Gauss Quadrature Formula For Airy Integrals With Singularities

In the research of physics,the application of Airy functions have been quite common.For example,in the problems of electromagnetic diffraction and propagation,quantum me-chanics,optics and so on,it is very important to conduct efficient numerical calculation for some problems related to Airy functions.Due to the oscillation property of Airy functions,traditional numerical integration methods,such as Gauss quadrature rule or the quadrature method based on polynomial interpolation,are used for Airy integral on an infinite interval.With the increase of oscillation frequency,even if a large number of quadrature nodes are used,it is difficult to obtain ideal numerical results.Therefore,the main purpose of this pa-per is devoted to the quadrature rules for a highly oscillatory Airy integral with singularities on the infinite interval.In the first chapter,the definition and application background of Airy oscillation func-tion and Airy integral transformation are introduced.Then we enumerate the application and research status of the oscillatory integral problems.In the second chapter,we derive the Gauss quadrature formula for a class of nonnega-tive weight functions w?z?=z_?K_v(z^2/3),and introduce two kinds of methods for calculating their nodes and weights.The first one is a method of solving equations by using the prop-erties of orthogonal polynomials.The second method is based on the modified Chebyshev algorithm to obtain the three term recursive relations,and then construct the Jacobi matrix.The eigenvalues of the Jacobi matrix is the nodes in the quadrature formula,and the weights can be expressed by the eigenvectors of the matrix.In the third chapter,we mainly discuss the calculation of Airy oscillatory integrals with singularities on the infinite interval with the following form:???using the properties of Airy function,the integral is transformed into non-oscillating and oscillating parts.Based on the discussion in the second chapter,the corresponding quadrature formula and error term are given by using the correlation formula of Airy function and Bessel function.Numerical examples are provided to illustrate the accuracy of the proposed methods.