Accurate algorithms are proposed for approximation of integrals involving highly oscillatory Bessel function of the first kind over finite and infinite domains. Accordingly, Bessel oscillatory integrals having high oscillatory behavior are transformed into oscillatory integrals with Fourier kernel by using complex line integration technique. The transformed integrals contain an inner non-oscillatory improper integral and an outer highly oscillatory integral. A modified meshfree collocation method with Levin approach is considered to evaluate the transformed oscillatory type integrals numerically. The inner improper complex integrals are evaluated by either Gauss–Laguerre or multi-resolution quadrature. Inherited singularity of the meshfree collocation method at $x=0$ is treated by a splitting technique. Error estimates of the proposed algorithms are derived theoretically in the inverse powers of $\omega$ and verified numerically.

提出了精确算法来逼近涉及有限域和无限域上第一类高度振荡贝塞尔函数的积分。因此，通过使用复线积分技术，将具有高振荡行为的贝塞尔振荡积分转化为具有傅立叶核的振荡积分。变换后的积分包含一个内部非振荡非正常积分和一个外部高振荡积分。考虑使用Levin 方法改进的无网格配置方法对变换后的振荡型积分进行数值评估。内部不适当的复积分由高斯-拉盖尔或多分辨率正交计算。无网格搭配方法的继承奇异性$X=0$采用分裂技术处理。所提出算法的误差估计是从理论上推导出的逆幂$\omega$ 并进行数值验证。