Solution of boundary value problems with moving boundaries by using an approximate method for constructing solutions of integro – differential equations

The problem of oscillations of objects with moving boundaries, formulated as a differential equation with boundary and initial conditions, is a non-classical generalization of a problem of hyperbolic type. To facilitate the construction of a solution to this problem and justify the choice of a solution form, equivalent integro-differential equations are constructed with symmetric and time-dependent kernels and integration limits varying in time. The method for constructing solutions of integro-differential equations is based on the direct integration of differential equations in combination with the standard replacement of the desired function with a new variable. The method is extended to a wider class of model boundary value problems that take into account the bending stiffness of an oscillating object, the resistance of the environment, and the rigidity of the substrate. Particular attention is paid to the consideration of the most common in practice case when external disturbances act at the boundaries. The solution is made in dimensionless variables accurate to second-order values of smallness with respect to small parameters characterizing the speed of the border.

Generalized solvability and optimal control for an integro-differential equation of a hyperbolic type

We consider an integro-differential operator with Volterra type integral term. We provide a priory inequalities in negative norms for certain spaces. Further, using obtained inequalities we prove well-posedness (existence and uniqueness of the (weak) generalized solution) of the corresponding boundary value problem as well as a theorem on optimal control existence.

An algebraic criterion of the Darboux integrability of differential-difference equations and systems

The article investigates systems of differential-difference equations of hyperbolic type, integrable in sense of Darboux. The concept of a complete set of independent characteristic integrals underlying Darboux integrability is discussed. A close connection is found between integrals and characteristic Lie-Rinehart algebras of the system. It is proved that a system of equations is Darboux integrable if and only if its characteristic algebras in both directions are finite-dimensional.

Functional Inequalities for Metric-Preserving Functions with Respect to Intrinsic Metrics of Hyperbolic Type

We obtain functional inequalities for functions which are metric-preserving with respect to one of the following intrinsic metrics in a canonical plane domain: hyperbolic metric or some restrictions of the triangular ratio metric, respectively, of a Barrlund metric. The subadditivity turns out to be an essential property, being possessed by every function that is metric-preserving with respect to the hyperbolic metric and also by the composition with some specific function of every function that is metric-preserving with respect to some restriction of the triangular ratio metric or of a Barrlund metric. We partially answer an open question, proving that the hyperbolic arctangent is metric-preserving with respect to the restrictions of the triangular ratio metric on the unit disk to radial segments and to circles centered at origin.