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2023 | Buch

Differential Equations, Mathematical Modeling and Computational Algorithms

DEMMCA 2021, Belgorod, Russia, October 25–29


Über dieses Buch

This book contains reports made at the International Conference on Differential Equations, Mathematical Modeling and Computational Algorithms, held in Belgorod, Russia, in October 2021 and is devoted to various aspects of the theory of differential equations and their applications in various branches of science. Theoretical papers devoted to the qualitative analysis of emerging mathematical objects, theorems of the existence and uniqueness of solutions to the boundary value problems under study are presented, and numerical algorithms for their solution are described. Some issues of mathematical modeling are also covered; in particular, in problems of economics, computational aspects of the theory of differential equations and boundary value problems are studied. The articles are written by well-known experts and are interesting and useful to a wide audience: mathematicians, representatives of applied sciences and students and postgraduates of universities engaged in applied mathematics.


Some Classes of Quasilinear Equations with Gerasimov—Caputo Derivatives
The Cauchy problem for resolved with respect to the oldest derivative quasilinear multi-term equations in Banach spaces with the fractional Gerasimov—Caputo derivatives, with bounded linear operators at them and with locally Lipschitzian nonlinear operator is studied. Theorem on the local existence and uniqueness of a solution to the Cauchy problem is proved. This result applied to study of the so-called degenerate (i.e. with a degenerate linear operator at the oldest derivative) equations of the similar form. Using the reduction of a special initial value problem for a degenerate equation to the Cauchy problem for two non-degenerate equations on two subspaces under four types of additional conditions on nonlinear locally Lipschitzian operator four theorems on local unique solvability are proved. Abstract results are illustrated by initial-boundary value problems for partial differential systems of equations with Gerasimov—Caputo derivatives in time.
Vladimir E. Fedorov, Kseniya V. Boyko
On the Solvability of Initial Problems for Abstract Singular Equations Containing Fractional Derivatives
With the help of integral representations of the Poisson type, it is established that the Cauchy problem for a number of abstract singular equations with fractional derivatives reduces to a simpler problem for a non-singular equation.
Alexander Glushak
Local Bifurcations of Periodic Traveling Waves in the Generalized Weakly Dissipative Ginzburg-Landau Equation
In  this paper we consider a periodic boundary value problem for the generalized Ginzburg-Landau. The generalized version of the weakly dissipative Ginzburg-Landau equation differs from the traditional version by replacing the cubic nonlinearity with nonlinearity of arbitrary odd degree. We will show that the periodic boundary value problem has a countable set of solutions that are single-mode and periodic in the evolutionary variable. We will examine the stability question as well as local bifurcations of such solutions when they change stability. In this case, the two-dimensional attracting invariant tori bifurcate emerges when stability is lost from single-mode solutions. These are non-resonant tori that have appeared in the generic situation. The main results are obtained on the basis and development of methods of the theory of dynamical systems with an infinite-dimensional phase space. These include the method of invariant manifolds and normal forms, as well as the principle of self-similarity. This principle allows us to reduce the problem of bifurcations of a countable set of single-mode solutions to the analysis of the corresponding problem.
Anatoly Kulikov, Dmitry Kulikov
Towards Discrete Octonionic Analysis
In recent years, there is a growing interest in the studying octonions, which are 8-dimensional hypercomplex numbers forming the biggest normed division algebras over the real numbers. In particular, various tools of the classical complex function theory have been extended to the octonionic setting in recent years. However not so many results related to a discrete octonionic analysis, which is relevant for various applications in quantum mechanics, have been presented so far. Therefore, in this paper, we present first ideas towards discrete octonionic analysis. In particular, we discuss several approaches to a discretisation of octonionic analysis and present several discrete octonionic Stokes’ formulae.
Rolf Sören Kraußhar, Anastasiia Legatiuk, Dmitrii Legatiuk
Axiomatic Method for Constructing a Generalized Solution of a Mixed Problem for a Telegraph Equation
The paper presents an algorithm for constructing a rapidly converging series representing a generalized solution of a mixed problem for a telegraphic equation considered in a half-band. Reviewed the case of an essentially non-self-adjoint operator in a spatial variable. The system of root functions of a differential operator, in addition to its eigenfunctions, contains an infinite number of associated functions. The constructed series can be considered as a generalized d’Alembert formula. A new axiomatic A.P. Khromov’s method is applied to construct the solution. The proposed approach superseds the traditional method of separating variables for solving mixed problems, which usually results in to slowly converging series. For the problem under consideration, in general, the method of separating variables is not applicable, since the coefficient of the equation depends both on the spatial variable and on time.
Igor S. Lomov
Non-local Substitutions for Liouville Equations with Three and Four Independent Variables
We obtained the non-local transformations of the Cole—Hopf type, which translate the Liouville equations with three and four independent variables into the Bianchi equations. The solutions with arbitrary functions of these Liouville equations are constructed.
Aleksey Mironov, Lyubov Mironova
Convergence Rates of a Finite Difference Method for the Fractional Subdiffusion Equations
We consider the convergence of an effective numerical method of the subdiffusion equation with the Caputo fractional derivative in time. We investigate an implicit difference scheme and an explicit difference scheme by using the projection method in space and a finite difference method which was proposed by Ashyralyev in time. Combining the method of functional analysis and the technique of numerical analysis, we utilize the idea of layering in temporal direction to obtain that the local truncation error is \(O(n^{-\alpha })\). Then we prove that the implicit and explicit numerical methods converge at a rate of \(O(\tau ^\alpha )\) in time. Finally, a numerical experiment is given to confirm the \(\alpha \)-th order accuracy.
Li Liu, Zhenbin Fan, Gang Li, Sergey Piskarev
Degenerate Quasilinear Equations with Dzhrbashyan—Nersesian Derivatives and Applications
Quasilinear equations with Dzhrbashyan—Nersesyan derivatives in Banach spaces are studied. The existence of a unique classical solution for a Showalter type initial value problem is proved for equation, which contains a degenerate linear operator at the oldest derivative. This result and results for the corresponding degenerate linear equation, which were obtained by authors earlier, are applied to the consideration of initial boundary value problems for linearized and nonlinear systems of partial differential equations with the Dzhrbashyan—Nersesyan time derivative, which describes the dynamics of viscoelastic fluids.
Marina Plekhanova, Elizaveta Izhberdeeva
On a K-Homogeneous Metric
We consider a Riemannian metric which generates the Beltrami-Laplace operator coinciding with the B-elliptic operator up to a factor.
Marina V. Polovinkina, Igor P. Polovinkin
Biooscillators in Models of Genetic Networks
We study periodic attractors in a system of ordinary differential equations, which is used to model genetic regulatory networks. The systems of order two and four are considered, which posess the periodic attractors. The systems of order three and six are considered also.
Felix Sadyrbaev, Inna Samuilik, Valentin Sengileyev
Numerical Method for Problem of Scattering by a Small Thickness Dielectric Layer on a Perfectly Conductive Substrate
In this work we consider the problems of scattering of a monochromatic wave by a dielectric body in the form of a thin layer placed on a perfectly conducting base. For this case we formulate the boundary value problem for Maxwell’s equations with an impedance boundary condition and reduce it to a system of two boundary integral equations with weakly and strongly singular integrals on a perfectly conducting surface. Finally, we construct a numerical method for the considered problem which based on solution of these integral equations.
Alexey Setukha, Stanislav Stavtsev
Invariants of Dynamical Systems with Dissipation on Tangent Bundles of Low-Dimensional Manifolds
Tensor invariants (differential forms) for homogeneous dynamical systems on tangent bundles to smooth two-dimensional manifolds are presented in this paper. The connection between the presence of these invariants and the full set of the first integrals necessary for the integration of geodesic, potential and dissipative systems is shown. At the same time, the introduced force fields make the considered systems dissipative with dissipation of different signs and generalize the previously considered ones. We also represent the typical examples from rigid body dynamics.
Maxim V. Shamolin
B-subharmonic Functions
Considering different problems with Bessel operator we inevitably should obtain the main theorems of harmonic analysis for Laplace–Bessel operator. In this article we obtain condition of B-subharmonicity using the second Green’s formula for the Laplace–Bessel operator.
Elina Shishkina
Some Multi–dimensional Modified G- and H-Integral Transforms on -Spaces
This paper is devoted to the study of three classes of multidimensional integral transformations with Fox’ H-function and the Meijer’s G-function in kernels in weighted spaces integrable functions in the domain \(\textrm{R}^{n}_{+}=\textrm{R}^{1}_{+}\times \textrm{R}^{1}_{+}\times \cdots \times \textrm{R}^{1}_{+}\). Mapping properties such as the boundedness, the rang, the representation and the inversion of the considered transforms are established.
S. M. Sitnik, O. V. Skoromnik, M. V. Papkopvich
On Sufficient Conditions of the Faddeev–Marchenko Theorem
We study sufficient conditions, ensuring validity of the Faddeev–Marchenko fundamental theorem on restoration of the potential of the Sturm–Liouville equation on the entire axis along the given linear ratios between the Jost functions. These conditions are formulated in terms of so-called reflection coefficient within the framework of the corresponding weighted Holder spaces on the real line with power behavior at infinity.
B. D. Koshanov, A. P. Soldatov
Variational Approach to Construction of Piecewise-Constant Approximations of the Solution of Dynamic Reconstruction Problem
In the paper, the problem of dynamic reconstruction of controls and trajectories for deterministic control-affine systems is considered. The reconstruction is performed in real time using known discrete inaccurate measurements of the observed trajectory of the system. This trajectory is generated by an unknown measurable control that satisfies known geometric constraints. A well-posed statement of the problem is given. A solution is proposed using the variational approach developed by the authors. This approach uses auxiliary variational problem with regularized integral residual functional. The integrant of the functional is a d.c. function. The suggested algorithm reduces the reconstruction problem to integration of Hamiltonian systems of ordinary differential equations. This paper offers a method for construction of piecewise-constant approximations that satisfy the given geometric control constraints. The approximations converge almost everywhere to the desired control, and the reconstructed trajectories of the dynamical system converge uniformly to the observed trajectory.
Nina Subbotina, Evgenii Krupennikov
Discrete Operators and Equations: Analysis and Comparison
We develop a discrete variant of a theory of pseudo-differential equations and boundary value problems in canonical domains which are model situations for manifolds with non-smooth boundaries. Using digitization process for ordinary functional spaces we construct certain discrete functional spaces or spaces of functions of a discrete variable and define discrete pseudo-differential operators acting in such spaces. A main problem in which we are interested is to establish a correspondence between continual and discrete solutions of considered continual and discrete equations and in future boundary value problems. We have illustrated our considerations by certain examples of Calderon–Zygmund operators for which we have some interesting conclusions.
Alexander Vasilyev, Vladimir Vasilyev, Asad Esmatullah
Pseudo-Differential Equations in Spaces of Different Smoothness Exponents on Variables
We study a model elliptic pseudo-differential equation and simplest boundary value problems for a half-space and a special cone in Sobolev–Slobodetskii spaces which have different smoothness with respect to separate variables. Sufficient conditions for a unique solvability for such boundary value problems are described
Vladimir Vasilyev, Victor Polunin, Igor Shmal
Thermodynamic Limit in Vector Lattice Models
Classes of Gibbs random fields \(\textbf{u} ({x})\), \({x} \in {\mathcal Z}^d\) on finite sets \(\varLambda \subset {\mathcal Z}^d\), \(d \in {\mathcal N}\) with values in the space \({\mathcal R}^n\), \(n \in {\mathcal N}\) are studied. Each class is connected with the sequence \(\langle \varLambda ;\, \varLambda \subset {\mathcal Z}^d \rangle \) unboundedly expanding according to the definite rule when \(\varLambda \rightarrow {\mathcal Z}^d\). Each random field is generated by the Hamiltonian \(\textsf{H}_\varLambda [\textbf{u} ({z})]\). Classes of all functionals \(\textsf{H}_\varLambda [\textbf{u} ({z})]\) corresponding to sequence \(\langle \varLambda ;\, \varLambda \subset {\mathcal Z}^d \rangle \) form the Banach space \(\textrm{H}_\nu \). It is proved the existence of the limit statistical characteristic \(\ln \, Z_\varLambda /|\varLambda |\) in each class when \(\varLambda \rightarrow {\mathcal Z}^d\) which is the continuous functional in \(\textrm{H}_\nu \).
Yuri P. Virchenko
Family of Smooth Solutions of Hyperbolic Differential-Difference Equation
Three-parameter familie of solutions is constructed for hyperbolic differential-difference equation with shift operators of the general-type acting with respect to all spatial variables. We prove theorem showing that the solutions obtained are classical provided that the real part of the symbol of the corresponding differential-difference operator is positive. Classes of equations for which these conditions are satisfied is given.
Natalya V. Zaitseva
Differential Equations, Mathematical Modeling and Computational Algorithms
herausgegeben von
Vladimir Vasilyev
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