A continuation algorithm-Longman's method scheme for the integration of transcendental functions

Iago Cavalcante¹; Ana Carolina Azevedo Vasconcelos¹; Josué Labaki¹

¹ Universidade Estadual de Campinas

doi:10.20906/CPS/CILAMCE2017-0062

Resumo

This paper presents a method for numerical integration of indefinite integrals with oscillatory-decaying integrands. Integrands with oscillatory-decaying behavior are at the core of harmonic analysis. Problems approached through Hankel transforms, which involve Bessel functions, will invariably result in an indefinite integral of this type to be solved. The integral of Bessel functions and other such integrands between two of their consecutive roots varies in signal and decreases in amplitude in a way that matches the requirements for which Longman's integration method was designed. The difficulty in applying Longman's method for the present case of Bessel function is finding these roots, since Bessel functions do not present a constant period of oscillation. The present integration scheme consists of locating the roots of the integrand, and integrating between these roots according to Longman's technique. For the first part, the roots are located through a homotopy continuation procedure. The integration between the roots, called "volumes", can be performed with ordinary Gaussian quadratures. Longman's method is then used to approximate the value of the integral from the integral of the volumes. The combination of continuation algorithm and Longman's method (CALM) is illustrated with functions with or without constant periods of oscillation. The method is then used to integrate the Green's function for a transversely isotropic half-space, which is a type of problem that commonly arises in the solution of dynamic soil-interaction problems with the boundary element method.

Palavras-chave: Numerical integration; Continuation algorithms; Longman's method; Bessel functions