Fast integration of rapidly oscillatory functions (Q1919412)

The author considers the problem of computing integrals of rapidly oscillatory functions of the type \[ I= \int^b_a g(x)S(rx)dx, \tag{1} \] where \(S\) is an oscillatory function, \(r\) is a large parameter and \(a\) and \(b\) are real and finite. The problem of computing the integral (1) is replaced by a problem of finding a solution of a system of linear ordinary differential equations, with no boundary conditions. This last problem is solved by the collocation method. The particular case for integrals involving Bessel functions \(J_\gamma (rx)\) with large oscillation parameter \(r\) is used as an example.