In order to solve a parabolic variational inequality with a nonlocal spatial operator and a one-sided constraint on the solution, a numerical method based on the penalty method, finite elements, and the implicit Euler scheme is proposed and studied. Optimal estimates for the accuracy of the approximate solution in the energy norm are obtained.

The article is devoted to the definition and properties of the class of diffeomorphisms of
the unit disk D = { z : | z| < 1} on the complex plane C for which the harmonic measure of the
boundary arcs of the slit disk has a limited distortion, i.e. is quasiinvariant. Estimates for derivative
mappings of this class are obtained. We prove that such mappings are quasiconformal and are also
quasiisometries with respect to the pseudohyperbolic metric. An example of a mapping with the
specified property is given. As an application, a generalization of the Hayman–Wu theorem to this
class of mappings is proved.

In the present study, we explore a new mathematical formulation involving modified
couple stress thermoelastic diffusion (MCTD) with nonlocal, voids and phase lags. The governing
equations are expressed in dimensionless form for the further investigation. The desired equations
are expressed in terms of elementary functions by assuming time harmonic variation of the field
variables (displacement, temperature field, chemical potential and volume fraction field). The
fundamental solutions are constructed for the obtained system of equations for steady oscillation,
and some basic features of the solutions are established. Also, plane wave vibrations has been
examined for two dimensional cases. The characteristic equation yields the attributes of waves like
phase velocity, attenuation coefficients, specific loss and penetration depth which are computed
numerically and presented in form of distinct graphs. Some unique cases are also deduced. The
results provide the motivation for the researcher to investigate thermally conducted modified couple
stress elastic material under nonlocal, porosity and phase lags impacts as a new class of applicable
materials.

A new method for obtaining a generalized solution of a mixed boundary value problem for a parabolic equation with boundary conditions of the third kind and a continuous initial condition is proposed. Generalized functions are understood in the sense of a sequential approach. The representative of the class of sequences, which is a generalized function, is obtained using the function interpolation operator, constructed using solutions of the Cauchy problem. The solution is obtained in the form of a series that converges uniformly inside the domain of the solution.

In this article, we consider a class Ꝝ* consisting of functions, subharmonic in the unit disk and such that their compositions with some families of linear fractional automorphisms of the disk form normal families. We prove a theorem which states that for any function of class Ꝝ* the set of points of the unit circle can be represented as a union of Fatou points, generalized point Plesner, and a set of zero measure.

In this paper, a 2х2 block operator matrix H is considered as a bounded and self-adjointoperator in a Hilbert space. The location of the essential σ_ess(H) of operator matrix H is described via the spectrum of the generalized Friedrichs model, i.e. the two- and three-particle branches of the essential spectrum σ_ess(H) are singled out. We prove that the essential spectrum σ_ess(H)  consists of no more than six segments (components).

 A transformation model of the dynamic deformation of an elongated orthotropic composite rod-type plate, consisting of two sections (fastened and free) along its length, is proposed. In the free section, the orthotropic axes of the material do not coincide with the axes of the Cartesian coordinate system chosen for the plate, and in the fastened section, the displacements of points of the contact’s boundary surface (rigid connection) with the elastic support element are considered to be known. The constructed model is based on the use for the free section of the relations of the refined shear model of S.P. Timoshenko, compiled for rods in a geometrically nonlinear approximation without taking into account lateral strain deformations. For the section fastened on the elastic support element, a one-dimensional shear deformation model is constructed taking into account lateral strain deformations, which is transformed into another model by satisfying the conditions of kinematic coupling with the elastic support element with given displacements of the interface points with the plate. The conditions for the kinematic coupling of the free and fastened sections of the plate are formulated. Based on the Hamilton–Ostrogradsky variational principle, the corresponding equations of motion and boundary conditions, as well as force conditions for the coupling of sections, are derived. The constructed model is intended to simulate natural processes and structures when solving applied engineering problems aimed at developing innovative oscillatory biomimetic propulsors