Vol 60, No 9 (2024)

The paper considers invariant manifolds with a change in stability of differential systems with differentscale variables. The interest in such manifolds is explained by their widespread use in applied problems. The questions of the existence of continuous invariant manifolds with a change in stability are investigated in three critical cases.

A new class of Ito^ integral equations is considered, which contains many classical problems, for example, the Cauchy problem for differential equations of integer and fractional order with and without stochastic perturbations, as well as some less known and little-studied types of equations that have been introduced recently. The purpose of the study is to find sufficiently general conditions that guarantee the existence and the uniqueness of solutions to such equations, taking into account their specific features. The article therefore proposes to use a special generalized Lipschitz condition, which, due to its flexibility, allows one to obtain effective solvability criteria in terms of the right-hand sides of equations. Numerous examples are considered, covering in particular Ito^ differential equations of fractional order with aftereffect and without aftereffect, equations with fractional Wiener processes, Ito^ equations with several time scales, as well as their generalizations.

The logistic dynamics model developed by U. Dieckmann and R. Law is considered in the article. The analysis of a nonlinear integral equation describing the equilibrium state of a single-species community with a three-parameter closure of the third spatial moment is carried out in the case when the dispersion and competition kernels are piecewise constant functions. Sufficient conditions for the mentioned equation solvability are established.

For a homogeneous isotropic elastic half-plane with a stratified elastic coating we consider the Poincar´e– Steklov operator that maps normal stresses into normal displacements on part of the boundary. To construct the transfer function of this operator, the variational formulation of the boundary value problem for transforms of displacements is used. A definition is given and the existence and uniqueness are proved for a generalized solution of the variational problem. This problem is approximated by the finite element method. To numerically solve the resulting system of linear algebraic equations, the preconditioned conjugate gradient method is used. The developed computational algorithm was verified.

A quadrature formula has been constructed for calculating the hypersingular integral over a segment, which uses the ends of the segment partition intervals as nodes of piecewise constant interpolation of the integral density, as well as specially selected collocation points. A distinctive feature of the proposed quadrature formula is the ability to calculate the integral of functions that suffer a finite number of discontinuities of the first kind on the integration interval. On the basis of quadrature formula constructed, a numerical scheme for solving the characteristic hypersingular integral equation on non-regular grid is developed. Estimate of the rate of convergence of approximate solutions to exact ones is proved in the class of piecewise Ho¨lder functions.