On an approach to the study of the solvability of boundary value problems for the system of differential equations of the 3D theory of elasticity

We study the solvability of a boundary value problem for a system of second-order linear partial differential equations. A theorem on the existence of a solution of the problem is proved. The method used in the study is to reduce the original system of equations to a system of 3D singular integral equations, whose solvability can be proved with the use of the notion of symbol of a singular operator.     

A generalization of Bitsadze–Samarskii problem for mixed type equation

We study a boundary-value problem with Bitsadze–Samarskii conditions on boundary characteristic on a special inner curve and on a segment of degeneration of mixed type equation. Its solvability is proved by method of integral equations, and uniqueness of solution is established by means of the maximum principle.     

Weak solvability of interior transmission problems via mixed finite elements and Dirichlet-to-Neumann mappings

We study the weak solvability of an interior linear-nonlinear transmission problem arising in steady heat transfer and potential theory. For the variational formulation, we use a Dirichlet-to-Neumann mapping on the interface, which is obtained from the application of the boundary integral method to the linear domain, and we utilize a mixed finite element method in the nonlinear region. Existence and uniqueness of solution for the continuous formulation are provided and general approximation results for a fully discrete Galerkin method are derived. In particular, a compatibility condition between the mesh sizes involved is deduced in order to conclude the solvability and stability of this Galerkin scheme.     

Conservative finite-difference scheme for computer simulation of contrast 3D spatial-temporal structures induced by a laser pulse in a semiconductor

The numerical approach for computer simulation of femtosecond laser pulse interaction with a semiconductor is considered under the formation of 3D contrast time-dependent spatiotemporal structures. The problem is governed by the set of nonlinear partial differential equations describing a semiconductor characteristic evolution and a laser pulse propagation. One of the equations is a Poisson equation concerning electric field potential with Neumann boundary conditions that requires fulfillment of the well-known condition for Neumann problem solvability. The Poisson equation right part depends on free-charged particle concentrations that are governed by nonlinear equations. Therefore, the charge conservation law plays a key role for a finite-difference scheme construction as well as for solvability of the Neumann difference problem. In this connection, the iteration methods for the Poisson equation solution become preferable than using direct methods like the fast Fourier transform. We demonstrate the following: if the finite-difference scheme does not possess the conservatism property, then the problem solvability could be broken, and the numerical solution does not correspond to the differential problem solution. It should be stressed that for providing the computation in a long-time interval, it is crucial to use a numerical method that possessing asymptotic stability property. In this regard, we develop an effective numerical approach—the three-stage iteration process. It has the same economic computing expenses as a widely used split-step method, but, in contrast to the split-step method, our method possesses conservatism and asymptotic stability properties. Computer simulation results are presented.     

Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator

We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem.     

Mixed value problem for a nonlinear differential equation of fourth order with small parameter on the parabolic operator

We study the solvability of mixed value problem for one type of nonlinear partial differential equation, consisting superposition of parabolic and hyperbolic operators. By the method of separation variables we obtain the countable system of nonlinear integral equation. We use the method of successive approximations. It will be proved the convergence of obtained series. We study the continuously dependence of solution from small parameter.     

Solution of boundary-value problems for a degenerating elliptic equation of the second kind by the method of potentials

In this paper we study basic boundary value problems for one multidimensional degenerating elliptic equation of the second kind. Using the method of potentials we prove the unique solvability of the mentioned problems. We construct a fundamental solution and obtain an integral representation for the solution to the equation. Using this representation we study properties of solutions, in particular, the principle of maximum. We state the basic boundary value problems and prove their unique solvability. We introduce potentials of single and double layers and study their properties. With the help of these potentials we reduce the boundary value problems to the Fredholm integral equations of the second kind and prove their unique solvability.     

Variational method for proving the existence of a nonlocal solution of a boundary value problem for a quasilinear partial differential equation

We study the solvability of a boundary value problem for a quasilinear partial differential equation of the second kind. To this end, we use a variational method; namely, we prove the existence of a point of absolute minimum of a functional, and this point is a solution of the original problem.     

On Topological Index of Solutions for Variational Inequalities on Riemannian Manifolds

In this paper, based on the fixed point index theory for a class of -multivalued maps on absolute neighbourhood retracts, we introduce the notion of index of solvability for a variational inequality on a Riemannian manifold involving a multivalued vector field. We describe the main properties of this topological characteristic and use it to justify the existence of a solution for a variational inequality problem. As application, the problem of optimization of a non-smooth functional on a Hadamard manifold is considered.     

On the solvability of the Cauchy characteristic problem for a nonlinear equation with iterated wave operator in the principal part

We consider one multidimensional version of the Cauchy characteristic problem in the light cone of the future for a hyperbolic equation with power nonlinearity with iterated wave operator in the principal part. Depending on the exponent of nonlinearity and spatial dimension of equation, we investigate the problem on the nonexistence of global solutions of the Cauchy characteristic problem. The question on the local solvability of that problem is also considered.     

Proof of the solvability of a system of partial differential equations in the nonlinear theory of shallow shells of Timoshenko type

We study the solvability of a system of second-order partial differential equations under given boundary conditions. To prove the existence of a solution of the system, we reduce it to a single nonlinear partial differential equation whose solvability is proved with the use of the contraction mapping principle.     

The Cauchy Problem for Shilov Parabolic Equations

We establish well-posed solvability of the Cauchy problem for the Shilov parabolic equations with time-dependent coefficients whose initial data are tempered distributions. For a certain class of equations we state necessary and sufficient conditions for unique solvability of the Cauchy problem whose properties with respect to the spatial variable are typical for a fundamental solution. The results are characterized only by the order and the parabolicity exponent.     

On the exceptional case of the characteristic singular equation with Cauchy kernel

We study the exceptional case of the characteristic singular integral equation with Cauchy kernel in which its coefficients admit zeros or singularities of complex orders at finitely many points of the contour. By reduction to a linear conjugation problem, we obtain an explicit solution formula and solvability conditions for this equation in weighted Hölder classes.     

The boundary-value problem for Lavrent’Ev–Bitsadze equation with two internal lines of change of a type

We study the problem with boundary conditions of the first and second kind on the boundary of a rectangular domain for an equation with two internal perpendicular lines of change of a type. With the use of the spectral method we prove the unique solvability of the mentioned problem. The eigenvalue problem for an ordinary differential equation obtained by separation of variables is not self-adjoint, and the system of root functions is not orthogonal. We construct the corresponding biorthogonal system of functions and prove its completeness. This allows us to establish a criterion for the uniqueness of the solution to the problem under consideration. We construct the solution as the sum of the biorthogonal series.     

Rough Burgers-like equations with multiplicative noise

We construct solutions to vector valued Burgers type equations perturbed by a multiplicative space–time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise.     

Galerkin methods for a model of population dynamics with nonlinear diffusion

A numerical method is proposed to approximate the solution of a nonlinear and nonlocal system of integro-differential equations describing age-dependent population dynamics with spatial diffusion. We use a finite difference method along the characteristic age-time direction combined with finite elements in the spatial variable. Optimal order error estimates are derived for this approximation. © 1996 John Wiley & Sons, Inc.     

Linear differential-algebraic equations perturbed by Volterra integral operators

We study linear inhomogeneous vector ordinary differential equations of arbitrary order in which the matrix multiplying the highest derivative of the unknown vector function is singular in the domain where the equations are defined. We also study perturbations (not necessarily small) of such equations, which are linear integro-differential equations with a Volterra operator. We obtain sufficient conditions for the solvability of such equations and give representations of their general solutions; solvability and uniqueness conditions are also given for initial value problems for such equations. The influence of small perturbations of the free term and the initial data on the solution is considered. A numerical method is suggested. The results of numerical experiments are given.     

Approximate Solution of Optimization Problems for Infinite-Dimensional Singular Systems

An optimal control problem is considered for a system described by a singular equation of parabolic type. The study bases on a special regularization method. We establish existence of a solution to the regularized problem, as well as the corresponding necessary optimality conditions. The results enable us to find an approximate solution to the original problem even in the absence of solvability.     

On the numerical solution of a complete two-dimensional hypersingular integral equation by the method of discrete singularities

We study the solvability of a complete two-dimensional linear integral equation with a hypersingular integral understood in the sense of the Hadamard principal value. We justify the convergence of a quadrature-type numerical method for the case in which the equation in question is uniquely solvable. We present an application of the results to the numerical solution of the Neumann boundary value problem on a plane screen for the Helmholtz equation by the surface potential method.     

Solvability of singular integral system connected with seismic waves around the borehole

We study in this article a boundary‐value problem arising in the propagation of waves in an elastic half‐space covered by a layer with a vertical borehole. We first show a uniqueness theorem under some restrictions on the solution. For the existence, we use the direct integral equation method. We obtain a singular integral system on the half‐line. For the solvability, we reduce this system to an elliptic pseudodifferential equation and establish the Fredholm property. Finally, we compute the index of the associated operator for various values of Poisson's ratio. Copyright © 2006 John Wiley & Sons, Ltd.