Solution of boundary-value problems for elliptic equations in the space of distributions

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C ^∞ (S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.     

An analytic method of studying the nonlinearly elastic equilibrium of a rectangular plate of variable thickness under bending

We propose an approximate analytic method of solving three-dimensional boundary-value problems of the physically nonlinear theory of elasticity for thick rectangular plates of variable thickness subject to a transverse load. The method is used to seek a solution of this problem in the form of double power series in small dimensionless parameters. In arbitrary approximation the original problem is reduced to a sequence of linear inhomogeneous boundary-value problems for plates of constant thickness. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 36–40     

Numerical solution of the boundary-value problem for a nonlinear diffusion equation in image processing

Diffusion filtering methods involve solving an initial boundary-value problem for the diffusion equation in which the initial condition is specified by a function representing the filtered image. The output of this filter is the solution u(x, y, t) of the initial boundary-value problem at some fixed time t = T. In a previous study we have proposed a new version of the diffusion filtering method that ensures improved noise removal due to inclusion of a dependence of the diffusion coefficient on local image intensity. The present study analyzes the resulting finite-difference method for the initial boundary-value problem, examines its numerical implementation, and analyzes its efficiency on prototype and real images. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 35–43, 2006.     

On a method of solving physically nonlinear problems in the bending of thick rectangular plates

We study the three-dimensional boundary-value problem of the physically nonlinear theory of elasticity for bending of a homogeneous isotropic thick plate by transverse forces. We assume that the intensity of the load is such that the connection between the stresses and small strains within the elastic limit can be described by nonlinear relations in the form of H. Kauderer. We develop an approximate analytic method that makes it possible to reduce the original nonlinear boundary-value problem to a recursive sequence of corresponding linear boundary-value problems. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, No. 40, 1997, pp. 30–35.     

A variant of the coboundary element method for studying heat-conduction problems in bodies of complicated shape

We propose a scheme for constructing the solution of boundary-value problems for bodies of complicated shape by applying coboundary or boundary elements and integral transforms. We illustrate the method of applying the finite Fourier cosine transform in solving a mixed boundary-value problem for a body having the shape of a trapezoid. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 97–103.     

Smooth and convergent ε-approximations of the first boundary-value problem for equations of the Kelvin-Voight and oldroyd fluids

Smooth convergent ε-approximations (11)–(13) for the equations of Oldroyd (8) and Kelvin-Voight (9), (10) fluids are constructed. It is shown that the first initial boundary-value problem for two-dimensional system (11) and three-dimensional systems (12) and (13) for every ε>0 has a unique classical solution, and as ε → 0 these solutions converge to classical solutions of the first initial boundary-value problem for Eqs. (8), (9), and (10) respectively. Bibliography: 10 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 246–255. Translated by O. A. Ivanov.     

Numerical analysis of the dispersion of electroelastic waves in a cylinder

The wave problem of electroelastic waves in a cylinder is reduced to a spectral problem for a system of eight ordinary differential equations. We give an algorithm for numerical solution based on reducing the original boundary-value problem to four Cauchy problems and determining the roots of the dispersion equation by the method of bisection. One figure. Bibliogrpahy: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 149–153.     

Smooth convergent ε-approximations of the first initial boundary-value problem for the equations of motion of Kelvin-Voight fluids

Smooth convergent ε-approximations (33), (34) and (38), (39) are constructed for the equations of Kelvin-Voight fluids (1) and for the equations of Kelvin-Voight fluids of order L=1,2,…(2). Namely, it is shown that, for any ε>0, the first initial boundary-value problem for the three-dimensional perturbed systems (33), (34) and (38), (39) has a unique classical solution in the large. As ε→0, these solutions converge to the classical solutions of the first initial boundary-value problem for the equations of Kelvin-Voight fluids (1) and for the equations of Kelvin-Voight fluids of order L=1,2,… (2), respectively. Bibliography: 19 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 186–212. Translated by N. B. Lebedinskaya.     

Bounded solutions of families of systems of differential equations and their approximations

The paper considers the problem of finding a bounded solution of a one-parametric family of systems of ordinary differential equations. Using the parametrization method, the author proves necessary and sufficient conditions for the existence of a unique solution of the problem considered that is bounded on the whole axis in terms of a two-sided, infinite block-band matrix composed with respect to integrals over intervals of length h > 0 of the matrix of the system of differential equations. Also, the author constructs a family of two-point boundary-value problems on a finite interval that approximates the problem of finding the bounded solution and finds an interconnection between the correct solvability of the initial and approximating problems. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 29–47, 2005.     

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R ² with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H ^m (Ω), m ≥ 4.     

Introduction to the theory of functional differential equations and their applications. Group approach

In this work, we give an introduction to the theory of nonlinear functional differential equations of pointwise type on a finite interval, semi-axis, or axis. This approach is based on the formalism using group peculiarities of such differential equations. For the main boundary-value problem and the Euler-Lagrange boundary-value problem, we consider the existence and uniqueness of the solution, the continuous dependence of the solution on boundary-value and initial-value conditions, and the “roughness” of functional differential equations in the considered boundary-value problems. For functional differential equations of pointwise type we also investigate the pointwise completeness of the space of solutions for given boundary-value conditions, give an estimate of the rank for the space of solutions, describe types of degeneration for the space of solutions, and establish conditions for the “smoothness” of the solution. We propose the method of regular extension of the class of ordinary differential equations in the class of functional differential equations of pointwise type. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 8, Functional Differential Equations, 2004.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_tt - a²u_xx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.     

Symmetry analysis and exact solutions of one class of (1?+?3)-dimensional boundary-value problems of the Stefan type

We present the group classification of one class of (1 + 3)-dimensional nonlinear boundary-value problems of the Stefan type that simulate the processes of melting and evaporation of metals. The results obtained are used for the construction of the exact solution of one boundary-value problem from the class under study.     

A generalization of the fundamental theorem of spherical harmonic theory

We consider a boundary-value problem of the first kind for a self-adjoint differential operator with constant coefficients on a domain in ℝⁿ bounded by an ellipsoid; boundary conditions are defined by an arbitrary polynomial of degree N. It is proved that the solution of the problem is again a polynomial of degree ≤N. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

A study of the bending of a short beam

We consider the two-dimensional problem of static deformation ofa short beam under the action of a self-balanced load. We propose an approximate method of solution based on a variational approach and a special choice of the stress function. We prove that the resulting boundary-value problem for a system of ordinary differential equations is well-posed. For special cases of the boundary conditions we give an analysis of the solutions. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 65–73.     

On the regularity of solutions of boundary-value problem for quasilinear elliptic systems with quadratic nonlinearity

The partial regularity up to the boundary of a domain is established for a solution u ∈ H¹ (Ω) ∩ L_∞ (Ω) to the boundary-value problem for a second-order elliptic system with strong nonlinearity in the case of dimension n≥3. Bibliography: 12 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 47–69.     

Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I

We consider the periodic boundary-value problem u _tt − u _xx = g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u ⁰(x, t) + ũ(x, t), where u ⁰(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the period ω. We show that the relation obtained for a solution includes known results established earlier. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005.     

Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type

We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in boundary conditions in the corresponding spaces and are studied by the method of operator differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 443–452, April, 2008.     

A priori estimates for solutions of the first initial boundary-value problem for systems of fully nonlinear partial differential equations

We prove a priori estimates for a solution of the first initial boundary-value problem for a system of fully nonlinear partial differential equations (PDE) in a bounded domain. In the proof, we reduce the initial boundary-value problem to a problem on a manifold without boundary and then reduce the resulting system on the manifold to a scalar equation on the total space of the corresponding bundle over the manifold. St. Petersburg Architecture Building University, St. Petersburg. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 3, pp. 338–363, March, 1997.     

On existence of a harmonic variational flow subjected to two-sided conditions

We construct weak solutions to parabolic systems of the variational flow type related to a quadratic functional with the initial and boundary data from a suitable Sobolev space and subjected to two-sided conditions. We present an approach based on Rothe’s method. It is applicable to solving the Cauchy problem and initial boundary-value problem for many types of equations. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov, POMI, Vol. 243, 1997, pp. 324–337.