Modified Levin formulation for highly oscillatory Bessel integral transforms

Helmholtz equations appear in several model problems in engineering and applied sciences. The oscillatory Helmholtz equation solution contains integrals involving oscillatory Bessel functions of the second and third kinds. The existing quadratures, such as the Gaussian quadrature, make it difficult to compute these integrals for high-frequency regimes. The current work develops new stable algorithms based on Levin quadrature theory to precisely and accurately compute such integrals. We perform a coupling of the Levin approach with the global and compactly supported radial basis functions (CS-RBFs). It is well known that the global RBFs produce dense and ill-conditioned interpolation matrices, especially for large data points. Therefore, this work considers the CS-RBFs to produce sparse and well-conditioned matrices, while coupling the Levin method guarantees the best computation of the oscillatory Bessel integrals. We derived and validated some theoretical facts numerically by solving a few test examples.