Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia2686910.22363/2658-4670-2021-29-2-146-157Research ArticleCalculation of special functions arising in the problem of diffraction by a dielectric ballMalyshevKsaverii Yu.<p>engineer, Skobeltsyn Institute of Nuclear Physics</p>kmalyshev08102@mail.ruhttps://orcid.org/0000-0001-8823-9136Skobeltsyn Institute of Nuclear Physics Lomonosov Moscow State University2806202129214615728062021Copyright © 2021, Malyshev K.Y.2021<p style="text-align: justify;">To apply the incomplete Galerkin method to the problem of the scattering of electromagnetic waves by lenses, it is necessary to study the differential equations for the field amplitudes. These equations belong to the class of linear ordinary differential equations with Fuchsian singularities and, in the case of the Lneburg lens, are integrated in special functions of mathematical physics, namely, the Whittaker and Heun functions. The Maple computer algebra system has tools for working with Whittaker and Heun functions, but in some cases this system gives very large values for these functions, and their plots contain various kinds of artifacts. Therefore, the results of calculations in the Maple11 and Maple2019 systems of special functions related to the problem of scattering by a Lneburg lens need additional verification. For this purpose, an algorithm for finding solutions to linear ordinary differential equations with Fuchsian singular points by the method of Frobenius series was implemented, designed as a software package Fucsh for Sage. The problem of scattering by a Lneburg lens is used as a test case. The calculation results are compared with similar results obtained in different versions of CAS Maple. Fuchs for Sage allows computing solutions to other linear differential equations that cannot be expressed in terms of known special functions.</p>linear differential equationsWhittaker functionsHeun functionsлинейные дифференциальные уравненияфункции Уиттекерафункции Гойна<p>1. Introduction The problem of diffraction of a plane electromagnetic wave by a ball with an arbitrary radially symmetric filling allows the construction of an analytical solution by the incomplete Galerkin method [1]. Let us make use of a spherical coordinate system and assume that the dielectric constant of the ball</p>[A. G. Sveshnikov and I. E. Mogilevsky, Matematicheskiye zadachi teorii difraktsii [Mathematical problems of the theory of diffraction]. Moscow: MSU, 2010, In Russian.][G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Annalen der Physik, vol. 25, no. 3, pp. 377-445, 1908. DOI: 10.1002/andp.19083300302.][C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles. New York: John Wiley & Sons, 1983.][C. Mätzler, MATLAB Functions for Mie Scattering and Absorption, Institute of Applied Physics, University of Bern, June 2002. Research Report No. 2002-08 http ://arrc.ou.edu/~rockee/NRA_2007_ website/Mie-scattering-Matlab.pdf, 2002.][J. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory,” Journal of the Optical Society of America A: Optics Image Science and Vision, vol. 25, pp. 2980-2990, 2008. DOI: 10.1364/JOSAA.25.002980.][Waterloo Maple (Maplesoft), Symbolic and numeric computing environment Maple, https://www.maplesoft.com/, 2019.][F. G. Tricomi, Differential equations. London: Blackie & Sons ltd., 1961.][The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.4), https://www.sagemath.org, 2016.][National Institute of Standards and Technology (NIST), United States, Digital Library of Mathematical Functions. Version 1.1.1, https://dlmf. nist.gov, 2021.][A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics. A Unified Introduction with Applications. Springer Basel AG, 1988.][S. Y. Slavyanov and W. Lay, Special functions: unified theory based on singularities. Oxford: OUP, 2000.]