The Connection Problem for General Linear Ordinary Differential Equations at Two Regular Singular Points with Applications in the Theory of Special Functions

1980 ◽  
Vol 11 (5) ◽  
pp. 848-862 ◽  
Author(s):  
Reinhard Schäfke ◽  
Dieter Schmidt
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Special Functions ◽  
Singular Points ◽  
General Linear ◽  
Linear Ordinary Differential Equations ◽  
Connection Problem

scholarly journals Calculation of special functions arising in the problem of diffraction by a dielectric ball

2021 ◽  
Vol 29 (2) ◽  
pp. 146-157
Author(s):  
Ksaverii Yu. Malyshev
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Electromagnetic Waves ◽  
Special Functions ◽  
Test Case ◽  
Linear Ordinary Differential Equations ◽  
Heun Functions ◽  
Calculation Results ◽  
Linear Differential ◽  
Frobenius Series

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.

Sign in / Sign up

Export Citation Format

Share Document