Methods for solving the nonlinear equilibrium equation of a compressed elastic rod

A nonlinear boundary value problem on the equilibrium of a compressed elastic rod on nonlinear foundation is considered for cases of free pinching or pivotally supported of the ends. The problem is written as a nonlinear operator equation. Numerical and analytical methods for solving nonlinear boundary value problems are discussed: The Newton-Kantorovich method and the Lyapunov-Schmidt method. We also consider a problem linearized on a trivial solution (the eigenvalue problem), which has an exact solution (Euler) in the case of a hinge support, and for the case of pinching the ends of the rod, the solution formulas are obtained in the works of A. A. Esipov and V. I. Yudovich. The eigenvalue problem is also solved by numerical method. To determine the equilibria of a nonlinear boundary value problem for a given value of the compressive force, it is proposed to apply the Newton-Kantorovich method in combination with the numerical methods, using as initial approximations the asymptotic formulas of new solutions found using the Lyapunov-Schmidt method in the vicinity of the critical value closest to the current value of the compressive load. Numerical calculations are performed and conclusions are drawn about the effectiveness of the methods used.