Reciprocal Function Method for Cauchy Problems
with First-Order Poles
A. A. Belova,b,* and Corresponding Member of the RAS N. N. Kalitkinc,**
a Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991 Russia
b RUDN University, Moscow, 117198 Russia
c Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow,
125047 Russia
Correspondence to: *e-mail: aa.belov@physics.msu.ru
Correspondence to: **e-mail: kalitkin@imamod.ru
Received 5 November, 2019
Abstract—For the numerical solution of the Cauchy problem with multiple poles, we propose a reciprocal function method. In the case of first-order poles, it makes it possible to continue the solution through the poles and to determine the solution and the pole positions with good accuracy. The method allows one to employ conventional explicit and implicit schemes, for example, explicit Runge–Kutta schemes. A test problem with multiple poles is computed as an example. The proposed method is useful for construction of software for direct computation of special functions.
Keywords: Cauchy problem, singularities, continuation through a pole
DOI: 10.1134/S1064562420020040