Numerical method for solving second order nonlinear differential equation with arbitrary boundary conditions

The aim of this work consists in developing an algorithm and a numerical calculation program allowing to solving a nonlinear arbitrary 2nd order differential equation (DE) of with generalizED Cauchy boundary conditions (BC) over the interval [a1, a2]. The Dirichlet and Neumann BC becomes a particular case. The problem consists in transforming DE to a system of 6 nonlinear DE of the first order (FODE) with initial 2 conditions (IC). The resolution of this FODE system is made by the adaptation of the Runge Kutta method of order 4. The determination of the IC is made by the resolution of an algebraic system of 2 nonlinear equations, of which the resolution is made simultaneously by the Newton's method. For each Newton’s method iteration, a system of nonlinear algebraic equations is obtainED whose solution is made by the method of Gauss. The function y(x) and y’(x) are obtainED point by point in the interval [a1, a2]. The initial values ​​of Newton's algorithm can be taken by the physical model since DE can model a precise physical phenomenon. The validation of the results is justifiED by the consideration of a DE by modifying only the BC. The integral of the function [y(i-1)(x)]k in the interval [a1, a2], therefore, is evaluatED by Simpson's quadrature. The application is made by the consideration of an arbitrary example of 2nd DE by modifying only their BC. The validation is made by considering the compound pendulum problem.