A technique for the numerical solution of initial-value problems based on a class of Birkhoff-type interpolation method

In this paper a class of Birkhoff-type interpolation problems on arbitrary nodal points is studied. The explicit representation (characterization), the uniqueness and the error function are explicitly given. Furthermore, we apply the obtained Birkhoff-type interpolation method to find: (i) the numerical solution of high order initial-value problems (IVPs) and the corresponding error of this approximation, (ii) the approximation of some special functions with their explicit error functions, and (iii) new interpolatory type quadrature formulae   of precision degree at least m+n−1m+n−1 and m+kn−1(m,n,k∈N,n,k≥2)m+kn−1(m,n,k∈N,n,k≥2). Numerical examples are included to demonstrate the validity and applicability of the technique proposed in this paper and a comparison is made with the existing results. The results reveal that the new method is effective, simple and accurate.