The asymptotic solution of a bisingular Cauchy problem for systems of ordinary differential equations

The Cauchy problem for system of ordinary differential equations with a small parameter in the highest derivatives takes a unique place in mathematics. The aim of the research is to develop the asymptotic method of boundary functions for constructing complete asymptotic expansions of the solutions to such problems. The proposed generalized method of boundary functions differs from the matching method in the fact that the growing features of the outer expansion are actually removed from it and with the help of the auxiliary asymptotic series are fully included in the internal expansions. This method differs from the classical method of boundary functions in the fact that the boundary functions decay non-exponentially in power-mode nature. Using the proposed method, a complete asymptotic expansion of the solution to the Cauchy problem for bisingular perturbed linear inhomogeneous system of ordinary differential equations is built. Abuilt asymptotic series corresponds to the Puiseux series. The basic term of the asymptotic expansion of the solution has a negative fractional degree of the small parameter, which is typical for bisingular perturbed equations. The built expansion is justified by the method of differential inequality.