A GENOCCHI WAVELET METHOD FOR HIGHLY OSCILLATORY INTEGRALS WITH AN ENDPOINT SINGULARITY

Abstract

This study introduces an efficient high-order wavelet quadrature technique for computing highly oscillatory integrals affected by endpoint singular behavior of the form: 𝑙⁢𝑎⁢𝑡⁢𝑒⁢𝑥⁢∫1 0𝑓⁡(𝑥)⁢sin⁡( 𝜔 𝑥𝑟 )𝑑𝑥 and ∫1 0𝑓⁡(𝑥)⁢cos⁡( 𝜔 𝑥𝑟 )𝑑𝑥,𝜔 ≫1,𝑟 >0, where standard numerical methods often lose stability and accuracy due to infinite oscillation frequency as x?0^+. The method applies sixth-order Genocchi wavelets within a multiresolution structure to approximate the smooth component of the integrand, while the singular boundary region is treated separately using analytical asymptotic approximations and interval decomposition. Closed-form expressions for the oscillatory kernels are obtained through appropriate transformations. Tests on several benchmark examples demonstrate very high precision, with errors ranging from 10^(-9)to 10^(-13), and show clear improvements in both accuracy and efficiency compared to existing wavelet and hybrid approaches. A practical application to radar cross-section computation is presented.