Vol 60, No 9 (2024)
ORDINARY DIFFERENTIAL EQUATIONS
STABILITY CHANGE OF INVARIANT MANIFOLDS OF DIFFERENTIAL SYSTEMS WITH MULTI-SCALE VARIABLES
Abstract
The paper considers invariant manifolds with a change in stability of differential systems with differentscale variables. The interest in such manifolds is explained by their widespread use in applied problems. The questions of the existence of continuous invariant manifolds with a change in stability are investigated in three critical cases.
Differential Equations. 2024;60(9):1155–1166
1155–1166
INTEGRAL EQUATIONS
EXISTENCE AND UNIQUENESS OF SOLUTIONS OF NONLINEAR FUNCTIONAL INTEGRAL ITOˆ EQUATIONS
Abstract
A new class of Ito^ integral equations is considered, which contains many classical problems, for example, the Cauchy problem for differential equations of integer and fractional order with and without stochastic perturbations, as well as some less known and little-studied types of equations that have been introduced recently. The purpose of the study is to find sufficiently general conditions that guarantee the existence and the uniqueness of solutions to such equations, taking into account their specific features. The article therefore proposes to use a special generalized Lipschitz condition, which, due to its flexibility, allows one to obtain effective solvability criteria in terms of the right-hand sides of equations. Numerous examples are considered, covering in particular Ito^ differential equations of fractional order with aftereffect and without aftereffect, equations with fractional Wiener processes, Ito^ equations with several time scales, as well as their generalizations.
Differential Equations. 2024;60(9):1167–1189
1167–1189
VOLUME SINGULAR INTEGRAL EQUATIONS FOR PROBLEMS OF LOW-FREQUENCY SCATTERING OF ELECTROMAGNETIC WAVES IN ANISOTROPIC STRUCTURES
Abstract
This paper deals with volume singular integral equations describing the problems of low-frequency scattering of electromagnetic waves in bounded three-dimensional anisotropic structures. The spectrum of integral operators is studied. The domain of the operator spectrum on the complex plane for the low-frequency case is presented explicitly. The generalized method of simple iteration is described, for application of which it is necessary to know the area of the operator spectrum on the complex plane. The collocation method on a uniform grid is used to discretize the integral equations. This allows, using a fast discrete Fourier transform, to construct an algorithm for fast multiplication of the matrix of a system of linear equations by a vector. The results of numerical solution of the considered problems are given.
Differential Equations. 2024;60(9):1190–1204
1190–1204
INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
ON THE EXISTENCE OF EQUILIBRIUM IN THE DIECKMANN–LAW’S MODEL IN THE CASE OF THE PIECEWISE CONSTANT KERNELS
Abstract
The logistic dynamics model developed by U. Dieckmann and R. Law is considered in the article. The analysis of a nonlinear integral equation describing the equilibrium state of a single-species community with a three-parameter closure of the third spatial moment is carried out in the case when the dispersion and competition kernels are piecewise constant functions. Sufficient conditions for the mentioned equation solvability are established.
Differential Equations. 2024;60(9):1205–1215
1205–1215
INTEGRO-DIFFERENTIAL EQUATIONS IN THE PROBLEM OF ELECTROMAGNETIC WAVE SCATTERING ON A DIELECTRIC BODY COVERED WITH GRAPHENE
Abstract
We consider the determination of resonance frequencies of dielectric bodies coated with graphene. In the addressed problem statement, the graphene nonlinearity is not taken into account. The initial boundary-value problem for Maxwell’s equations is reduced to a system of integro-differential equations on the graphene surface. We prove the Fredholm property of this system under certain sufficient conditions and establish the discreteness of the spectrum of an operator-valued function corresponding to this system in a certain region of the complex plane of the circular frequency spectral parameter.
Differential Equations. 2024;60(9):1216–1224
1216–1224
NUMERICAL METHODS
CONSTRUCTION OF THE TRANSFER FUNCTION OF THE POINCARE´–STEKLOV OPERATOR FOR A COATED ELASTIC HALF-PLANE
Abstract
For a homogeneous isotropic elastic half-plane with a stratified elastic coating we consider the Poincar´e– Steklov operator that maps normal stresses into normal displacements on part of the boundary. To construct the transfer function of this operator, the variational formulation of the boundary value problem for transforms of displacements is used. A definition is given and the existence and uniqueness are proved for a generalized solution of the variational problem. This problem is approximated by the finite element method. To numerically solve the resulting system of linear algebraic equations, the preconditioned conjugate gradient method is used. The developed computational algorithm was verified.
Differential Equations. 2024;60(9):1225–1240
1225–1240
SOLVING OF ONE-DIMENSIONAL HYPERSINGULAR INTEGRAL EQUATION USING HAAR’S WAVELETS
Abstract
We constructed a numerical method for the one-dimensional hypersingular integral equation which uses sparse matrix approximations. This method has the same convergence order as conventional methods for hypersingular integral equations but the new method is more effective in both memory and arithmetic operations.
Differential Equations. 2024;60(9):1241–1260
1241–1260
A TWO-POINT COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF ONE-DIMENSIONAL HYPERSINGULAR INTEGRAL EQUATIONS ON NONUNIFORM PARTITIONS
Abstract
A quadrature formula has been constructed for calculating the hypersingular integral over a segment, which uses the ends of the segment partition intervals as nodes of piecewise constant interpolation of the integral density, as well as specially selected collocation points. A distinctive feature of the proposed quadrature formula is the ability to calculate the integral of functions that suffer a finite number of discontinuities of the first kind on the integration interval. On the basis of quadrature formula constructed, a numerical scheme for solving the characteristic hypersingular integral equation on non-regular grid is developed. Estimate of the rate of convergence of approximate solutions to exact ones is proved in the class of piecewise Ho¨lder functions.
Differential Equations. 2024;60(9):1261–1275
1261–1275
CONVERGENCE OF THE METHOD OF PIECEWISE LINEAR APPROXIMATIONS AND COLLOCATIONS FOR A TWO-DIMENSIONAL HYPERSINGULAR INTEGRAL EQUATION ON A SET WITH BOUNDARY
Abstract
A hypersingular integral equation on a convex bounded set on the plane with an integral understood in the sense of a finite part in the sense of Hadamard is considered. Equations of this type, in particular, arise when solving the Neumann boundary value problem for the Lapalse and Helmholtz equations on a flat screen in the case where the solution is sought in the form of a double layer potential. To numerically solve the equation, a numerical scheme is used based on piecewise linear approximation of the unknown function on a triangular conformal mesh and the collocation method. The uniform convergence of numerical solutions to an exact solution on a grid when the maximum cell diameter tends to zero has been proven.
Differential Equations. 2024;60(9):1276–1296
1276–1296