Abstract
In this paper, we introduce a new numerical scheme for approximation of highly oscillatory integrals having Bessel kernel. We transform the given integral to a special form having improper nonoscillatory Laguerre type and proper oscillatory integrals with Fourier kernels. Integrals with Laguerre weights over [0, ∞) will be solved by Gauss-Laguerre quadrature and oscillatory integrals with Fourier kernel can be evaluated by meshless-Levin method. Some numerical examples are also discussed to check the efficiency of proposed method.
Highlights
In this paper, we are concern with highly oscillatory integrals of the form [1] ∫ I[g] =b a g(x) Jμdx, (1)where g(x) is a smooth function, Jμ is Bessel function of first kind of order μ and κ is parameter of frequency
In [9], different approaches are presented for different types of oscillatory integrals, the method designed for Bessel type oscillatory integrals is based on Lagrange’s identity
In [10], the oscillatory integrals over infinite positive domain [0, ∞) are evaluated by “integration summation with extrapolation” (ISE) method. [14], proposed Filon-type method and Clenshaw-Curtis-Filon-type method based on Fast Fourier transform and fast computation of modified moments for evaluation of highly oscillatory integrals containing Bessel functions
Summary
Where g(x) is a smooth function, Jμ (κx) is Bessel function of first kind of order μ and κ is parameter of frequency. For large value of κ the integral become highly oscillatory and cannot be approximate by usual quadrature rules To handle this type of problems, we formulate special numerical schemes. [14], proposed Filon-type method and Clenshaw-Curtis-Filon-type method based on Fast Fourier transform and fast computation of modified moments for evaluation of highly oscillatory integrals containing Bessel functions. In [15] the authors proposed meshless procedure for approximation of oscillatory integrals with Bessel kernel, the case of singularity is handled with multi-resolution quadrature based on Haar wavelet quadrature and hybrid function. [2], has transformed Bessel oscillatory integrals to special type integrals with Fourier kernel and integrals with Laguerre weights and used Levin type method with Gauss-Laguerre quadrature for approximation. For transformation purpose the same approach can be used while for approximation we use meshless-Levin method based on Gauss-Laguerre quadrature
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