We propose an effective method for solving Cauchy problem for an ordinary differential equation with multiple poles of an integer order. The method provides through calculation of a pole for both single pole and chain of poles. The method uses a special algorithm for finding the multiplicity of each pole. This multiplicity is used to define the generalized reciprocal function for which the K-th order pole of the initial function is a prime zero. Calculating such a zero is not difficult, so the proposed method provides high accuracy even near the poles. After passing this zero, the calculation of the initial function resumes. Using this method on a sequence of poles permits to find a numerical solution simultaneously with a posteriori estimation of its error. The method is illustrated with test examples.

Cauchy problem, singularities, continuation through pole

Mathematical modelling in actual problems of science and technics

Belov Alexander Alexandrovich,  ,  Faculty of Physics M.V. Lomonosov MSU; PFUR

Kalitkin Nikolaj Nikolaevich,  ,  orcid.org/0000-0002-0861-1792,  KIAM RAS