Runge Kutta method for third order differential equation with generalized Cauchy boundary conditions

The paper focuses in developing a numerical method allowing to solving a third order nonlinear differential equation (DE) with generalized Cauchy boundary conditions (BC). The problem consists in transforming the third order DE, to a system of twelve nonlinear first order differential equation (FODE) with three initials conditions (IC). The resolution of this system of FODE 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 three nonlinear equations, of which the resolution is made simultaneously by the Newton's algorithm. 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 these successive derivatives 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 application is made by the consideration of an arbitrary example of 3rd DE by modifying only their BC. The validation is made by considering the 3th Falkner-Skan DE of the boundary layer.