Abstract
The Hankel transform is widely used to solve various engineering and physics problems, such as the representation of electromagnetic field components in the medium, the representation of dynamic stress intensity factors, vibration of axisymmetric infinite membrane and displacement intensity factors which all involve this type of integration. However, traditional numerical integration algorithms cannot be used due to the high oscillation characteristics of the Bessel function, so it is particularly important to propose a high precision and efficient numerical algorithm for calculating the integral of high oscillation. In this paper, the improved Gaver-Stehfest (G-S) inverse Laplace transform method for arbitrary real-order Bessel function integration is presented by using the asymptotic characteristics of the Bessel function and the accumulation of integration, and the optimized G-S coefficients are given. The effectiveness of the algorithm is verified by numerical examples. Compared with the linear transformation accelerated convergence algorithm, it shows that the G-S inverse Laplace transform method is suitable for arbitrary real order Hankel transform, and the time consumption is relatively stable and short, which provides a reliable calculation method for the study of electromagnetic mechanics, wave propagation, and fracture dynamics.
Highlights
The integration of high oscillation function generally exists in the application fields of aeronautics, seismic imaging, electromagnetic mechanics, and so on
The G-S inverse Laplace transform method (G-SILTM) is mainly used to solve the transient sounding forward problem, and here we introduce it to solve arbitrary real-order
The numerical solutions of the examples are in good agreement with the analytical solutions, which fully demonstrates the correctness of the algorithm, but the algorithm requires convergence of the integrals sF (s)ds
Summary
The integration of high oscillation function generally exists in the application fields of aeronautics, seismic imaging, electromagnetic mechanics, and so on. Evaluating infinite integrals including Bessel functions of arbitrary order This method requires that the non-oscillating term must be monotonic, and its accuracy is related to the segmentation interval, which is not efficient. Yu[18] modified the Filon-type method based on special functions and gave a high-precision solution of the integral a f ( )Jv ( r)d , but more stringent restrictions for f ( ) and its derivative are required in the algorithm. The above numerical algorithms of the Hankel transform have disadvantages such as complex processing, low computational efficiency, higher requirements for the integrand function (excluding the Bessel factor), and a small range of application of the Hankel transform order. The G-SILTM has the advantages of high efficiency, simplicity, wide application range, and low requirements for the integrand function (excluding Bessel factor), so we develop an algorithm to calculate the arbitrary real-order Hankel transform
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