The author uses numerical differentiation formulas in deriving numerical methods for solving ordinary and partial differential equations. The automatic differentiation and the regularization method on numerical differentiation are familiar techniques to obtain the derivatives of a function whose values are only obtained empirically at a set of points.
First, explicit difference formulas of arbitrary order for approximating first and higher derivatives for unequally or equally spaced data, are proposed. Then, an th generalized Vandermonde determinant is introduced and its basic properties are discussed. A linear algebraic system of Vandermonde type is solved analytically in terms of generalized Vandermonde determinants.
A comparison with other implicit finite difference formulas based on interpolating polynomials shows that the new explicit formulas are very easy to implement for numerical approximation of arbitrary order to first and higher derivatives, and they need less computation time and storage and can be directly used for designing difference schemes for ordinary and partial differential equations. Numerical results suggest that the new explicit difference formulae are very efficient .