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.

scholarly journals Asymptotics of Solutions of Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely Distant Point

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2249
Author(s):  
Maria Korovina
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Principal Symbol ◽  
Classical Problem ◽  
Multiple Roots ◽  
Asymptotics Of Solutions ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential ◽  
Poincaré Problem ◽  
Irregular Singular Points

This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier.

A method of solving second order simultaneous linear differential equations using the mallock machine

1940 ◽  
Vol 36 (2) ◽  
pp. 204-208
Author(s):  
M. V. Wilkes
Keyword(s):  
Differential Equations ◽  
Linear Equation ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential

The paper describes a method of integrating differential equations by means of the Mallock linear equation machine. The method, which is based on one described by Hartree, is applicable to pairs of second order simultaneous linear ordinary differential equations in two unknowns. Any required degree of accuracy in the solution can be obtained.

scholarly journals Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I

2021 ◽  
Vol 41 (1) ◽  
pp. 71-94
Author(s):  
Manabu Naito
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Ordinary Differential Equations ◽  
Nonoscillatory Solutions ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential

We consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t \to \infty\).

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