Modern Problems in Applied Analysis

In this paper we describe briefly recent new results about the degenerate equations of the p-Laplacian type in a bounded cone. We shall consider the Dirichlet problem for such equation with the strong nonlinear right part as well as the Robin problem for such equation with singular nonlinearity in the right part. Such problems are mathematical models occurring in reaction-diffusion theory, non-Newtonian fluid theory, non-Newtonian filtration, the turbulent flow of a gas in porous medium, in electromagnetic problems, in heat transfer problems, in Fick’s law of diffusion et al. The aim of our investigations is the behavior of week solutions to the problem in the neighborhood of an angular or conical boundary point of the bounded cone. We establish sharp estimates of the type |u(x)| = O(|x|^ϰ) for the weak solutions u of the problems under consideration.

The article gives an overview about recently developed spatial generalizations of the Kolosov-Muskhelishvili formulae using the framework of hypercomplex function theory. Based on these results, a hypercomplex version of the classical Kelvin solution is obtained. For this purpose a new class of monogenic functions with (logarithmic) line singularities is studied and an associated two step recurrence formula is proved. Finally, a connection of the function system to the Cauchy-kernel function is established.

Understanding of viscoelastic response of a periodontal membrane under the action of short-term and long-term loadings is important for many orthodontic problems. A new analytic model describing behavior of the viscoelastic periodontal ligament after the tooth root translational displacement based on Maxwell approach is suggested. In the model, a tooth root and alveolar bone are assumed to be a rigid bodies. The system of differential equations for the plane-strain state of the viscoelastic periodontal ligament is used as the governing one. The boundary conditions corresponding to the initial small displacement of the root and fixed outer surface of the periodontal ligament in the dental alveolus are utilized. A solution is found numerically for fractional viscoelasticity model assuming that the stress relaxation in the periodontal ligament after the continuing displacement of the tooth root occurs approximately within five hours. The character of stress distribution in the ligament over time caused by the tooth root translational displacement is evaluated. Effect of Poisson’s ratio on the stresses in the viscoelastic periodontal ligament is considered. The obtained results can be used for simulation of the bone remodelling process during orthodontic treatment and for assessment of optimal conditions of the orthodontic load application.

In this paper, we investigate a Dirichlet type problem, known as Riquier problem, for higher order linear complex differential equations in the unit polydisc of \(\mathbb {C}^2\). After deriving a Green’s function, we present the solution for a model equation with homogeneous boundary conditions. Afterwards we obtain the solution of a linear equation for Riquier boundary value problem on the unit polydisc in \(\mathbb {C}^2\).

We present and qualitatively analyze a stochastic microscopic model of redistribution of individuals inside a domain which can be thought as representing an elevator. The corresponding mesoscopic model is also derived.

This paper is devoted to boundary value problems for elastic problems modelled by the biharmonic equation in two-dimensional composites. All the problems are studied via the method of complex potentials. The considered boundary value problems for analytic functions are reduced to functional-differential equations. Applications to calculation of the effective properties tensor are discussed.

The boundary value problem of the theory of elasticity for a finite anisotropic plate with random cracks has been solved. Stress intensity factors and energy flows near the tips of cracks are determined as a linear functional on solutions to a system of singular integral equations. It is shown, that in case of the normal distribution of the cracks, the statistical characteristics (mathematical expectations and dispersions) of the distraction (stress intensity factors and energy flows) have also the normal law distribution.

Mixed boundary value problems for the two-dimensional Laplace’s equation in a domain D are reduced to the Riemann-Hilbert problem Re \(\overline {\lambda (t)}\psi (t) = 0\), t ∈ ∂D, with a given Hölder continuous function λ(t) on ∂D except at a finite number of points where a one-sided discontinuity is admitted. The celebrated Keldysh-Sedov formulae were used to solve such a problem for a simply connected domain. In this paper, a method of functional equations is developed to mixed problems for multiply connected domains. For definiteness, we discuss a problem having applications in composites with a discontinuous coefficient λ(t) on one of the boundary components. It is assumed that the domain D is a canonical domain, the lower half-plane with circular holes. A constructive iterative algorithm to obtain an approximate solution in analytical form is developed in the form of an expansion in the radius of the holes.

We present a boundary integral method for solving a certain class of Riemann-Hilbert problems known as the general conjugation problem. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann kernel. We present also an alternative proof for the existence and uniqueness of the solution of the general conjugation problem.

The aim of this work is to describe new interesting examples of non-smooth manifolds and elliptic pseudo-differential operators acting in functional spaces on such manifolds. Fredholm properties for these operators are studied by factorization methods, and these are based on several complex variables.

We propose an introductory study of gravity driven Stokesian flow past the wavy bottom, based on Adler’s et al. papers. In examples the waviness is described by a sinus function and its amplitude is small, up to O(ε ²). A correction to Hagen-Poiseuille’s type free-flow solution is found. A contribution of capillary surface tension is discussed.

We study a first order differential system subject to a nonlocal condition. Our goal in this paper is to establish conditions sufficient for the existence of positive solutions when the considered problem is at resonance. The key tool in our approach is Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima. We conclude the paper with several examples illustrating the main result.