Effective Calculation of Oscillatory Bessel Transform with Strong Singularity
In order to study the calculation problem of oscillatory Bessel integral with strong singularity,a new integration formula is constructed by the numerical steepest descent method.Firstly,Taylor polynomial is used to divide the integral into three integrals based on the additivity of the integral interval.Two of them can be transformed into Fourier type integrals according to the relationship between Bessel function and Hankel functions.Then,according to Cauchy theorem,Fourier type integrals can be transformed into infinite integrals,which can be effectively calculated by Gauss Laguerre quadrature rules.Secondly,with the help of the relationship between Meijer G function and Bessel function,another integral is represented by an explicit expression.Finally,the error analysis of the proposed quadrature formula is carried out.Numerical examples have verified the correctness of error analysis,and the efficiency and the accuracy of numerical methods.
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