Modified quadrature rules based on a generalised mixed interpolation formula

In the present paper, based on a recently developed generalised mixed interpolation formula, which integrates exactly any linear combination of polynomials up to (n − 2) degree and two other functions U1(kx) and U2(kx), representing two linearly independent solutions of a general second-order linear differential equation, of the form y″(x) + kq(kx)y′(x)+k2p(kx)y(x) = 0, where k is a free parameter various quadrature rules have been derived. The formulae that we have derived can be called the generalised modified Newton-Cotes formulae (GMNCF) of the “closed” type. They are obtained by replacing the integrand by an interpolation function of the form aU1(kx) + bU2(kx) + ∑i=0n−2 cixi, used for equally spaced nodes xj = jh. The truncation errors involved in the present quadrature formulae are also examined. Several numerical examples are handled by the generalised modified rules and the utility of the error formulae is also tested in these examples.