On the solvability and limiting properties of some systems of partial differential equations with a small parameter in the principal part

In this paper, we consider linear systems of partial differential equations involving a small parameter as the coefficient of one of higher derivatives and establish a relationship between solutions of the singularly perturbed problem and solutions of the limit system in which the perturbation parameter is equal to zero. We examine the influence of the matrix pencil composed of the coefficients of the equations on the solvability of both original and limit problems and state sufficient conditions for the passage to the limit in terms of the parameter from the perturbed system to the limit system. Using vector-matrix methods, we obtain explicit formulas for solutions of the problems considered.

small parameter, Cauchy problem, limit problem, matrix pencil, regularity index