Research On Numerical Approximation Of Bessel Functions

Bessel functions are widely appied in wave propagation,static potential,signal processing,and other applications.Bessel functions are considered as a function of complex form,which can not be represented by elementary functions.Although the power series and asymptotic series expansion are available to approximately represent Bessel functions,the performance are numerically unstable and impractically slow.Due to their importance in application,how to improve the efficiency and accuracy of approximations on Bessel functions has significant academic values.Main works of this paper are as follows:1.The numerical approximation on Bessel functions of the first kind of integer order has been studied.Based on their approximately periodic behavior,we apply two variants of Prony's method on Bessel functions of the first kind of integer order. The Prony-like methods in cosine or sine version yield approximations as sums of sinusoidal functions of Bessel functions of the first kind of integer order.In the symbolic computation software Maple,we compute the approximations for different orders and over different intervals,and compare these approximations with those obtained through the Fourier method.Experiments show that Prony-like methods perform much better than the Fourier method.2.By comparing with other numerical methods,the advantages of Prony-like methods on Bessel functions of the first kind of integer order are further highlighted.Besides,we present the comparison on power series and asymptotic series expansions for Bessel functions,the experiments show that Prony-like methods are significantly better than the power series and asymptotic series.3.We optimize the Prony-like methods to improve the approximation efficiency and accuracy:(1)In order to avoid the complicated calculation process of constructing Hankel matrixs and solving generalized eigenvalues,the nodes in Prony-like methods have been replaced by Chebyshev points,which can simplify the algorithm.Using the proposed method,we can still obtain good approximations for Bessel functions of the first kind of integer order.(2)We propose a new sampling algorithm to obtain coefficients in the Prony-like methods,which can improve the accuracy of the approximation results for Bessel functions.