On methods for evaluating a class of highly oscillatory integral

In this paper we consider two alternative strategies for evaluating the highly oscillatory integrals similar to those which can occur when the boundary integral method is employed to solve problems in acoustics. In the first case we shall look at a method based on interpolating the oscillatory and non-oscillatory parts of the integrand separately. In the second method we shall apply a change of variables to the integral which transforms it into a standard Fourier type integral. However, this method places some restrictions on the function in the oscillatory part of the integrand, and will require us to find the inverse of the function in the oscillatory part. Both methods will be looked at in terms of their accuracy and computational efficiency. The methods will then be used in the solution of a test integral equation.