Regularized computation of oscillatory integrals with stationary points

Ability to calculate integrals of rapidly oscillating functions is crucial for solving many problems in optics, electrodynamics, quantum mechanics, nuclear physics, and many other areas. The article considers the method of computing oscillatory integrals using the transition to the numerical solution of the system of ordinary differential equations. Using the Levin's collocation method, we reduce the problem to solving a system of linear algebraic equations. In the case where the phase function has stationary points (its derivative vanishes on the interval of integration), the solution of the corresponding system becomes an ill-posed task. The regularized algorithm presented in the article describes the stable method of integration of rapidly oscillating functions at the presence of stationary points. Performance and high accuracy of the algorithms are illustrated by various examples. © 2018

Publisher
Elsevier B.V.
Language
English
Pages
22-27
Status
Published
Volume
26
Year
2018
Organizations
  • 1 Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6, Miklukho-Maklaya str., Moscow, 117198, Russian Federation
  • 2 Department of Applied Physics, Peoples’ Friendship University of Russia (RUDN University), 6, Miklukho-Maklaya str., Moscow, 117198, Russian Federation
Keywords
Chebyshev differentiation matrix; Ill conditioned matrices; Integration of rapidly oscillating functions; Levin collocation method; Regularization; Stable methods for solving systems of linear algebraic equations
Date of creation
19.10.2018
Date of change
01.03.2021
Short link
https://repository.rudn.ru/en/records/article/record/6662/
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