Fast Hankel Transforms Algorithm Based on Kernel Function Interpolation with Exponential Functions

Huaiqing Zhang,Yu Chen,Xin Nie

doi:10.1155/2014/105469

Abstract

The Pravin method for Hankel transforms is based on the decomposition of kernel function with exponential function. The defect of such method is the difficulty in its parameters determination and lack of adaptability to kernel function especially using monotonically decreasing functions to approximate the convex ones. This thesis proposed an improved scheme by adding new base function in interpolation procedure. The improved method maintains the merit of Pravin method which can convert the Hankel integral to algebraic calculation. The simulation results reveal that the improved method has high precision, high efficiency, and good adaptability to kernel function. It can be applied to zero-order and first-order Hankel transforms.

Highlights

The Hankel transforms (HT) arise naturally in the discussion of problems posed in cylindrical coordinates and [1], as a result of separation of variables, involving Bessel functions
The stratified model and cylindrical coordinates are widely used in geophysical research, and the HT arise in forward and inverse calculation with zero or first order
The second is the back-projection and projection-slice methods [4], which carry out the HT as a double integral by means of one of the standard integral representations of the Bessel functions

Summary

Introduction

The Hankel transforms (HT) arise naturally in the discussion of problems posed in cylindrical coordinates and [1], as a result of separation of variables, involving Bessel functions. Analytical evaluations are rare and their numerical computations are difficult because of the oscillatory behavior of the Bessel function and the infinite length of the interval. To overcome these difficulties, various different techniques are available in the literature. In [6], the authors used Filon quadrature philosophy to evaluate zero-order Hankel transforms They separated the integrand into the product of slowly varying component and a rapidly oscillating one. Later in 2004, Guizar-Sicairos and Gutierrez-Vega [8] obtained a powerful scheme to calculate the HT of order n by extending the zeroorder HT algorithm of Yu to higher orders Their algorithm is based on the orthogonality properties of Bessel functions. Numerical examples and comparison were given to illustrate the proposed algorithms

Calculation Principles Based on Kernel Function Decomposition

Method

Numerical Examples

Findings

Conclusions