A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for the general second-order non-linear differential equation yN = f(x, y, y′) subject to mixed two-point boundary conditions. An interesting feature of our method is that each discretization of the differential equation at an interior grid point is based on just three evaluations of f. We establish, under appropriate conditions, O(h4)-convergence of the finite difference scheme. In the case of linear differential equations, our finite difference scheme leads to tridiagonal linear systems. Numerical examples are considered to demonstrate computationally the fourth order of the method.