Ашық рұқсат
Рұқсат берілді
Тек жазылушылар үшін
Том 60, № 6 (2024)
ORDINARY DIFFERENTIAL EQUATIONS
EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO MIXED-TYPE STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY THE FRACTIONAL BROWNIAN MOTIONS WITH HURST INDICES 𝐻 > 1/4
Аннотация
In this paper there is investigated the problem of unique solvability of the Caushy problem for the mixed type stochastic differential equation driven by the standard Brownian motion and fractional Brownian motions with Hurst indices 𝐻 > 1/4. We prove a theorem on the existence and uniqueness of strong solutions to the mentioned mixed type stochastic differential equations.
Differencial'nye uravneniya. 2024;60(6):723-735
723-735
SPECIFIED GLOBAL POINCARE–BENDIXSON ANNULUS WITH THE LIMIT CYCLE OF THE RAYLEIGH SYSTEM
Аннотация
In the work of A. Grin and K. Schneider [1] two algebraic transversal ovals, which form the Poincare–Bendixson annulus 𝐴(𝜆), are analytically constructed. This annulus contains the unique limit cycle of the Rayleigh equation ¨ 𝑥+𝜆( ˙ 𝑥2/3−1) ˙ 𝑥+𝑥 = 0 for all values of the parameter 𝜆 > 0. In the constructed annulus 𝐴(𝜆) inner boundary consists of zero-level set of the Dulac–Cherkas function and outer boundary represents the diffeomorphic image corresponding boundary of such an annulus for the van der Pol system. In this paper new methods of construction two Dulac–Cherkas functions are worked out. With help of these functions a better inner boundary of the Poincare–Bendixson annulus 𝐴(𝜆) depending on the parameter is found. Also, for the Rayleigh system, a procedure for direct finding a polynomial whose zero-level set contains a transversal oval used as the outer boundary 𝐴(𝜆) is proposed. Then we determine the interval for 𝜆 where the better outer boundary of the annulus is a closed contour, which composed from two arcs of the constructed oval and two arcs of non-closed curves belonging to zerolevel set of one from Dulac–Cherkas functions. Thus, finally the specified global Poincare–Bendixson annulus for the limit cycle of the Rayleigh system is presented.
Differencial'nye uravneniya. 2024;60(6):736-746
736-746
PARTIAL DERIVATIVE EQUATIONS
GROUP ANALYSES, REDUCTIONS AND EXACT SOLUTIONS OF MONGE–AMPERE EQUATION OF MAGNETIC HYDRODYNAMICS
Аннотация
We study the Monge–Amp`ere equation with three independent variables which occurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partial derivative equation is carried out. An eleven-parameter transformation preserving the form of the equation is found. A formula is obtained that makes it possible to construct multiparametric families of solutions based on simpler solutions. Two-dimensional reductions leading to simpler partial differential equations with two independent variables. One-dimensional reductions are described, which make it possible to obtain self-similar and other invariant solutions that satisfy ordinary differential equations. Exact solutions with additive, multiplicative and generalized separation of variables are constructed, many of which admit representation in elementary functions. The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by strongly nonlinear partial differential equations.
Differencial'nye uravneniya. 2024;60(6):750-763
750-763
EXISTENCE OF A RENORMALIZED SOLUTION OF A QUASI-LINEAR ELLIPTIC EQUATION WITHOUT THE SIGN CONDITION ON THE LOWEST TERM
Аннотация
The paper considers a second-order quasilinear elliptic equation with an integrable right-hand side. Restrictions on the structure of the equation are formulated in terms of the generalized 𝑁-function. Unlike the author’s previous works, there is no sign condition for the low-order term of the equation. In non-reflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain, the existence of a renormalized solution to the Dirichlet problem of this equation is proven.
Differencial'nye uravneniya. 2024;60(6):764-785
764-785
CONTROL THEORY
THE EXISTENCE OF OPTIMAL SETS FOR LINEAR VARIATIONAL EQUATIONS AND INEQUALITIES
Аннотация
This paper considers an optimal control problem in which the controlled process is described by a linear functional equation in a Hilbert space, and the control action is a change of space. Sufficient conditions for the existence of a solution are obtained. The results are generalized to the case when the controlled process is described by a linear variational inequality.
Differencial'nye uravneniya. 2024;60(6):786-797
786-797
SOLUTION OF THE SPECTRUM ALLOCATION PROBLEM FOR A LINEAR CONTROL SYSTEM WITH CLOSED FEEDBACK
Аннотация
The method for constructing a feedback matrix to solve the problem spectrum allocation (spectrum control, pole assignment) for linear dynamic system is given. A new proof of the well-known theorem about the connection between the complete controllability of a dynamic system with existence of the feedback matrix is formed in the process of construction by the cascade decomposition method. The entire set of arbitrary elements that influence the non-uniqueness of the matrix is identified. Examples of constructing a feedback matrix in the case of a real spectrum and in the presence of complex conjugate eigenvalues, as well as for the case of multiple eigenvalues, are given. The stability of the specified spectrum under small perturbations of the system parameters with a fixed feedback matrix is studied.
Differencial'nye uravneniya. 2024;60(6):798-816
798-816
NUMERICAL METHODS
EXPLICIT-IMPLICIT SCHEMES FOR CALCULATING DYNAMICS OF ELASTOVISCOPLASTIC MEDIA WITH SOFTENING
Аннотация
The paper examines the dynamic behavior of elastoviscoplastic media under the action of an external load. For the case of a linear viscosity function and a nonlinear softening function, an explicit-implicit calculation scheme has been constructed that makes it possible to obtain a numerical solution to the original semilinear hyperbolic problem. This approach does not involve the use of the method of splitting into physical processes. Despite this, an explicit computational algorithm was obtained that can be effectively implemented on modern computing systems.
Differencial'nye uravneniya. 2024;60(6):817-829
817-829
USING OPERATOR INEQUALITIES IN STABILITY RESEARCH OF DIFFERENCE SCHEMES FOR NONLINEAR BOUNDARY PROBLEMS WITH NONLINEARITY OF UNLIMITED GROWTH
Аннотация
The article develops the theory of stability of linear operator schemes for operator inequalities and nonlinear nonstationary initial-boundary value problems of mathematical physics with nonlinearities of unlimited growth. Based on sufficient conditions for the stability of A.A. Samarsky’s two-layer and three-layer difference schemes, the corresponding a priori estimates for operator inequalities are obtained under the condition of the criticality of the difference schemes under consideration, i.e. when the difference solution and its first time derivative are non-negative at all nodes of the grid region. The results obtained are applied to the analysis of the stability of difference schemes that approximate the Fisher and Klein-Gordon equation with a nonlinear right-hand side.
Differencial'nye uravneniya. 2024;60(6):830-843
830-843
CHRONICLE
844-864