  A mathematical model of the heat process in one-dimensional domain governed
by a cylindrical heat equation with a heat source on the axis $z=0$ and
nonlinear thermal coefficients is considered. The developed model is
particularly applicable for analyzing temperature variations on electrical
contact surfaces, where precise thermal management is crucial for ensuring
optimal performance and preventing overheating. To solve the mathematical
model, we employ a solution method based on similarity transformations. This
technique allows us to reduce the problem to an ordinary differential problem,
which is equivalent to a nonlinear integral equations system. To ensure the
existence and uniqueness of the solution, we employ the fixed point theory in a
Banach space, providing a rigorous mathematical foundation for our analysis.
