FEATURES OF THE CONSTRUCTION OF UNIFORM ASYMPTOTIC SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH A TURNING POINT WITH POSITIVE COEFFICIENTS OF THE MATRIX

Abstract

The work is devoted to the analysis of the coefficients of the singular operator of the Orr-Sommerfeld type in vector form including the turning point. The system of singularly perturbed differential equations with a small parameter at the highest derivative is investigated. We consider the case when the spectrum of the limit operator contains multiple and identically equal zero elements. Using the method of essentially singular functions, the uniform asymptotic solution of the system is constructed. For the case of a stable turning point, the asymptotics of the solutions of the system are constructed in the sector that contains the turning point. The asymptotics of the first two solutions for the homogeneous problem are constructed using the Airy functions and their derivatives. The third formal solution of a homogeneous system for this case presents certain difficulties. Therefore, taking into account the specified conditions, in order to construct the uniform asymptotics of the solution for the given system, we used the partial solution of the heterogeneous system as the third solution of the homogeneous system.