Vol 40, No 7 (2019)

Since the second half of the twentieth century, wide studies of Sobolev-type equations are undertaken. These equations contain items that are derivatives with respect to time of the second order derivatives of the unknown function with respect to space variables. They can describe nonstationary processes in semiconductors, in plasm, phenomenons in hydrodynamics and other ones. It is important to notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existense and uniqueness of solutions, their asymptotics and blow-up are known. But there are few results about exact solutions of Sobolev-type equations. There are books and papers about exact solutions of partial equations, but they are devoted mainly to classical equations, where the first or second order derivative with respect to time or the derivative with respect to time of the first order derivative of the unknown function with respect to the space variable is equal to a stationary expression. Therefore it is interesting to study exact solutions of Sobolev-type equations.

In the present paper, three nonclassical nonlinear partial equations are studied. Some results about existense, uniqueness and blow-up of their solutions are already known. Here, we construct some classes of their exact solutions with the help of Maple System. We use the method of travelling waves, the method of short-cut decompositions and construction of solutions of some special forms. Also, we discuss the way of realizing of these investigations with the help of Maple system of computer mathematics.

The article focuses on the problem of defining paradigmatic relations between definitions of certain fields in mathematical physics. The ultimate goal is to outline the hierarchical relations between the terms that can be used when searching on the mathematical resources along with additional classification parameters set in secondary documents. A thesaurus entry is selected as an information model. The thesaurus was formed by analyzing the original works of classics of mathematical analysis and differential calculus, and a representative list of articles was organized for that purpose. Following the example of thesaurus on the ‘problem of mixed type equations’ domain, a way of employing formulas in a mathematical article search is proposed. The paper covers a work script of a user, who is familiar with the subject domain and deals with papers done with the help of TeX-notation. A natural document indexing mechanism is set by key words in cited secondary documents. Such an approach helps to specify the search query with mathematical notation regardless the source language. The semantic links effect is based on usage of terms from the mathematical subject domain thesaurus stored with formulas that serve as a background for a mathematical search. It results in lower level of information noise and reduced search time.

It is proposed a procedure for constructing a non-integrable variational form, allowing to extend the procedure of formal construction of variational models of mechanics of deformable media to irreversible deformation processes. We introduce the non-integrable variational forms that determine possible dissipation channels depending on the list of generalized variables for a particular model of media. The problems of filtering of Biot models are considered and it is proved that non-integrable variational forms allow us to construct variational hydrodynamic models of Darcy, Navier-Stokes and Navier-Stokes-Darcy. It is shown that the equations of the Navier—Stokes—Darcy model contain the Helmholtz operator, which will allow to take into account the scale effects associated with the boundary layers.

We study attractors existence of weak solutions to viscoelastic media with memory motion model in non-autonomous case. The theory of trajectory attractors for non-invariant trajectory spaces is applied and the existence of uniform trajectory attractor and uniform global attractor for this system is proved. The proof of existence theorems is based on the approximation-topological method.

The connection is considered between integrals and series involving polygamma ψ (z) and zeta ζ (z, s) functions, Bernoulli B_n (z) and Euler E_n (z) polynomials, and Bernoulli B_n and Euler E_n numbers.

Using the uniform approach, linear differential equations of elliptic, hyperbolic and parabolic types with variable coefficients depending on coordinates and time are considered. It is shown that the solution of the initial equation can be expressed by means of an integral formula through the solution of the accompanying equation of the same type, but with constant coefficients. It is considered that the solution of the accompanying equation is known. From the integral formula, assuming smoothness of the accompanying solution, an equivalent representation of the solution of the initial equation is obtained in the form of series in all possible derivatives of the solution of the accompanying equation. For coefficients at derivatives a system of recurrent equations is obtained, which can be solved analytically at some cases.

A non-singular solution of the gradient elastic fracture mechanics for the cracks of Modes I and II is constructed in this paper. Previously, a similar problem was solved only for cracks of Mode III. A generalized theory of elasticity is used, in which the governing equation in displacements is represented as a product of the Lamé operator and the Helmholtz operator, and the classical boundary value problem for the total stress is completely distinguished in the problems of crack mechanics. As a result, the determination of local stress fields reduces to solving the Helmholtz equations with the known right-hand side of the equations. The Papkovich-Neuber representation in a complex form is used to construct a solution of the mechanics of cracks. We used also the method of radial multipliers, which allows us to construct fundamental solutions of the Helmholtz equations corresponding to analytical functions with a fractional exponent and, as a result, to find solutions that compensate for the singularities.

The non-linear bending of slender heavy beam resting on rigid plane and subjected to the end point force is studied. Solution of the problem is derived by using series expansion technique. Approximate closed-form analytical solution for the length of the separated segment of elastica is found. It is shown that obtained simple solution can be used as the approximate lower bound for the evaluation of the force that should be applied to the beam’s end to provide the prescribed length of separation, while the known classical solution of linear theory can be used as the upper bound.