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## Numerical Methods for Constructing Solutions of Functional Differential Equations of Pointwise Type

The article discusses construction of traveling wave type solutions for the Frenkel-Kontorova model on the propagation of longitudinal waves. For the first time, based on the existence and uniqueness theorem of traveling wave type solutions, as well as the approximation theorem, a complete family of traveling wave type solutions is constructed in the form of subfamilies of bounded solutions (horizontals) and unbounded solutions (verticals).

This paper deals with the implementation of numerical methods for searching for traveling waves for Korteweg-de Vries-type equations with time delay. Based upon the group approach, the existence of traveling wave solution and its boundedness are shown for some values of parameters. Meanwhile, solutions constructed with the help of the proposed constructive method essentially extend the class of systems, possessing solutions of this type, guaranteed by theory. The proposed method for finding solutions is based on solving a multiparameter extremal problem. Several numerical solutions are demonstrated.

A one-parameter family of Mackey-Glass type differential delay equations is considered. The existence of a homoclinic solution for suitable parameter value is proved. As a consequence, one obtains stable periodic solutions for nearby parameter values. An example of a nonlinear functions is given, for which all sufficient conditions of our theoretical results can be verified numerically. Numerically computed solutions are shown.

The system considered in this paper consists of two equations $(k=1,2)$ $\dot x(t)=(-1)^{k-1} (0\le t<\infty),\, k(0)=1,\,x(0)=0,\,x(t)\not\in\{0,1\}(-1\le t<0),$ that change mutually in every instant $t$ for which $x(t-\tau)\in\{0,1\}$, where $\tau={\rm const}>0$ is given. In this paper the behavior of the solutions is characterized for every $\tau\in(\frac{4}{3},\frac{3}{2})$, i. e. in case not covered in \cite{ADM}; as it was noted there, this behavior turned out to be more complex then when $\tau\in(3/2,\infty)$. Thus the behavior of the solutions of this system with critical set $K=\{0,1\}$ is characterized for every $\tau>0$.

We consider linear boundary-value problems for systems of functional differential equations when the number of boundary conditions is greater than the dimension of the system. We allow the boundary conditions to be fulfilled approximately. We propose an approach based on theorems whose conditions allow the verification by special reliable computing procedures.

For equations of mathematical physics, which are the Euler-Lagrange equation of the corresponding variational problem, an important class of solutions are traveling wave solutions (soliton solutions). In turn, soliton solutions for finite-difference analogs of the equations of mathematical physics are in one-to-one correspondence with solutions of induced functional differential equations of pointwise type (FDEPT). The presence of a wide range of numerical methods for constructing FDEPT solutions, as well as the existence of appropriate existence and uniqueness theorems for the solution, a continuous dependence on the initial and boundary conditions, the "rudeness" of such equations, allows us to construct soliton solutions for the initial equations of mathematical physics. Within the framework of the presented work, on the example of a problem from the theory of plastic deformation the mentioned correspondence between solutions of the traveling wave type and the solutions of the induced functional differential equation will be demonstrated.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.