Mathematical Software for Modified Bessel Functions

Abstract

The high-quality mathematical software for the computation of modified Bessel functions of the second kind with integer, imaginary and complex order and real argument is elaborated. The value of function may be evaluated with high precision for given value of the independent argument x and order r. These codes are addressed to the wide audience of scientists, engineers and technical specialists. The tables of these functions are published. This software improves significantly the capability of computer libraries. These functions arise naturally in boundary value problems involving wedge-shaped or cone-shaped geometries. They are fundamental to mathematical modeling activities in applied science and engineering. Methods of mathematical and numerical analysis are adapted for the creation of appropriate algorithms for these functions, computer codes are written and tested. Power series, Tau method and numerical quadratures of trapezoidal kind are used for the construction of subroutines. New realization of the Lanczos Tau method with minimal residue is proposed for the constructive approximation of the second order differential equations solutions with polynomial coefficients. A Tau method computational scheme is applied for the constructive approximation of a system of differential equations solutions related to the differential equation of hypergeometric type. Various vector perturbations are discussed. Our choice of the perturbation term is a shifted Chebyshev polynomial with a special form of selected transition and normalization. The minimality conditions for the perturbation term are found for one equation. They are sufficiently simple for verification in a number of important cases. Tau method’s approach gives a big advantage in the economy of computer’s time. The mathematical software for kernels of Lebedev type index transforms – modified Bessel functions of the second kind with complex order is elaborated in detail. The software for new applications of Lebedev type integral transforms and related dual integral equations for the numerical solution of problems of mathematical physics is constructed. The algorithm of numerical solution of some mixed boundary value problems for the Helmholtz equation in wedge domains is developed. Observed examples admitting complete analytical solution demonstrate the efficiency of this approach for applied problems.