Operations on oscillatory functions Original Research Article

We introduce a procedure which allows modifying some standard algorithms with the aim of making them tuned on oscillatory functions. We consider the case when the involved functions are of form y(x) = f1(x) sin(ωx) + f2(x) cos (ωx) where f1(x) and f2(x) are smooth functions, to obtain tuned formulae for the first and second derivatives, for the Simpson quadrature formula and for the Numerov algorithm for differential equations. The expressions of the parameters of the new formulae are written in a way which makes them tuned also for functions of form y(x) = f1(x) sinh(λx) + f2(x) cosh(λx). The proposed procedure is general enough to allow amending many other classical algorithms.