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Solution of the Cauchy Problem for One Degenerate Equation with the Dzhrbashyan–Nersesyan Fractional Derivative Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 B. Yu. Irgashev
Abstract A solution of the Cauchy problem is obtained for one degenerate equation with the Dzhrbashyan–Nersesyan fractional derivative, particular solutions of which are represented using the Kilbas–Saigo function.
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On One Cauchy Problem for a Hyperbolic Differential-Difference Equation Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 N. V. Zaitseva
Abstract We provide a formulation of the Cauchy problem in a strip for a two-dimensional hyperbolic equation containing a superposition of a differential operator and a shift operator with respect to the spatial variable varying along the entire real axis. The solution of the problem using integral Fourier transforms is constructed in explicit form.
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On the Smoothness of the Poisson Potential for Second-Order Parabolic Systems on the Plane Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 E. A. Baderko, K. D. Fedorov
Abstract We consider the solution of the Cauchy problem in a strip on the plane for a homogeneous second-order parabolic system. The coefficients of the system satisfy the double Dini condition. The initial function is continuous and bounded along with its first and second derivatives. Using the Poisson potential, the nature of the smoothness of this solution is studied and the corresponding estimates
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Pseudodifferential Equations and Boundary Value Problems in a Multidimensional Cone Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 V. B. Vasil’ev
Abstract We consider a special boundary value problem in the Sobolev–Slobodetskii space for a model elliptic pseudodifferential equation in a multidimensional cone. Taking into account the special factorization of the elliptic symbol, we write the general solution of the pseudodifferential equation that contains an arbitrary function. To determine it unambiguously, some integral condition is added
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Analysis of a Multipoint Boundary Value Problem for a Nonlinear Matrix Differential Equation Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 A. N. Bondarev, V. N. Laptinskii
Abstract For a nonlinear differential matrix equation, we study a multipoint boundary value problem by a constructive method of regularization over the linear part of the equation using the corresponding fundamental matrices. Based on the initial data of the problem, sufficient conditions for its unique solvability are obtained. Iterative algorithms containing relatively simple computational procedures
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On the Features of Numerical Solution of Coefficient Inverse Problems for Nonlinear Equations of the Reaction–Diffusion–Advection Type with Data of Various Types Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 D. V. Lukyanenko, R. L. Argun, A. A. Borzunov, A. V. Gorbachev, V. D. Shinkarev, M. A. Shishlenin, A. G. Yagola
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On Singular Heat Equation Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 E. L. Shishkina, A. K. Yusupova
Abstract In physics, the singular heat equation with the Bessel operator is used to explain the basic process of heat transport in a substance with spherical or cylinder symmetry. This paper examines the solution of the Cauchy problem for the heat equation with the Bessel operator acting in the space variable. We obtain some properties of the solution and consider the normalized modified Bessel function
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Initial–Boundary Value Problems for Homogeneous Parabolic Systems in a Semibounded Plane Domain and Complementarity Condition Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 S. I. Sakharov
Abstract We consider initial–boundary value problems for homogeneous parabolic systems with coefficients satisfying the double Dini condition with zero initial conditions in a semibounded plane domain with nonsmooth lateral boundary. The method of boundary integral equations is used to prove a theorem on the unique classical solvability of such problems in the space of functions that are continuous
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Existence of an Anti-Perron Effect of Change of Positive Exponents of the Linear Approximation System to Negative Ones under Perturbations of a Higher Order of Smallness Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 N. A. Izobov, A. V. Il’in
Abstract We prove the existence of a two-dimensional linear system \(\dot {x}=A(t)x \), \(t\geq t_0\), with bounded infinitely differentiable coefficients and all positive characteristic exponents, as well as an infinitely differentiable \(m\)-perturbation \(f(t,y) \) having an order \(m>1 \) of smallness in a neighborhood of the origin \(y=0 \) and an order of growth not exceeding \(m \) outside it
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Cauchy Problem for the Loaded Korteweg–de Vries Equation in the Class of Periodic Functions Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 A. B. Khasanov, T. G. Khasanov
Abstract The inverse spectral problem method is applied to finding a solution of the Cauchy problem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions. A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded terms and a derivation of an analog of Dubrovin’s system of differential equations are proposed. It is shown that the sum
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Searching for Parameters of a Model with the Best Local Controllability Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 M. A. Velishchanskiy, V. N. Chetverikov
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Discrete Equations, Discrete Transformations, and Discrete Boundary Value Problems Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 E. B. Afanas’eva, V. B. Vasil’ev, A. B. Kamanda Bongay
Abstract We study the solvability of discrete elliptic pseudodifferential equations in a sector of the plane. Using special factorization of the symbol, the problem is reduced to a boundary value problem for analytic functions of two variables. A periodic analog of one integral transformation is obtained that was used to construct solutions of elliptic pseudodifferential equations in conical domains
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On the Solvability of Linear Differential Operators on Vector Bundles over a Manifold Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 M. S. Smirnov
Abstract A necessary and sufficient condition is established for the closedness of the range or surjectivity of a differential operator acting on smooth sections of vector bundles. For connected noncompact manifolds it is shown that these conditions are derived from the regularity conditions and the unique continuation property of solutions. An application of these results to elliptic operators (more
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Existence of Sub-Lorentzian Longest Curves Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 Yu. L. Sachkov
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On the Existence of Feedback Control for One Fractional Voigt Model Diff. Equat. (IF 0.6) Pub Date : 2024-02-26 A. V. Zvyagin, E. I. Kostenko
Abstract We study the feedback control problem for a mathematical model that describes the motion of a viscoelastic fluid with memory along the trajectories of the velocity field. We prove the existence of an optimal control that delivers a minimum to a given bounded and lower semicontinuous cost functional.
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Algorithms for Robust Inversion of Dynamical Systems Diff. Equat. (IF 0.6) Pub Date : 2024-01-25 E. I. Atamas’, A. V. Il’in, S. K. Korovin, V. V. Fomichev
Abstract A new methodology for solving inverse dynamics problem is developed. The methodology is based on using a mathematical model of a dynamical system and robust stabilization methods for a system under uncertainty. Most exhaustively the theory is described for linear finite-dimensional time-invariant scalar systems and multiple-input multiple-output systems. The study shows that with this approach
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On the Variation of the Nonlinearity Parameter in the “Super-Twisting” Algorithm Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 V. V. Fomichev, A. O. Vysotskii
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Hopf Bifurcation in a Predator–Prey System with Infection Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 A. P. Krishchenko, O. A. Podderegin
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On the Construction of the Graph of Discrete States of a Switched Affine System Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 A. S. Fursov, P. A. Krylov
Abstract The problem of constructing the graph of states of a switched affine system closed by a static state feedback is considered. To solve this problem, a constructive algorithm based on the study of the consistency of systems of linear algebraic inequalities is proposed.
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On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 A. A. Davydov, Kh. A. Khachatryan, H. S. Petrosyan
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On Nonlinear Boundary Value Problems for Differential Inclusions Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 A. V. Arutyunov, Z. T. Zhukovskaya, S. E. Zhukovskiy
Abstract We consider autonomous differential inclusions with nonlinear boundary conditions. Sufficient conditions for the existence of solutions in the class of absolutely continuous functions are obtained for these inclusions. It is shown that the corresponding existence theorem applies to the Cauchy problem and the antiperiodic boundary value problem. The result is used to derive a new mean value
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Behavior of Trajectories of a Four-Dimensional Model of HIV Infection Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 A. N. Kanatnikov, O. S. Tkacheva
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Bounded Real Lemma for the Anisotropic Norm of Time-invariant Systems with Multiplicative Noises Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 A. V. Yurchenkov, I. R. Belov
Abstract We consider a discrete-time-invariant system with multiplicative noise with implementation in the state space. The exogenous disturbance is chosen from the class of time-invariant ergodic sequences of nonzero colorness. We consider the level of mean anisotropy of the exogenous disturbance to be bounded by a known value. Conditions for the anisotropic norm to be bounded by a given number are
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On a Positional Control Problem for a Nonlinear Equation with Distributed Parameters Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 V. I. Maksimov
Abstract We consider a guaranteed control problem for a nonlinear distributed equation of diffusion type. The problem is essentially to construct a feedback control algorithm ensuring that the solution of a given equation tracks the solution of a similar equation subjected to an unknown disturbance. The case in which a discontinuous unbounded function can be a feasible disturbance is studied. We solve
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Solvability of Linear Differential Equations Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 V. S. Mokeichev, A. M. Sidorov
Abstract We propose a new approach to the solvability of ordinary as well as partial differential equations in the theory of linear differential equations and also in the theory of integral equations.
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On the Stability of Periodic Solutions of a Model Navier–Stokes Equation in a Thin Layer Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 E. S. Boldyreva
Abstract We study the existence and stability of periodic solutions of the model Navier–Stokes equation in a thin three-dimensional layer depending on the existence and stability of periodic solutions of a special limit two-dimensional equation.
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On a Problem of Calculating the Solvability Set for a Linear System with Uncertainty Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 A. A. Melnikova, P. A. Tochilin
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On the Relationship Between the Pontryagin Maximum Principle and the Hamilton–Jacobi–Bellman Equation in Optimal Control Problems for Fractional-Order Systems Diff. Equat. (IF 0.6) Pub Date : 2023-12-29 M. I. Gomoyunov
Abstract We consider the optimal control problem of minimizing the terminal cost functional for a dynamical system whose motion is described by a differential equation with Caputo fractional derivative. The relationship between the necessary optimality condition in the form of Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives is
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Mixed Problems with Integral Conditions for Hyperbolic Equations with the Bessel Operator Diff. Equat. (IF 0.6) Pub Date : 2023-12-15 N. V. Zaitseva
Abstract The paper considers nonlocal problems with integral conditions for hyperbolic equations with the Bessel differential operator whose statement substantially depends on the intervals where the parameter occurring in this operator varies. The well-posedness of these problems is studied according to a unified scheme based on the classical method of separation of variables, which is also used to
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On Asymptotics of the Spectrum of an Integral Operator with a Logarithmic Kernel of a Special Form Diff. Equat. (IF 0.6) Pub Date : 2023-12-01
Abstract We study the asymptotic behavior of the spectrum of an integral operator similar to an integral operator with a logarithmic kernel depending on the sum of arguments. By a simple change of variables, the corresponding equation is reduced to an integral equation of convolution type defined on a finite interval (as is well known, such equations in the general case cannot be solved by quadratures)
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On Exact Solutions of a Multidimensional System of Elliptic Equations with Power-Law Nonlinearities Diff. Equat. (IF 0.6) Pub Date : 2023-12-01
Abstract Equations and systems of elliptic type with power-law nonlinearities are considered. Such equations are found in modeling distributed robotic formations, as well as in chemical kinetics, biology, astrophysics, and many other fields. The problem of constructing multidimensional exact solutions is studied. It is proposed to use a special type of ansatz that reduces the problem to solving systems
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System without Characteristic Directions with a Nonanalytic Center Condition Diff. Equat. (IF 0.6) Pub Date : 2023-12-01
Abstract A real autonomous differential system of the fifth degree with a degenerate singular point without characteristic directions is obtained. The necessary and sufficient condition for the center at a given point is determined by a function that is not analytic at the boundary point of the set of system parameters for which the singular point of the system is monodromic. An asymptotic representation
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Dirichlet Problem on the Half-Line for an Abstract Euler–Poisson–Darboux Equation Containing Powers of an Unbounded Operator Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 A. V. Glushak
Abstract We consider an abstract Euler–Poisson–Darboux equation containing powers of an unbounded operator that is the generator of a Bessel operator function. Sufficient conditions for the unique solvability of the Dirichlet problem on the half-line are obtained. The question concerning the convergence of the solution to zero at infinity is investigated. Examples are given.
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Gellerstedt Problem with a Nonlocal Oddness Boundary Condition for the Lavrent’ev–Bitsadze Equation Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 T. E. Moiseev
Abstract We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with the oddness boundary condition on the boundary of the ellipticity domain. All eigenvalues and eigenfunctions are obtained in closed form. It is proved that the system of eigenfunctions is complete in the elliptic part of the domain and incomplete in the entire domain. The unique solvability of the problem is also proved;
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Analytical Solution of Mixed Problems for the One-Dimensional Ionization Equations in the Case of Constant Velocities of Atoms and Ions Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 M. B. Gavrikov, A. A. Taiurskii
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Partial Stability of Systems of Itô Linear Delay Differential Equations Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 R. I. Kadiev
Abstract We study the moment stability of solutions in part of the variables with respect to the initial data for systems of Itô linear delay differential equations using a modified regularization method based on the choice of an auxiliary equation and an application of the theory of nonnegatively invertible matrices. For these systems, sufficient stability conditions are obtained in terms of nonnegative
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On the Properties of the Root Vector Function Systems of a $$2m $$ th-Order Dirac Type Operator with an Integrable Potential Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 E. C. Ibadov
Abstract We consider a Dirac type operator with matrix coefficients. Estimates for the root vector functions are established, and criteria for the Bessel property and the unconditional basis property of the root vector function systems of this operator in the space \(L_{2}^{2m}(G) \), where \(G=(a,b)\subset \mathbb {R} \) is a finite interval, are obtained.
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On the Multiple Spectrum of a Problem for the Bessel Equation of an Integer Order with Squared Spectral Parameter in the Boundary Condition Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 N. Yu. Kapustin
Abstract The problem for the Bessel equation of an integer order with complex physical and spectral parameters in the boundary condition is considered. The spectral parameter enters the boundary condition quadratically. The question of the basis property of the system of eigenfunctions in the case of the appearance of a multiple eigenvalue is studied.
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Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 A. B. Khasanov, Kh. N. Normurodov, U. O. Khudayorov
Abstract The inverse spectral problem method is used to integrate the nonlinear Liouville equation in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution of the nonlinear Liouville equation is introduced. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class
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Some Theoretical Aspects of the Neural Network Approach to Stabilization of Switched Interval Systems Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 A. S. Fursov, Yu. M. Mosolova
Abstract We consider the problem of stabilization of a switched interval linear system with slow switchings that are inaccessible to observation. It is proposed to look for a solution in the class of variable structure controllers. To ensure the functionality of such a controller, it is necessary to construct an observer of the switching signal. This paper is devoted to some theoretical issues related
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Influence of Nonisolated Singularities in a Lower-Order Coefficient of the Bitsadze Equation on the Statement of Boundary Value Problems Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 A. B. Rasulov
Abstract We study the influence of nonisolated singularities (i.e., singularities along closed lines lying inside the domain) in the lower-order coefficients of the Bitsadze equation on the statement of boundary value problems. We discover that the conditions on the boundary of the domain in the Riemann–Hilbert problem are not sufficient for the solution; therefore, we consider a problem that combines
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Existence of Two Solutions of the Inverse Problem for a Mathematical Model of Sorption Dynamics Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 A. M. Denisov, Zhu Dongqin
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One-Dimensional Inverse Problem for Nonlinear Equations of Electrodynamics Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 V. G. Romanov
Abstract For the system of nonlinear electrodynamics equations, we consider the problem of determining the medium conductivity coefficient multiplying the nonlinearity. It is assumed that the permittivity and permeability are constant and the conductivity depends only on one spatial variable \(x \), with this conductivity being zero on the half-line \(x<0 \). For a mode in which only two electromagnetic
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Construction of Integral Representations of Fields in Problems of Diffraction by Penetrable Bodies of Revolution Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 Yu. A. Eremin, V. V. Lopushenko
Abstract Based on integral representations with densities distributed along a segment of the symmetry axis, a representation of the solution of the boundary value problem of plane wave diffraction by a local penetrable body of revolution with smooth surface is constructed and justified. The resulting integral representation allows one to avoid resonances of the interior domain when analyzing the scattering
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Approximation to the Sturm–Liouville Problem with a Discontinuous Nonlinearity Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 D. K. Potapov
Abstract We consider a continuous approximation to the Sturm–Liouville problem with a nonlinearity discontinuous in the phase variable. The approximating problem is obtained from the original one by small perturbations of the spectral parameter and by approximating the nonlinearity by Carathéodory functions. The variational method is used to prove the theorem on the proximity of solutions of the approximating
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On a Control Problem for a System of Implicit Differential Equations Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 E. S. Zhukovskiy, I. D. Serova
Abstract We consider the differential inclusion \(F(t,x,\dot {x})\ni 0 \) with the constraint \(\dot {x}(t)\in B(t) \), \(t\in [a, b]\), on the derivative of the unknown function, where \(F\) and \(B \) are set-valued mappings, \(F:[a,b]\times \mathbb {R}^n\times \mathbb {R}^n\times \mathbb {R }^m\rightrightarrows \mathbb {R}^k \) is superpositionally measurable, and \( B:[a,b]\rightrightarrows \mathbb
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Integral Equations of Volterra Typewith Two Boundary and One Interior Singular Point Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 N. Rajabov, L. N. Rajabova
Abstract Explicit solutions of model and nonmodel integral equations of Volterra type with two boundary and one interior singular point are obtained, and the properties of the resulting solutions are studied. The well-posed statement of problems with conditions specified on singular manifolds is found in the case where the solution of the model equation contains an arbitrary constant.
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On the Existence of an Infinite Spectrum of Damped Leaky TE-Polarized Waves in an Open Inhomogeneous Cylindrical Metal–Dielectric Waveguide Coated with a Graphene Layer Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 Yu. G. Smirnov, E. Yu. Smolkin
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Regularity of the Pressure Function for Weak Solutions of the Nonstationary Navier–Stokes Equations Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 E. V. Amosova
Abstract We study the nonstationary system of Navier–Stokes equations for an incompressible fluid. Based on a regularized problem that takes into account the relaxation of the velocity field into a solenoidal field, the existence of a pressure function almost everywhere in the domain under consideration for solutions in the Hopf class is substantiated. Using the proposed regularization, we prove the
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Fredholm Integral Equation for Problems of Acoustic Scattering by Three-Dimensional Transparent Structures Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 A. B. Samokhin, A. S. Samokhina, I. A. Yurchenkov
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Classical Solution of the Second Mixed Problem for the Telegraph Equation with a Nonlinear Potential Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 V. I. Korzyuk, J. V. Rudzko
Abstract For the telegraph equation with a nonlinear potential, we consider a mixed problem in the first quadrant in which the Cauchy conditions are specified on the spatial semiaxis and the Neumann condition is set on the temporal semiaxis. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integral equations. The solvability of these
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On the Existence of a Solution of a Boundary Value Problem on a Graph for a Nonlinear Equation of the Fourth Order Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 R. Ch. Kulaev, A. A. Urtaeva
Abstract A fourth-order nonlinear differential equation on a network that is a model of a system of Euler–Bernoulli rods is considered. Based on the monotone iteration method, the existence of a solution of a boundary value problem on a graph for this equation is established using the positiveness of the Green’s function and the maximum principle for the corresponding linear differential equation.
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Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 V. E. Kruglov
Abstract A system of third-order recurrence relations for the coefficients of polynomial eigenfunctions (PEFs) of a differential equation is solved. A recurrence relation for three consecutive PEFs and a formula for differentiating PEFs are obtained. We consider differential equations one of which generalizes the Hermite and Laguerre differential equations and the other is a generalization of the Jacobi
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Estimates of Integrally Bounded Solutions of Linear Differential Inequalities Diff. Equat. (IF 0.6) Pub Date : 2023-11-23 V. S. Klimov
Abstract We study integrally bounded solutions of the differential equation \(\mathscr {A}(x)=z \), where \(\mathscr {A} \) is a linear differential operator of order \(l \) defined on functions \(x\colon \mathbb {R} \to H \) (\(\mathbb {R}=(-\infty ,\infty \)) and \(H \) is a finite-dimensional Euclidean space). The right-hand side \(z \) is an integrally bounded function on \(\mathbb {R} \) ranging
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On the Existence of Global Compactly Supported Weak Solutions of the Vlasov–Poisson System with an External Magnetic Field Diff. Equat. (IF 0.6) Pub Date : 2023-11-01
Abstract We consider the first mixed problem for the Vlasov–Poisson system with an external magnetic field in a domain with piecewise smooth boundary. This problem describes the kinetics of a two-component high-temperature plasma under the influence of a self-consistent electric field and an external magnetic field. The existence of global weak solutions is proved. In the case of a cylindrical domain
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On the Monotonicity of Solutions of Nonlinear Systems with Respect to the Initial Conditions Diff. Equat. (IF 0.6) Pub Date : 2023-10-06 L. I. Rodina, M. S. Woldeab
Abstract We consider the autonomous system of differential equations \(\dot x=f(x) \) and the solution \(\varphi (t,x) \) of this system with the initial condition \(\varphi (0,x)=x \). Sufficient conditions for the following monotonicity property of solutions with respect to the initial conditions are obtained: if \(x(0),y(0)\in {\mathbb {R}}^n\) and \( x(0)\leq y(0) \), then \(\varphi (t,x(0))\leq
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Representation of the Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball Diff. Equat. (IF 0.6) Pub Date : 2023-10-06 V. V. Karachik
Abstract We define the elementary solution of the polyharmonic equation, with the help of which an explicit representation of the Green’s function of the Dirichlet problem for the polyharmonic equation in the unit ball is given for all space dimensions except for some finite set. On the basis of the obtained Green’s function, the solution of the homogeneous Dirichlet problem in the unit ball is constructed
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The Problem of Two-Dimensional String Vibrations with a Nonlinear Condition Diff. Equat. (IF 0.6) Pub Date : 2023-10-06 M. B. Zvereva
Abstract We study a model of small spatial transverse vibrations of a string where the deviation of any of its points from the equilibrium is characterized by two coordinates. It is assumed that in the course of vibrations one end of the string is inside a bounded closed convex set \(C \) belonging to a plane \(\pi \) perpendicular to the segment along which the string is stretched. In turn, the set
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Solution of a Singularly Perturbed Mixed Problem on the Half-Line for a Parabolic Equation with a Strong Turning Point of the Limit Operator Diff. Equat. (IF 0.6) Pub Date : 2023-10-06 A. G. Eliseev, T. A. Ratnikova, D. A. Shaposhnikova
Abstract We study singularly perturbed problems in the presence of spectral singularities of the limit operator using S.A. Lomov’s regularization method. In particular, a regularized asymptotic solution is constructed for a singularly perturbed inhomogeneous mixed problem on the half-line for a parabolic equation with a strong turning point of the limit operator. Based on the idea of asymptotic integration
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Existence and Stability of Solutions with Internal Transition Layer for the Reaction–Diffusion–Advection Equation with a KPZ-Nonlinearity Diff. Equat. (IF 0.6) Pub Date : 2023-10-06 N. N. Nefedov, A. O. Orlov
Abstract We study a boundary value problem for a quasilinear reaction–diffusion–advection ordinary differential equation with a KPZ-nonlinearity containing the squared gradient of the unknown function. The noncritical and critical cases of existence of an internal transition layer are considered. An asymptotic approximation to the solution is constructed, and the asymptotics of the transition layer