Numerical solution of the Cauchy problem with singularities

For numerical solution of the Cauchy problem with multiple singularities, we consider two distinct approaches. Firstly, we provide schemes permitting direct computation through the singularity. These schemes are implicit ones. However, this approach provides only the first order of accuracy. Secondly, we propose the reciprocal function method. For the first order poles, this method allows to continue the solution through the pole providing calculation of the solution itself and the pole position with good accuracy. One can imply traditional explicit and implicit schemes, e.g., the Runge-Kutta methods. We provide an example of calculation of a problem with multiple poles. The proposed method is useful for numerical calculation of special functions.