Valentyn Sobchuk, Iryna Zelenska, Vasyl Bobochko
abstract:
This paper investigates the case where the principal matrix contains negative
components that significantly affect the asymptotic behavior of solutions.
Constructive conditions for the existence of an asymptotic solution to a system
of singularly perturbed fourth-order differential equations with a differential
turning point are established. An algorithm for constructing the corresponding
approximate solution is proposed. Applying the method of essentially singular
functions, an asymptotic representation of the solution is derived that reflects
the specific structural features of the problem. Particular attention is given
to the case where the spectrum of the limiting operator contains multiple
eigenvalues and zero spectral elements. The analysis conducted provides a deeper
understanding of the behavior of solutions at critical points and lays the
foundation for further studies of related classes of problems.
Mathematics Subject Classification: 34M60, 34E20
Key words and phrases: Asymptotic solution, singularly perturbed system of differential equations, turning point, essentially singular functions, space of resonance-free solutions, uniform asymptotics, singular point, perturbation