Numerical integration of multi-dimensional highly oscillatory, gentle oscillatory and non-oscillatory integrands based on wavelets and radial basis functions

In this paper Haar wavelets (HWs), hybrid functions (HFs) and radial basis functions (RBFs) are used for the numerical solution of multi-dimensional mild, highly oscillatory and non-oscillatory integrals. Part of this study is extension of our earlier work [9] and [47] to multi-dimensional oscillatory and non-oscillatory integrals. Second part of the study is focused on coupling Levin's approach [30] with meshless methods. In first part of the paper, application of the numerical algorithms based on HWs and HFs [9] is extended to integrals having a varying oscillatory and non-oscillatory integrands defined on circular and rectangular domains. In second part of the paper, we propose a meshless method based on multiquadric (MQ) RBF for highly oscillatory multi-dimensional integrals. The first approach is directly related to numerical quadrature with wavelets basis. Like classical numerical quadrature, this approach does not need any intermediate numerical technique. The second approach based on meshless method of Levin's type converts numerical integration problem to a partial differential equation (PDE) and subsequently finding numerical solution of the PDE by a meshless method. The computational algorithms thus derived are tested on a number of benchmark kernel functions having varying oscillatory character or integrands with critical points at the origin. The novel methods are compared with the existing methods as well. Accuracy of the methods is measured in terms of absolute and relative errors. The new methods are simple, more efficient and numerically stable. Theoretical and numerical convergence analysis of the HWs and HFs is also given.