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卷 65, 编号 1 (2025)
General numerical methods
COLLOCATION-VARIATIONAL APPROACHES TO SOLVE THE VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND NUMERICALLY
摘要
Linear Volterra equations of the first kind are considered. A class of problems that have a single solution is identified, and collocation-variational methods are proposed to solve them numerically. The essence of these algorithms is that the approximate solution is found at the nodes of a uniform grid (the collocation condition) that yield an underdetermined system of linear algebraic equations. The system thus obtained is supplemented by the condition of minimum of the objective function, which approximates the squared norm of the approximate solution. As a result, a quadratic programming problem is obtained, viz. the objective function (the squared norm of the approximate solution) is quadratic, and the constraints (the collocation conditions) are equalities. This problem is solved by the method of Lagrange multipliers. Sufficiently simple third-order methods are considered in detail. The calculation results for test problems are given. Further development of this approach to solve other classes of integral equations numerically is discussed.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):3-9
3-9
Ordinary differential equations
ON SPECTRAL APPROXIMATIONS FOR THE STABILITY ANALYSIS OF BOUNDARY LAYERS
摘要
Approximation of spectral and boundary-value problems arising in the stability analysis of incompressible boundary layers is considered. As an alternative to the collocation method with mappings, the Galerkin–collocation method based on Laguerre functions is adopted. A robust numerical implementation of the latter method is discussed. The methods are compared within the stability analysis of the Blasius and Ekman layers. The Galerkin-collocation method demonstrates an exponential convergence rate for scalar stability characteristics and has a number of advantages.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):10-22
10-22
Partial Differential Equations
THE FIRST INITIAL BOUNDARY VALUE PROBLEM FOR PARABOLIC SYSTEMS IN A SEMI-BOUNDED DOMAIN WITH CURVILINEAR LATERAL BOUNDARY
摘要
The first initial boundary value problem for a second-order parabolic system in a semi-bounded domain on the plane is considered. The coefficients of the system satisfy the double Dini condition. The function defining the lateral boundary of the domain is continuously differentiable on the closed interval. When the right-hand side of the boundary condition of the first kind is continuously differentiable and the initial function is continuous and bounded together with its first and second derivatives, it is established that the solution of the problem is continuous and bounded in the closure of the domain together with its higher order derivatives. The corresponding estimates are proved. An integral representation of the solution is given. If the lateral boundary of the domain has “corners” and the boundary function has a piecewise continuous derivative, it is proved that, despite the lateral boundary and the boundary function being non-smooth, the higher order derivatives of the solution are continuous everywhere in the closure of the domain, except the corner points, and are bounded.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):23-35
23-35
NONLINEAR METHOD OF ANGULAR BOUNDARY FUNCTIONS UNDER THE INFLUENCE OF THE INFLECTION POINT
摘要
In the rectangle Ω = {(x, t) | 0 < x < 1, 0 < t < T} the initial boundary value problem for the singularly perturbed parabolic equation ε2 (︂ a2 ∂2u ∂x2 − ∂u ∂t )︂ = F(u, x, t, ε), (x, t) ∈ Ω, u(x, 0, ε) = φ(x), 0 ≤ x ≤ 1, u(0, t, ε) = ψ1(t), u(1, t, ε) = ψ2(t), 0 ≤ t ≤ T. is considered. It is assumed that at the angular points (k, 0) of the rectangle Ω, where k = 0 or 1, the function F(u) = F(u, k, 0, 0) takes the form F(u) = u3 − u30 , где u0 = u0(k) < 0. The nonlinear method of angular boundary functions is used to construct the asymptotics of the solution to the problem. Earlier, we considered the case when the boundary value of φ at the angular points is separated from the inflection point u = 0 by the condition u0(k) < φ(k) ≤ u0(k) 2 < 0, at which functions of the “simplest” form suitable in the entire domain in question fitted to the role of barrier functions. In this work, the case u0(k) 2 < φ(k) < 0 is considered, where the domain has to be divided into parts, the barrier functions have to be constructed in each subdomain taking into account their continuous junction at the common boundaries of the subdomains, and then the piecewise continuous lower and upper solutions have to be smoothed. As a result, a complete asymptotic expansion of the solution when ε → 0 is obtained and its uniformity in the closed rectangle is justified.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):36-49
36-49
SOLVABILITY THEORY OF SPECIAL INTEGRODIFFERENTIAL EQUATIONS IN THE CLASS OF GENERALIZED FUNCTIONS
摘要
A linear integrodifferential equation with a special differential operator in the principal part is studied. For its approximate solution in the space of generalized functions, a special generalized version of the collocation method is proposed and justified.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):50-61
50-61
STUDYING DYNAMIC PROCESSES IN AN ELASTIC LAYER ON THE SURFACE OF A COMPRESSIBLE FLUID
摘要
An interesting phenomenon discovered during earthquakes occurring in one area of the southern part of Azerbaijan is studied. Taking into account the rare features of this part of the Earth’s crust, the occurring event was modeled in the form of a mathematical problem of the dynamic theory of elasticity, which revealed the cause of the phenomenon involved.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):62-68
62-68
FEYNMAN–KAC FORMULAS FOR SOLUTIONS OF NONSTATIONARY PERTURBED EVOLUTION EQUATIONS
摘要
A bijective mapping of the space of operator-valued functions into the set of complex-valued finite additive cylindrical measures on the space of trajectories is constructed and studied. The conditions under which the Cauchy problem for the first order equation with a variable operator generates a two-parameter evolutionary family of operators are found. A representation of the solution to the Cauchy problem with a variable perturbed generator by means of a continuum integral of the perturbation-defined functional on the trajectory space over a cylindrical pseudomeasure specified by an unperturbed two-parameter evolutionary family of operators is obtained.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):69-87
69-87
Mathematical physics
MODELING UPPER MANTLE CONVECTION IN THE SUBDUCTION ZONE
摘要
A model of upper mantle convection in the subduction zone of a cold lithospheric plate (subduction) into the Earth’s upper strata is developed. The issues of constructing the initial distributions of model variables are discussed. Computational schemes for solving the model equations are given. Calculation of dynamics of mantle convection and reorganization of its structure are performed in the vorticity-current function variables, and dynamics of the plate subduction is calculated on the basis of the smoothed-particle hydrodynamics method (SPH). A series of computational experiments are performed.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):88-96
88-96
SPATIAL OPTIMAL DISTURBANCES OF THREE-DIMENSIONAL AERODYNAMIC BOUNDARY LAYERS
摘要
In the present paper, we propose a numerical method for modeling the downstream propagation of optimal disturbances in compressible boundary layers over three-dimensional aerodynamic configurations. At each integration step, the method projects the numerical solution of governing equations onto an invariant subspace of physically relevant eigenmodes; and the numerical integration is performed along the lines of disturbance propagation. The propagation of optimal disturbances is studied in a wide range of parameters for two configurations: a boundary layer over a swept wing of finite span, and a boundary layer over a prolate spheroid. It is found that the dependence of the disturbance energy amplification on the spanwise wavenumber has two local maxima. It is discussed how to combine the developed method with the modern approaches, which are designed to predict the onset of laminar-turbulent transition using the eN-method.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):97-109
97-109
AN IDEAL-FLUID FLOW THROUGH A NEAR-WALL FIXED GRANULAR LAYER IN THE FORM OF SEMI-INFINITE STEP
摘要
The problem on the flow of an ideal fluid along a flat surface in the presence of a fixed granular layer on it in the form of a semi-infinite step of finite thickness consisting of an infinite number of identical spherical granules statistically uniformly distributed in the layer is considered. The problem is solved based on using the previously developed method of the self-consistent field, which allows studying the effects of hydrodynamic interaction of a large number of spherical particles in flows of an ideal fluid, including in the presence of external boundaries, and obtaining the averaged dynamic characteristics of such flows. In the first approximation in the volume fraction of granules in a layer, an analytical function is obtained that describes the averaged velocity field of the fluid both inside and outside this layer.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):110-119
110-119
Computer science
STABLE MATCHINGS, CHOICE FUNCTIONS, AND LINEAR ORDERS
摘要
A model of stable edge subsets (“matchings”) in a bipartite graph G = (V, E) is considered, in which preferences for vertices of one side (“firms”) are given by choice functions with standard properties of consistency, substitutability, and cardinal monotonicity, and preferences for vertices of the other side (“workers”) are given by linear orders. For such a model,we give a combinatorial description of the structure of rotations and propose an algorithm for constructing a rotation poset with a time complexity estimate O(|E|2) (including calls to oracles associated with choice functions). As a consequence, a “compact” affine representation of stable matchings can be obtained and related problems can be solved efficiently.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(1):120-138
120-138