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Vol 61, No 9 (2025)
PARTIAL DERIVATIVE EQUATIONS
CLASSICAL SOLUTION OF THE FIRST MIXED PROBLEM FOR THE INHOMOGENEOUS LIOUVILLE EQUATION IN A HALF-STRIP
Abstract
The question of existence and uniqueness of the classical solution of the mixed problem in a half-strip for the nonlinear Liouville equation is investigated. The solution is presented in implicit form as a solution of an integral equation, the solvability of which is proved using the Leray-Schauder theorem, and the corresponding a priori estimate is obtained using energy methods.
Differential equations. 2025;61(9):1155–1166
1155–1166
WEIGHTED HARDY-LITTLEWOOD SPACES IN PIECEWISE SMOOTH DOMAINS
Abstract
New weighted spaces of the Hardy–Littlewood type of continuous functions defined in domains with piecewise smooth boundaries are introduced, and their structural properties are described. It is established that these spaces contain functions representable by generalized integrals of the Cauchy type with homogeneous kernels of degree –1. The latter circumstance opens up the possibility of using these spaces for solving elliptic systems of the first and second orders with continuous coefficients.
Differential equations. 2025;61(9):1167–1182
1167–1182
INTEGRAL EQUATIONS
ON ELLIPTIC PROBLEMS AND INTEGRAL EQUATIONS
Abstract
A model pseudo-differential equation in Sobolev–Slobodetskii space is considered in a cone which is a direct product of low dimensional cones. Under existence of a special factorization for symbol a general solution can be written, it includes arbitrary functions from certain Sobolev–Slobodetskii space. A certain example in three-dimensional space is considered, the unknown functions can be determined with help of Dirichlet conditions on a piece of a boundary by reducing to a system of linear integral equations.
Differential equations. 2025;61(9):1183-1194
1183-1194
SOLVABILITY OF A SYSTEM OF NONLINEAR INTEGRAL EQUATIONS WITH PIECEWISE CONSTANT KERNELS
Abstract
The model of population dynamics of a single-species biological community proposed by W. Dieckmann and R. Law is considered. Changes in the population are described by a system of integro-differential equations, which characterizes the dynamics of spatial moments and in the state of equilibrium is reduced to a nonlinear integral equation. The solvability of this equation is studied, according to which the solution of the original system is written out, for which a nonlinear integral operator is constructed and the problem of finding its fixed point is solved. Sufficient conditions for the existence of a nontrivial solution are established. An analytical example of the values of biological parameters that satisfy these conditions is given.
Differential equations. 2025;61(9):1195-1206
1195-1206
BOUNDARY VALUE PROBLEM OF TWO-DIMENSIONAL FILTRATION OF A LIQUID IN AN INHOMOGENEOUS LAYER WITH A BOUNDARY CONDITION OF DISCONTINUITY OF THE FIRST KIND
Abstract
The first boundary value problems (internal and external) of a two-dimensional filtration flow of a liquid in a non-uniform porous layer of variable thickness and permeability are investigated. The given discrete sources of the flow are located in the flow region and are modeled by singularities (isolated special points) of a complex potential. The boundary of the flow region is modeled by an arbitrary smooth (piecewise smooth) closed contour. A function is specified on the boundary that characterizes the pressure distribution on it and has discontinuities of the first kind. A method for regularizing (smoothing) the boundary condition is proposed, which allows reducing the problems to a boundary singular integral equation with a weak singularity and a smooth right-hand side. This regularization method is applied to the solution of a boundary value problem, simulating the operation of a system of wells in a heterogeneous layer (formation) of soil, on the supply contour of which a given pressure distribution (generalized potential) suffers discontinuities of the first kind.
Differential equations. 2025;61(9):1207-1217
1207-1217
1218–1231
ON THE EXISTENCE OF SOLUTIONS OF TWO-DIMENSIONAL HYPERSINGULAR INTEGRAL EQUATION IN THE CLASS OF FUNCTIONS WITH A SINGULARITY ON THE BOUNDARY OF A DOMAIN
Abstract
A two-dimensional hypersingular integral equation in a convex bounded domain whose boundary is a smooth curve is considered. The equation contains an integral operator with an integral understood in the sense of a finite part according to Hadamard. The question of the existence of solutions having a power singularity in the neighborhood of the boundary of the domain is considered: the solution is sought in a class of functions represented as the ratio of a smooth function and the root of the distance from a point to the boundary. It is proved that when an integral operator with a power polar singularity of the third order acts on a function from the class in which the solution is sought, a function arises that is continuous in the sense of Helder on the entire domain. Further, the existence of solutions of the hypersingular equation having the specified power singularity in the neighborhood of the boundary of the domain is proved. A boundary condition is presented under which such a solution is unique.
Differential equations. 2025;61(9):1232-1253
1232-1253
ON THE FREDHOLM PROPERTY OF A HYPERSINGULAR INTEGRAL OPERATOR IN A SCATTERING PROBLEM OF ELECTROMAGNETIC WAVES ON A SLAB COVERED WITH GRAPHENE
Abstract
The paper focuses on a boundary-value problem for a system of two homogeneous Helmholtz equations with mixed boundary conditions arising in the theory of electromagnetic waves scattering on structures covered with two-dimensional materials. The problem is reduced to a boundary integral equation on the whole line involving a hypersingular integral operator. Introducing new variables and sought-for function, one passes to the integral equation on segment. To find an approximate solution of the obtained equation the collocation method using Fourier-Chebyshev series to present a solution is suggested; hypersingular integrals are calculated analytically. In the main part of the paper we prove the Fredholm property of the hypersingular operator involved in the integral equation under consideration.
Differential equations. 2025;61(9):1254–1271
1254–1271
NUMERICAL METHODS
LOCAL COMPUTATIONAL ALGORITHMS FOR THE SYSTEM OF FIRST-ORDER EQUATIONS WITH MEMORY EFFECTS
Abstract
We consider the Cauchy problem for a system of integro-differential equations of the first order with difference kernels in finite-dimensional Hilbert spaces. This class of equations arises in the mathematical modeling of a wide range of nonstationary processes taking into account memory effects, including, in particular, the system of Maxwell's equations. For numerical solution, a method of reducing the original nonlocal problem to an equivalent system of local differential equations of the first order on the basis of approximation of kernels by a finite sum of exponential functions is proposed. Two-level operator-difference schemes are proposed, for which the stability of the initial data and the right-hand side is analyzed. The performed theoretical analysis demonstrates the correctness of the proposed approach.
Differential equations. 2025;61(9):1272–1285
1272–1285
NUMERICAL METHOD FOR SOLVING DIFFRACTION PROBLEMS FOR PAIRED NANOPARTICLES TAKING INTO ACCOUNT QUANTUM EFFECTS
Abstract
A numerical method for solving the boundary value problem of diffraction on a system of paired nanoparticles with a subnanometer gap has been developed and implemented. The boundary value problem of diffraction includes a system of Maxwell's equations and mesoscopic boundary conditions with Feibelman parameters. The solution to the problem is constructed using the discrete source method with the location of sources for the internal field in the complex plane. A complete mathematical justification of the method is provided. For a pair of gold nanoparticles, a numerical analysis of the influence of quantum effects on the field intensity in a subnanometer gap is performed. It is found that taking into account quantum effects has a significant influence on the field characteristics. In particular, a decrease in the plasmon resonance amplitude can reach 65%, and its position shift to the long-wave region reaches 25 nm.
Differential equations. 2025;61(9):1286–1296
1286–1296