A class of non-uniform mesh three point arithmetic average discretization for y″ = f(x, y, y′) and the estimates of y′

We propose two new non-uniform mesh three point arithmetic average discretization strategy of order two and three, to solve non-linear ordinary differential equation y″ = f(x, y, y′), a < x < b and the estimates of first-order derivative y′, where y = y(x), subject to the essential boundary conditions y(a) = A, y(b) = B. Both methods are compact and directly applicable to singular problems. There is no need to discuss any special scheme for the singular problems. Error analysis of a proposed method is discussed. Numerical experiments are performed to study the convergence behaviors and efficiency of the proposed methods.