Finite difference methods for a class of two-point boundary value problems with mixed boundary conditions

We discuss the construction of three-point finite difference aproximations for the class of two-point boundary value problems: p(x)y′]′ = f(x, y), α₀y(a) - α₁y′(a) = A, β₀y(b) + β₁y′(b) = B.We first establish an identity from which general three-point finite difference approximations of various orders can be obtained. We then consider in detail obtaining fourth-order methods based on three evaluations of f. We obtain a family of fourth-order discretizations for the differential equations; appropriate discretizations for the boundary conditions are also obtained for use with fourth-order methods. We select the free parameters available in this discretizations which lead to a “simplest” fourth-order method. This method is described and its convergence is established; numerical examples are given to illustrate this new fourth-order method.