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Some gaps in two of my publications dated 1958 and 1959 years are indicated, and the way of closing them is given. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

The paper is concerned with error identities for a class of parabolic equations. One side of such an identity is a natural measure of the distance between a function in the corresponding energy class and the exact solution of the problem in question. Another side is either directly computable or serves as a source of fully computable error bounds. Particular forms of the identities can be viewed as analogs of the hypercircle identity well known for elliptic problems. It is shown that identities possess an important consistency property. Therefore, the identities and the corresponding error estimates can be used in quantitative analysis of direct and inverse problems associated with parabolic equations. The first part of the paper deals with linear parabolic equations. A class of nonlinear problems is considered in the second part. In particular, this class includes problems whose spatial parts are presented by the α-Laplacian operator. [ABSTRACT FROM AUTHOR]

The paper is concerned with the asymptotic expansion of solutions to the following 2 × 2 Dirac type system: L y = - iB - 1 y ′ + Q x y = λ y , B = b 1 0 0 b 2 , y = col y 1 , y 2 , 0.1 with a smooth matrix potential Q ∈ W 1 n 0 , 1 ⊗ C 2 × 2 and b1 < 0 < b2. If b2 = −b1 = 1, this equation is equivalent to one dimensional Dirac equation. We apply these formulas to get the asymptotic expansion of the characteristic determinant of the boundary value problem associated with the above equation subject to the general two-point boundary conditions. This expansion directly yields new completeness result for the system of root functions of such BVP with nonregular boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear boundary-value problem proposed as the simplest model for describing oscillations in a gas flow. We analyze the stability of the trivial (zero) equilibrium state and find a critical value of the speed of the incoming gas flow. Exact solutions of the problem are found in the form of time-periodic functions and their stability is examined. All the results are obtained analytically based on the qualitative theory of infinite-dimensional dynamical systems. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the smoothness of generalized solutions of the first boundary-value problem for a strongly elliptic functional differential equation containing orthotropic contraction transformations of the arguments of the unknown function in the leading part. The problem is considered in a disc, the coefficients of the equation are constant. Orthotropic contraction is understood as different contraction in different variables. Conditions for the conservation of smoothness on the boundaries of neighboring subdomains formed by the action of the contraction transformation group onto the disc are found in explicit form for any right-hand side from the Lebesgue space. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can fail near the boundary of these subdomains even for an infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary ∂Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one-dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. Also there was obtained the smoothness (in Sobolev spaces W 2 k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding ε-neighborhoods of certain points. However, the smoothness (in Hölder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Hölder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Hölder space. [ABSTRACT FROM AUTHOR]

In this paper, we investigate qualitative properties of solutions of boundary-value problems for strongly elliptic differential-difference equations. Earlier results establish the existence of generalized solutions of these problems. It was proved that smoothness of such solutions is preserved in some subdomains but can be violated on their boundaries even for infinitely smooth function on the right-hand side. For differential-difference equations on a segment with continuous right-hand sides and boundary conditions of the first, second, or the third kind, earlier we had obtained conditions on the coefficients of difference operators under which there is a classical solution of the problem that coincides with its generalized solution. Also, for the Dirichlet problem for strongly elliptic differential-difference equations, the necessary and sufficient conditions for smoothness of the generalized solution in Hölder spaces on the boundaries between subdomains were obtained. The smoothness of solutions inside some subdomains except for ε-neighborhoods of angular points was established earlier as well. However, the problem of smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations remained uninvestigated. In this paper, we use approximation of the differential operator by finite-difference operators in order to increase the smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in the scale of Sobolev spaces inside subdomains. We prove the corresponding theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem for a system described by a parabolic linear partial differential equation with the Neumann boundary condition. We impose some constraints on the control. The performance functional has the integral form. The control time T is fixed. The initial condition is not given by a known function but belongs to a certain set (incomplete information about the initial state). To obtain optimality conditions for the Neumann problem, a generalization of the Dubovitskii–Milyutin theorem was applied. The problem formulated in this paper describes the process of optimal heating for which we do not have exact information about the initial temperature of the heating object. We also present an example in which admissible controls and one of the initial conditions are given by means of the norm constraints. [ABSTRACT FROM AUTHOR]

In this paper, we consider the boundary-value problem for the Kuramoto–Sivashinsky equation with homogeneous Neumann conditions. The problem on the existence and stability of second-kind equilibrium states was studied in two ways: by the Galerkin method and by methods of the modern theory of infinite-dimensional dynamical systems. Some differences in results obtained are indicated. [ABSTRACT FROM AUTHOR]

The paper discusses how to verify the quality of approximate solutions to partial differential equations constructed by deep neural networks. A posterior error estimates of the functional type, that have been developed for a wide range of boundary value problems, are used to solve this problem. It is shown, that they allow one to construct guaranteed two-sided estimates of global errors and get distribution of local errors over the domain. Results of numerical experiments are presented for elliptic boundary value problems. They show that the estimates provide much more reliable information on the quality of approximate solutions generated by networks than the loss function, which is used as a quality criterion in the Deep Galerkin method. [ABSTRACT FROM AUTHOR]

The this paper, we introduce a pair of Sobolev spaces with special Jacobi–Gegenbauer weights, in which the general boundary-value problem for a class of ordinary integro-differential equations characterized by the positivity of the difference of orders of the inner and outer differential operators is well-posed in the Hadamard sense. Based on this result, we justify the general polynomial projection method for solving the corresponding problem. An application of general results to the proof of the convergence of the polynomial Galerkin method for solving the Cauchy problem in the Sobolev weighted space is given. The convergence rate of the method is characterized in terms of the best polynomial approximations of an exact solution, which automatically responds to the smoothness properties of the coefficients of the equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problems on the solvability and constructing solutions of one nonlocal boundary-value problem for the multidimensional, fourth-order, integro-differential Benney–Luke equation with degenerate kernel and spectral parameters. For various values of spectral parameters, necessary and sufficient conditions of the existence of a solution are obtained. The Fourier series for solutions of the problem corresponding to various sets of spectral parameters are obtained. For regular values of spectral parameters, the absolute and uniform convergence of the series and the possibility of their termwise differentiation with respect to all variables are proved. The problem is also examined studied for cases of irregular values of spectral parameters. [ABSTRACT FROM AUTHOR]

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form ∂ z ¯ f + af + b f ¯ = c , where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that ∂ z ¯ f = g with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type ∂ z ¯ f z = h z q f z . On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media. [ABSTRACT FROM AUTHOR]

The presented paper is devoted to the study of the well-known Hilbert boundary-value problem for semi-linear Beltrami equations with arbitrary boundary data that are measurable with respect to logarithmic capacity. Namely, we prove here the corresponding results on the existence, regularity, and representation of its nonclassical solutions with a geometric interpretation of boundary values as the angular (along the nontangential paths) limits in comparison with the classical approach in PDE. For this purpose, we apply completely continuous operators by Ahlfors–Bers, first of all to obtain solutions of semi-linear Beltrami equations, generally speaking with no boundary conditions, and then to derive their representation through the solutions of the Vekua-type equations and the so-called generalized analytic functions with sources. Besides, we obtain similar results for nonclassical solutions of the Poincaré boundary-value problem on directional derivatives and, in particular, of the Neumann problem with arbitrary measurable data to semi-linear equations of the Poisson type. The obtained results are applied to some problems of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states, and stationary burning in anisotropic and inhomogeneous media. [ABSTRACT FROM AUTHOR]

The investigation of differential-algebraic boundary-value problems was originated in the works by Weierstrass, Luzin, and Gantmakher. The works by Campbell, Boyarintsev, Chistyakov, Samoilenko, Perestyuk, Yakovets, Boichuk, Ilchmann, and Reis were devoted to the systematic study of differential-algebraic boundary-value problems. At the same time, the investigation of differential-algebraic boundary-value problems is closely connected with the analysis of linear boundary-value problems for ordinary differential equations originated in the works by Poincaré, Lyapunov, Krylov, Bogolyubov, Malkin, Myshkis, Grebenikov, Ryabov, Mitropolskii, Kiguradze, Samoilenko, Perestyuk, and Boichuk. The investigation of linear differential-algebraic boundary-value problems is connected with numerous applications of the corresponding mathematical models to the theory of nonlinear oscillations, mechanics, biology, radio-engineering, and the theory of stability of motion. Thus, the problem of generalization of the results obtained by Campbell, Samoilenko, and Boichuk to the case of nonlinear integrodifferential boundary-value problems unsolved with respect to the derivative is quite actual. In particular, this is true for the problem of finding necessary and sufficient conditions for the existence of solutions of nonlinear integrodifferential boundary-value problems unsolved with respect to the derivative. In the present paper, we establish conditions for the existence of solutions of the nonlinear integrodifferential boundary-value problem unsolved with respect to the derivative and present a constructive scheme for their finding. [ABSTRACT FROM AUTHOR]

In this paper, we construct boundary-domain integral equations (BDIEs) of the Dirichlet and mixed boundary value problems for a two-dimensional variable-coefficient Helmholtz equation. Using an appropriate parametrix, these problems are reduced to several BDIE systems. It is shown that the BVPs and the formulated BDIE systems are equivalent. Fredholm properties and unique solvability and invertibility of BDIE systems are investigated in appropriate Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of nonlinear hybrid fractional differential equations involving ψ -Caputo fractional derivative. We establish the existence and uniqueness for a new class of impulsive fractional boundary value problems with nonlocal and boundary hybrid conditions. Our main theorem is demonstrated using fixed point theorems, including the Banach fixed point theorem and the Leray-Schauder alternative fixed point theorem. Some examples are also constructed to demonstrate the application of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we examine the synthesis problem for a distributed thermal control system in the case where a Fredholm integral operator is involved in the boundary-value problem. We use the method proposed by A. I. Egorov and develop it based on the Bellman scheme. Using the notions of a generalized solution of a boundary-value problem and the Fréchet differential for the Bellman functional, we obtain a partial integro-differential equation. For synthesizing a distributed optimal control, we propose a numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we prove coercive a priori estimates of solutions of a Dirichlet-type boundary-value problem in a strip for a certain higher-order degenerate elliptic equation containing weighted derivatives of a special form up to the order 2m and ordinary partial derivatives up to the order 2k−1 under the condition 2m > 2k −1. At the boundary of the strip, Dirichlet-type conditions are imposed. A coercive a priori estimate for solutions of the problem considered in special weighted Sobolev-type spaces is obtained. [ABSTRACT FROM AUTHOR]

One of directions in the development of Lomov's regularization method is the approach related to holomorphic regularization of singularly perturbed problems, which allows one to construct solutions to such problems in the form of series in powers of a small parameter that converge in the usual sense. For boundary-value problems, the problem of pseudo-holomorphic continuation of solutions is very urgent. In this paper, we examine a boundary-value problem for a Tikhonov system and give conditions for the existence of its pseudo-holomorphic solution. [ABSTRACT FROM AUTHOR]

The paper deals with the so-called M-transform, which maps divergence-free vector fields in Ω T:= { x ∈ Ω| dist( x, ∂Ω) < T}, Ω ⊂⊂ $$ \mathbb{R} $$3, to the space of transversal fields. The latter space consists of vector fields in Ω T tangential to the equidistant surfaces of the boundary ∂Ω. In papers devoted to the dynamical inverse problem for the Maxwell system, in the framework of the BC-method, the operator M T was defined for T < Tω, where Tω depends on the geometry of Ω. This paper provides a generalization for arbitrary T. It is proved that M T is partially isometric, and its intertwining properties are established. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

The paper is concerned with a new way of deriving computable estimates for the difference between the exact solutions of elliptic variational inequalities and arbitrary functions in the corresponding energy space that satisfy the main (Dirichlét) boundary conditions. Unlike the method derived earlier, the estimates are obtained by certain transformations of variational inequalities without using duality arguments. For linear elliptic and parabolic problems, this method was suggested by the author in previous papers. The present paper deals with two different types of variational inequalities (also called variational inequalities of the first and second kind). The techniques discussed can be applied to other nonlinear problems related to variational inequalities. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation lawsut+fux=0,ut=0=u0x,where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form∑tjinteraction timeσαj−σαj′αjαj′αj+αj′≤CfTot.Var.u02,where αj and αj′ are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form).The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which:• all the main ideas of the construction are presented;• all the technicalities of the proof in the general setting [8] are avoided. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author's knowledge, there were no results in this direction for the higher-order differential operators with distribution coefficients (generalized functions) in a general form. The technique of this paper relies on the recently obtained regularization and the Birkhoff-type solutions for differential operators with distribution coefficients. Our results have applications to the theory of inverse spectral problems as well as a separate significance. [ABSTRACT FROM AUTHOR]

On the base of boundary-value, spectral and initial-boundary value problems studied earlier for the case of single domain, we consider corresponding problems generated by a sesquilinear form for two domains. Arising operator pencils with corresponding operator coefficients acting in a Hilbert space and depending on two parameters are studied in detail. In the perturbed and unperturbed cases, we consider two situations where one of the parameters is spectral and the other is fixed. In this paper, we use the superposition principle that allows us to present the solution of the original problem as a sum of solutions of auxiliary boundary-value problems containing inhomogeneity either in the equation or in one of the boundary conditions. The necessary and sufficient conditions for the correct solvability of boundary-value problems on a given time interval are obtained. Theorems on properties of the spectrum and on the completeness and basicity of the system of root elements are proved. [ABSTRACT FROM AUTHOR]

In the paper, we investigate the question of obtaining an algebraic condition for the exponential stability of the numerical solution of the upwind difference scheme for the mixed problem posed for one-dimensional symmetric t-hyperbolic systems with constant coefficients and with dissipative boundary conditions. An a priori estimate for the numerical solution of the boundary-value difference problem is obtained. This estimate allows us to state the exponential stability of the numerical solution. A theorem on the exponential stability of a numerical solution of a boundary-value difference problem is proved. Easily verifiable algebraic conditions for the exponential stability of a numerical solution are given. The convergence of the numerical solution is proved. [ABSTRACT FROM AUTHOR]

In this paper, anisotropic classes of well-posed Cauchy problems and boundary-value problems for systems of convolutions equations are obtained. For a particular case of differential equations, a hypersurface of conjugate orders of the corresponding polynomial is used, and various classes of well-posed problems are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose an abstract scheme for the study of special operators and apply this scheme to examining elliptic pseudo-differential operators and related boundary-value problems on manifolds with nonsmooth boundaries. In particular, we consider cases where boundaries may contain conical points, edges of various dimensions, and even peak points. Using the constructions proposed, we present well-posed formulations of boundary-value problems for elliptic pseudo-differential equations on manifolds discussed in Sobolev–Slobodecky spaces. [ABSTRACT FROM AUTHOR]

Abstract. This paper is devoted to the study of the solvability of boundary-value problems for differential equations of the form α 0 t + α 1 t ∆ u tt − B u t − Cu = f x t , where Δ is the Laplace operator acting with respect to spatial variables and B and C are also secondorder differential acting with respect to spatial variables. A feature of the equations considered is the condition that the functions α0(t) and α1(t) may not possess the fixed sign property on the range (0, T) of the temporal variable; in particular, the operator α0(t)+α1(t)Δ may be irreversible at any point of the interval (0, T), including any strictly inner segments. For problems considered, we prove theorems on the existence and uniqueness of regular solutions (i.e., solutions possessing all generalized derivatives in the Sobolev sense). [ABSTRACT FROM AUTHOR]

In this paper, the Green functions of the first boundary-value problem for the fractional diffusion-wave equation in multidimensional (bounded and unbounded) hyper-rectangular domains are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we study a boundary-value problem with discontinuous conjugation conditions on the line of type changing for a model equation of mixed hyperbolic-parabolic type with degeneration of order in the hyperbolicity domain. In the parabolic domain, the equation is the fractional diffusion equation, whereas in the hyperbolic domain it is the loaded one-speed transfer equation. We prove the uniqueness and existence theorem and propose an explicit solution of the problem in the parabolic and hyperbolic domains. [ABSTRACT FROM AUTHOR]

In this paper, a boundary-value problem for the inhomogeneous Aller–Lykov moisture transfer equation with a fractional Riemann–Liouville time derivative is examined. The equation considered is a generalization of the Aller–Lykov equation obtained by introducing the fractal rate of change of humidity, which explains the appearance of flows directed against the potential of humidity. The existence of a solution to the first boundary-value problem is proved by the Fourier method. Using the method of energy inequalities for solutions of the problem, we obtain an a priori estimate in terms of the fractional Riemann–Liouville derivative, which implies the uniqueness of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we present the formulation and a method for solving the problem of freezing living biological tissue with a flat circular cryoapplicator. The model is a two-dimensional boundary-value problem of Stefan type with nonlinear heat sources of a special type that provide the actually observed spatial localization of the temperature field. Some results of computer simulation are presented. [ABSTRACT FROM AUTHOR]

In this paper, we present a method of computer calculation of Green functions in the form of generalized power series for third-order linear differential equations admitting regular singularities. For specific boundary-value problems, we construct Green functions by using the software proposed. [ABSTRACT FROM AUTHOR]

This paper is a brief review of results in the theory of covering mappings of metric spaces and vector metric spaces and its applications to implicit differential equations. For the Cauchy problem and boundary-value problems, we obtain existence conditions, estimates of solutions, and conditions of the continuous dependence of solutions on the parameters of the equation and initial and boundary conditions. [ABSTRACT FROM AUTHOR]

In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals. [ABSTRACT FROM AUTHOR]

Conditions for the solvability of the linear boundary-value problem for systems of differential-algebraic equations with the variable rank of the leading-coefficient matrix and the corresponding solution construction procedure have been found. [ABSTRACT FROM AUTHOR]

We study the solvability of boundary value problems nonlocal with respect to the spatial variable with the generalized Samarskii–Ionkin condition for parabolic equations where x ∈ (0, 1), t ∈ (0, T) and h(t), a(x), c(x, t), f(x, t) are given functions. If a(x) is positive, then the function h(t) can have different signs at different points of [0, T] or even vanish on a set of positive measure in [0, T]. We prove the existence and uniqueness of regular solutions, i.e., solutions possessing all weak derivatives (in the sense of Sobolev) occurring in the corresponding equation. The obtained results are new even for the classical Samarskii–Ionkin problem for the heat equation. [ABSTRACT FROM AUTHOR]

In this paper, we obtain solvability conditions for the periodic boundary-value problem for a certain first-order differential equation unsolved with respect to the derivative. These condition were obtained by using the theorem on implicit operators. [ABSTRACT FROM AUTHOR]

In the Sturm-Liouville boundary value problem the Dirichlet boundary condition at the point zero has not yet been studied in sufficient details. In this paper the author gives the necessary and sufficient conditions for two sequences { μ n } n = 0 ∞ and { a n } n = 0 ∞ to be correspondingly the eigenvalues and norming constants of the Sturm-Liouville problem with Dirichlet condition at zero. In particular, the constructive solution of inverse problem is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem of determining u(x, t) and k(t) functions in the one-dimensional integro-differential heat equation with the nonlocal initial-boundary and over-determination conditions. The unique solvability of the direct problem is rigorously proved using the Fourier method and Schauder principle. To investigate the solvability of the inverse problem, we first consider an auxiliary inverse boundary value problem, which is equivalent to the original one. Then using the Fourier method, the problem is reduced by an equivalent closed system of integral equations with respect to unknown functions. The existence and uniqueness theorem for this system of integral equations is proved by contraction mappings principle. [ABSTRACT FROM AUTHOR]

This paper is concerned with the derivation of computable and guaranteed upper and lower bounds of the difference between exact and approximate solutions of a boundary value problem for static Maxwell equations. Our analysis is based upon purely functional argumentation and does not invoke specific properties of the approximation method. For this reason, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of the functional type) have been derived earlier for elliptic problems. Bibliography: 24 titles. [ABSTRACT FROM AUTHOR]

Contrary to the opinion that has prevailed for the last several decades about the incorrectness of the inverse–MEEG problems (see, for example, the paper of D. Sheltraw and E. Coutsias in Journal of Applied Physics, 94, No. 8, 5307–5315 (2003)), in this note it is shown that this problem is absolutely well posed: it has a unique solution, but in a special class of functions (different from those considered by biophysicists). The solution has the form q = q0 + p0δ|∂Y, where q0 is an ordinary function defined in the domain of the region Y occupied by the brain, and p0δ|∂Y is a δ-function on the boundary of the domain Y with a certain density p0. Moreover, the operator of this problem realizes an isomorphism of the corresponding function spaces. This result was obtained due to the fact that: (1) Maxwell's equations are taken as a basis; (2) a transition was made to the equations for the potentials of the magnetic and electric fields; (3) the theory of boundary value problems for elliptic pseudodifferential operators with an entire index of factorization is used. This allowed us to find the correct functional class of solutions of the corresponding integral equation of the first kind. Namely: the solution has a singular boundary layer in the form of a delta function (with some density) at the boundary of the domain. From the point of view of the MEEG problem, this means that the sought-for current dipoles are also concentrated in the cerebral cortex. [ABSTRACT FROM AUTHOR]

This is the first paper of a conceived series under the common title “The boundary control method in inverse problems.” The aim of the series is to expound systematically an approach to inverse problems based upon its relationship with control theory. The 1d-variant of the method is shown with the example of the classical problem of recovering the density of an inhomogeneous string, and both dynamical and spectral statements of the problem are considered. The paper is written in such a way as to serve as an introduction to the multidimensional BC-method: the basic tools and constructions are amenable to further generalization to multidimensional problems. Bibliography: 31 titles. [ABSTRACT FROM AUTHOR]

With reference to the homogenization theory, as well as problems of vibrations in physics and mechanics, we note a growing interest in the analysis of domains whose boundaries are singularly perturbed. We consider solids containing large clusters of defects that may possess unusual dynamic responses to external loads. We review the method of the mesoscale uniform asymptotic approximations, which use the Green kernels and take into account different scales (defined relative to the size of individual inclusions), together with boundary layer fields. The method provides an efficient analytical tool, as well as solvers for hybrid numerical schemes aimed at solutions of boundary value and spectral problems for mesoscale structures, containing large clusters of defects. We discuss examples that include solids with many small inclusions, where a small parameter, the relative size of an inclusion, may compete with a large parameter, representing an overall number of inclusions. We note that in some cases, the approach of mesoscale asymptotic approximations provides a powerful alternative to the conventional homogenization algorithms for solids with large clusters of small inclusions. [ABSTRACT FROM AUTHOR]

The numerical solution of boundary value problems by the finite element method leads to a system of equations with a matrix that depends on the triangulation mesh. A change in the geometric configuration of the mesh leads to the changes in the corresponding matrix and hence to the changes in the rate of convergence of the process of successive approximations to the numerical solution of the problem. The paper shows how it is possible to speed up the rate of convergence of the iterative solution process without changing the position of the grid vertices. A class of matrices is found whose mesh optimization leads to Delaunay triangulation. An example of using the results of the work in the numerical solution of the magnetic field is given. [ABSTRACT FROM AUTHOR]

By using the Krasnoselskii theorem, we obtain general conditions for the unique solvability of boundaryvalue problems for (non)linear fractional functional-differential equations. [ABSTRACT FROM AUTHOR]

An example of solving a boundary-value problem for a homogeneous Monge-Ampère equation is given, which produces a Bellman function for an extremal problem on the space BMO. The paper contains a step-by-step instruction for calculation of this function. Cases of rather complicated foliations are considered. This illustrates the technique elaborated in a paper by Ivanishvili, Stolyarov, Vasyunin, and Zatitskiy. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

In the present paper, we study an optimal sphere rolling problem on the plane (without slew and slip) with predefined boundary-value conditions. To solve it, we use methods from the optimal control theory. The controlled system for sphere orientation is represented via the rotation quaternion. Asymptotics of extremal paths on a sphere rolling along small-amplitude sine waves is found. [ABSTRACT FROM AUTHOR]