Extended quadrature rules for oscillatory integrands Original Research Article

We consider the integral of a function y(x), View the MathML source and its approximation by a quadrature rule of the form View the MathML source i.e., by a rule which uses the values of both y and its derivatives up to p-th order at the nodes of the quadrature rule. We focus only on the case when the nodes are assumed known and present the procedure to calculate the weights. Two cases are actually examined: (i) y(x) is a polynomial and (ii) y(x) is an ω dependent function of the form y(x)=f1(x)sin(ωx)+f2(x)cos(ωx) with smoothly varying f1 and f2. For the latter case, the weights wk(j) (j=0,1,…,p) are ω dependent. A series of specific properties for this case is established and a numerical illustration is given.