand and predict material properties.
It's similar to the reason we can't find easy analytical solutions to other complex systems (like models trained via Machine Learning), there are just too many complex factors and interactions to take into account.
