Discrete method for solving a three-point boundary-value problem for a third-order equation

Coupled equations are used to develop a method for solving boundary-value problems for second- and third-order equations. With the use of the factorization method, a three-point boundary-value problem for a third-order equation is reduced to a system of first- and second-order equations. In order to solve the second-order equation, a discrete problem is constructed, which is then used to solve the main problem. This method is peculiar because discrete (difference) boundary-value problems are constructed without using approximations of differential operators. The method is generalized to solve higher-order equations.