Numerical algorithms using reproducing kernels for oscillatory boundary value problems

As is known, it is perceived as a difficult problem to obtain an effective approximate solution to boundary value problems (BVPs) with highly oscillatory solutions. Traditional reproducing kernel methods (RKMs) are not effective for highly oscillatory BVPs although the RKM is a useful approach to approximation theory. The aim of the letter is to propose a novel class of RKMs to solve highly oscillatory second-order BVPs. To this end, we begin with the variation-of-constants formula (VCF). We then derive RKMs for highly oscillatory BVPs. Our simulations confirm the high accuracy of the introduced techniques.