Abstract For the boundary value problem of diffraction of an external disturbance by a local penetrable body with smooth surface, we construct an integral representation of the solution based on a linear combination of single and double layer potentials with densities distributed over a common auxiliary interior surface. A complete mathematical justification of this representation is given.

Abstract We consider a nonlinear Sobolev equation of the third order describing the electric field potential in a conductor. Six classes of exact solutions are constructed, and their qualitative behavior is analyzed.

Abstract We consider the three-dimensional problem of diffraction of a monochromatic electromagnetic wave by a system of objects of various physical nature, including dielectric bodies (domains), perfectly conducting bodies, and perfectly conducting screens. The perfectly conducting bodies may be placed in an exterior medium or immersed into the dielectric domains. In addition, the perfectly conducting

Abstract We consider a generalized Richards equation with power-law nonlinearities modeling filtration in porous media. Conditions are derived under which the problem can be reduced to the linear heat equation or to nonlinear equations with known solutions. The families of explicit exact solutions that can be expressed via elementary functions or Lambert’s \(W \)-function are found. Some examples illustrating

Abstract We study singular modes of volume singular integral equations describing problems of scattering of electromagnetic wave in anisotropic dielectric structures. An explicit form of these modes is obtained for a certain class of anisotropic media. Example of real media in which singular modes can exist are considered.

Abstract We consider the problem of diffraction of a waveguide wave by an impedance rod in a rectangular waveguide with perfectly conducting walls and the problem of diffraction of a plane two-dimensional electromagnetic wave and the field of a point source by an evenly spaced array formed by infinite cylinders of arbitrary cross-section with perfectly and well conducting walls. Both problems are reduced

Abstract We consider the problem on the analytic continuation of the solution of the system of vibration equations in elasticity theory in a spatial domain based on the values of the solution and the stresses on part of the boundary of this domain, i.e., a Cauchy problem. The problem is ill posed. If the part of the domain on which the Cauchy data are given is real analytic, then the problem has a

Abstract We study inverse problems of magnetostatics that arise when designing two-dimensional multilayered shielding and cloaking devices. It is assumed that the device to be designed has the form of a circular shell consisting of finitely many layers, each filled with a homogeneous isotropic medium. Using an optimization method, the inverse problems under consideration are reduced to finite-dimensional

Abstract A quadrature formula for the direct value of the normal derivative of the single layer potential with smooth density defined on a closed or nonclosed surface is obtained. Single layer potentials for the Laplace and Helmholtz equations are considered. Numerical tests confirm that our quadrature formula gives considerably higher accuracy than the standard quadrature formula. It can be used when

Abstract We state and study the first and second boundary value problems and the transmission problem for complex potentials of plane filtration flows in porous media. In the general case, the flow sources are arbitrary discrete sources that can be located both on the boundaries and outside the medium boundaries. If the boundaries are canonical (modeled by a straight line or a circle), then the solutions

Abstract We consider a system of nonlinear integral equations arising when studying the inverse coefficient problem for a system of partial differential equations. The inverse problem is to determine two coefficients of the system based on additional information on one of the solution components. We prove the local existence and uniqueness of the solution of the nonlinear system of integral equations

Abstract We study a linear integro-differential equation of the third kind with a coefficient having power-order zeros. To solve this equation approximately in the space of generalized functions, we propose and justify a generalized version of the subdomain method based on special Kantorovich polynomials.

Abstract We study abstract integro-differential equations that are operator models of problems in viscoelasticity. We present results based on an approach related to the study of one-parameter semigroups for linear evolution equations. The presented approach can also be used to study other integro-differential equations containing integral terms of the Volterra convolution form. A method is given for

Abstract For a generalized Cauchy–Riemann equation on the plane with a coefficient that has finitely many singular points and with a continuous free term, we find an integral representation of the solution in the class of functions bounded at infinity.

Abstract We consider linear completely integrable Pfaffian systems of differential equations with matrices whose entries are analytic functions. The property of complete controllability of the system in a neighborhood of a regular point is studied. A sufficient condition for a Pfaffian system to have this property is derived. Whether the system has the complete controllability property is determined

Abstract We study abstract integro-differential equations serving as operator models for problems arising in viscoelasticity. The results are based on an approach related to the study of one-parameter semigroups for linear evolution equations. The approach proposed can be used when investigating other integro-differential equations containing integral terms of the form of Volterra convolution.

Abstract For the solutions of the linear differential inequality \(\mathscr {L}(u)\geq 0 \), where \(\mathscr {L} \) is a linear differential operator of order \(l \) defined on functions of one variable, we establish estimates of the form \(\|u; W^l (J^\delta )\|\leq C(\delta )\|u; L(J)\| \), where \(J=[a,b]\subset \mathbb {R} \), \(0<3\delta

Abstract We consider the leaky wave problem for an open inhomogeneous metal–dielectric circular-section waveguide. The problem is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The inhomogeneity of the dielectric filling and the fact that the spectral parameter occurs in the matching conditions necessitate defining the solution of

Abstract We consider a linear homogeneous \(n\)th-order differential equation, \(n\in \mathbb {N}\), with constant bounded operator coefficients in a Banach space. Under some conditions on the (real and complex) roots of the corresponding characteristic equation, we obtain a formula expressing the general solution of the differential equation via the operator functions given by the exponential, sine

Abstract We study the basis properties of root functions of a spectral problem describing the bending vibrations of a homogeneous rod with a longitudinal force acting in its cross sections. Both rod ends are elastically fixed and either there is a lumped mass or a follower force acts on each of the ends. We establish a sufficient condition for the basis property of the system of root functions of this

Abstract We consider the Cauchy problem for general linear systems of complex partial differential equations in scales of Banach spaces of vector functions of the exponential type with an integral metric. Necessary and sufficient conditions for the well-posed solvability of this problem are obtained. Thus, we describe the structure of linear systems of complex partial differential equations for which

Abstract We consider the well-known Bazykin–Svirezhev model describing the predator–prey interaction. This model is a system of two nonlinear ordinary differential equations with a small parameter multiplying one of the derivatives. The existence and stability of a so-called relaxation cycle in such a system are studied. A peculiar feature of such a cycle is that as the small parameter tends to zero

Abstract For the Gellerstedt equation with a singular coefficient, we prove theorems on the uniqueness and existence of a solution of the problem with local and nonlocal conditions on parts of the boundary characteristic and with discontinuous transmission conditions on the degeneration line.

Abstract We study the existence of oscillatory modes in a controlled nonlinear system closed by a feedback in the form of a bistable hysteretic switch. The system is a simplified mathematical model of a thermal energy harvester. Conditions on the controller and on the system parameters are established that guarantee the existence of oscillatory modes in this system.

Abstract We study the problem of electromagnetic wave diffraction on a junction of two rectangular waveguides. The boundary value problem for the system of Maxwell equations is reduced to a vector pseudodifferential equation in special Sobolev spaces. Sufficient conditions are obtained for the existence of a unique solution of the boundary value problem and the integro-differential equation.

Abstract We consider the analytic properties of solutions to equations of arbitrary order in the generalized hierarchy of the second Painlevé equation. The local properties of solutions, the Bäcklund transformations, and rational solutions and their representations via generalized Yablonskii–Vorob’ev polynomials are studied.

Abstract For the solutions of a nonlinear second-order differential-operator equation and the corresponding three-level operator-difference scheme, we establish sufficient boundedness conditions as well as sufficient conditions for the stability with respect to perturbations of the operators. These results are applied to studying the coefficient stability of a mixed problem for a hyperbolic equation

Abstract We propose methods for solving an exterior boundary value problem for the Laplace equation based on the main integral Green formula. The main technique is the one of setting an artificial integral boundary condition with iterative improvement. It is shown that iterative methods converge at the rate of a geometric progression. The applicability of the methods for solving exterior problems is

Abstract It has recently been established that all fundamental solutions of a multidimensional singular elliptic equation can be expressed via the well-known multivariate Lauricella hypergeometric function. In the present paper, we prove that the generalized Holmgren problem for an elliptic equation with several singular coefficients has a unique solution and find this solution in closed form. When

Abstract This work belongs in the field of mathematical modeling and numerical studies of equilibrium configurations of plasma, magnetic field, and electric current in Galathea traps with current-carrying conductors immersed in a plasma volume. Models of permissible configurations in the annular neighborhood of a straight conductor that are not in contact with its surface are constructed and investigated

Abstract We consider an adaptive interpolation algorithm for numerical integration of systems of ordinary differential equations (ODEs) with interval parameters and initial conditions. At each time moment, a piecewise polynomial function of a prescribed degree is constructed in the course of algorithm operation that interpolates the dependence of the solution on particular values of interval uncertainties

Abstract For a system of partial differential equations, we consider the inverse problem of determining one of the coefficients based on additional information on one of the solution components. An iterative method is proposed for calculating the unknown coefficient based on the reduction of the inverse problem to a nonlinear operator equation. The convergence of the iterative method and the existence

Abstract We consider a quasi-hydrodynamic regularized model of the phase field type that describes the dynamics of an isothermal compressible two-phase viscous mixture with allowance for interphase effects. The dissipativity of the model, i.e., the lack of growth in the complete energy of the closed system as it tends to an equilibrium state, is discussed. A spatially discrete approximation to the

Abstract We propose an approach to modeling nonlinear membranes deformation by the method of hyperelastic nodal forces. The method is based on interpolation properties of barycentric coordinates and the principle of minimum potential energy; this permits one to derive all necessary formulas in an analytical, concise representation. To verify the approach, we run a number of numerical benchmark tests

Abstract We study periodic boundary value problems for a second-order linear partial differential equation with constant coefficients in a half-plane under various assumptions about the roots of the characteristic equation. If the imaginary parts of these roots are of the same sign, then we consider a boundary value problem of the type of the Hilbert problem, and if they are of opposite signs, then

Abstract We consider the problem of mathematical modeling of the current distribution in Josephson structures based on semiclassical equations of the microscopic theory of superconductivity (the Usadel equations). These equations are a system of quasilinear elliptic equations for Green’s functions \(\Phi _\omega (r)\) and \(G_\omega (r) \) and the pairing potential \(\Delta (r) \), which is determined

Abstract We consider a multidimensional singularly perturbed stationary diffusion model with a cubic nonlinearity. For models of this type, a modified asymptotic method of boundary functions, which extends the classical asymptotic analysis methods to the case of multidimensional problems, and the asymptotic method of differential inequalities, which is based on the comparison principle, are used to

Abstract We consider a multidimensional hyperbolic quasi-gasdynamic system of differential equations of the second order in time and space linearized on a constant solution (with an arbitrary velocity). For the linearized system with constant coefficients, we study the implicit three-level weighted and two-level vector difference schemes. The important domination property of the operator of viscous

Abstract Computational practice widely employs inhomogeneous time approximations in which one part of the problem operator is taken from the lower time level and the other, from the upper one. We discuss questions of constructing explicit–implicit schemes for the approximate solution of the Cauchy problem for a first-order evolution equation based on additive two-component splitting of the main problem

Abstract We study the homogeneous Dirichlet problem for a second-order elliptic equation with a nonlinearity discontinuous in the state variable in the resonance case. A class of resonance problems that does not overlap with the previously investigated class of strongly resonance problems is singled out. Using the variational method, we establish a theorem on the existence of at least three nontrivial

Abstract Based on earlier-established necessary nonlocal conditions that must be satisfied by all regular solutions of a fractional diffusion–wave equation in a rectangular domain, we propose a method for determining the well-posedness of nonlocal boundary value problems for this equation and a method for finding their solutions.

Abstract For an integro-differential equation of the convolution type defined on the half-line \([0,\infty ) \) with a power nonlinearity and an inhomogeneity in the linear part, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space \(C[0,\infty ) \). It is shown that the solution can be found by a

Abstract Bitsadze–Samarskii boundary conditions are obtained for a regular elliptic-parabolic volume potential given in a cylindrical domain based on the spectral decomposition of Green’s function of the Holmgren problem.

Abstract The notions of semilalgebra and its associated subalgebra are introduced. Linear spaces of row-column, skew-angle, and acute-angle matrices that are subalgebras or semialgebras of the full matrix algebra are also defined. Conditions for the splittability and solvability by quadratures of a matrix linear ordinary differential equation of the first order with two-sided multiplication \(dX/dt=\sum

Abstract In the Cartesian product \(\mathbb {R}^n\times (0,+\infty ) \), we consider an integro-differential heat equation with an integral term of the convolution type on the right-hand side. The direct problem is the Cauchy problem about determining the temperature of a medium given a known initial heat distribution (for the zero value of the time variable \(t \)). The inverse problem consists in

Abstract We consider an operator semigroup arising in the study of an integro-differential equation modeling small transverse vibrations of a viscoelastic pipeline with allowance for Kelvin–Voigt damping. The explicit form of the generator of this semigroup is established, its spectrum is studied, and the norm of its resolvent is estimated. Based on these results, the analyticity of this semigroup

Abstract For the beam vibration equation, we study the inverse problems of finding the right-hand side (the source of vibrations) and the initial conditions. The solutions of the problems are constructed in closed form as sums of series, and the corresponding existence and uniqueness theorems are proved. The problem of small denominators arises when justifying the convergence of the series. In this

Abstract In the Cartesian product \((\mathbb {R}^n\setminus B)\times \mathbb {R} \), where \(B \) is a ball in \(\mathbb {R}^n \), \(n\geq 3\), we consider a system of two semilinear parabolic equations with bounded measurable time-periodic coefficients and with power nonlinearities. Exact conditions on the nonlinearity exponents under which this system does not have global positive time-periodic solutions

Abstract Boundary value problems for a second-order ordinary differential equation with a nonsmooth potential are considered. An asymptotic representation for an operator semigroup is established based on a previously derived theorem on the eigenvalue asymptotics for the Hill operator. This semigroup is used to describe weak solutions of a mixed problem for a parabolic equation.

Abstract For a two-dimensional hyperbolic equation with a differential-difference operator acting with respect to the spatial variable, we construct a one-parameter family of global solutions. We prove that these solutions are classical for all values of the real parameter provided that the real part of the symbol of the difference operator occurring in the equation is positive. We present a class

Abstract We consider expressions \(\mathcal {L}_k[y]\) that are ordered products of \(k\), \(k\in \mathbb {N} \), linear symmetric quasidifferential expressions of the second order with matrix coefficients. Asymptotic formulas as \(x\to +\infty \) are derived for one of the fundamental solution systems of the equation \(\mathcal {L}_k[y]=\lambda y \), \(\lambda \in \mathbb {C} \), \(\lambda \ne 0\)

Abstract We propose a new approach to studying the global asymptotic stability of an equilibrium position of a system of ordinary differential equations with respect to part of the variables. New functions, different from Lyapunov ones, are used to study state trajectories of systems of ordinary differential equations and prove the global stability theorem. An application of the theorem to proving

Abstract Under appropriate conditions, we prove that for two arbitrary solutions \(s_1 \) and \(s_2 \) of the Dirichlet problem lying between lower and upper functions there exists a continuous mapping \(z \) of the interval \([0,1] \) into the solution set of the Dirichlet problem such that \(z(0)=s_1 \) and \(z(1)=s_2 \).

Abstract We study the Lyapunov asymptotic stability of the stationary solution of the spatially one-dimensional initial–boundary value problem for a nonlinear singularly perturbed differential equation of the reaction–diffusion–advection (RDA) type in the case where the advection and reaction terms undergo a discontinuity of the first kind at some interior point of an interval. Sufficient conditions

Abstract We study the Dirichlet problem for a nonlocal integro-functional-differential equation of the composite type. Theorems on the existence and uniqueness of a twice continuously differentiable solution are proved.

Abstract We consider an initial–boundary value problem for the beam vibration equation, which is a fourth-order nonlinear equation with two independent variables. It is shown that under certain conditions on the initial data this problem can be reduced to the Cauchy problem for a countable system of quasilinear ordinary differential equations. Using the method of energy inequalities, we prove that

Abstract We study the Bessel and basis properties of root vector functions of a Dirac type operator with an integrable potential. Criteria for the Bessel and basis properties are established, and a theorem on the equivalent basis property in \(L_{2}^{2m}(0,2\pi )\) is proved.

Abstract We construct a fundamental solution for an ordinary differential equation with a distributed-order differentiation operator. The properties of the solution are studied. A Lagrange formula is obtained for distributed-order differential operators. A well-posed form of the Cauchy problem is found for the equation in question, and the solution of the problem is written using the Lagrange formula

Abstract A boundary control problem is studied for a model loaded equation of the hyperbolic type. Necessary and sufficient conditions on the initial and terminal functions ensuring the existence of boundary controls are established. Under these conditions, an explicit analytical form of the desired controls is found.

Abstract For one-dimensional non-selfadjoint Schrödinger and Dirac operators with periodic complex-valued potentials belonging to the class \(L_2 \), asymptotic representations for spectral gaps are obtained in terms of Fourier coefficients of the potentials and estimates for the gap lengths are given.