An Iterative FEM for Solving Wave Problems of a Halfspace

An iterative finite element method for solving wave problems of a halfspace is presented in this paper. The halfspace is first truncated by introducing a fictive finite boundary on which some fictive boundary conditions must be imposed. A finite computational domain is in each iteration subjected to actual boundary conditions on real boundary and to fictive Dirichlet or Neumann boundary conditions on the fictive boundary. The radiation condition is satisfied by using DtN operator. The DtN operator is not introduce in the finite element formulation on the fictive boundary so any finite elements can be used. The method is simple and specially useful for computing higher harmonics.     

Degenerate boundary conditions for the diffusion operator

We describe all degenerate two-point boundary conditions possible in a homogeneous spectral problem for the diffusion operator. We show that the case in which the characteristic determinant is identically zero is impossible for the nonsymmetric diffusion operator and that the only possible degenerate boundary conditions are the Cauchy conditions. For the symmetric diffusion operator, the characteristic determinant is zero if and only if the boundary conditions are falsely periodic boundary conditions; the characteristic determinant is identically a nonzero constant if and only if the boundary conditions are generalized Cauchy conditions.     

Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

We study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.     

On the conditions of solvability and the resolvent of a second-order deferential boundary operator in a space of vector functions

A boundary differential operator generated by the Sturm-Liouville differential expression with bounded operator potential and nonlocal boundary conditions is considered. The conditions for a considered operator to be a Fredholm and solvable operator are established and its resolvent is constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 517–524, April, 1995.     

On the Maxwell system under impedance boundary conditions with memory

We consider an initial boundary value problem for the system of the Maxwell equations in a bounded domain with smooth boundary on a finite time interval with new boundary conditions with memory. In appropriate function spaces, we define and study the nonselfadjoint operator that is generated by the Maxwell operator under a boundary condition with memory. Using the operator method, we prove an existence and uniqueness theorem for a solution to the initial boundary value problem.     

Solvability conditions of a boundary value problem with operator coefficients and related estimates of the norms of intermediate derivative operators

Sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions of the boundary value problem for a certain class of second-order elliptic operator differential equations on a semiaxis are obtained. The boundary condition at zero involves an abstract linear operator. The solvability conditions are established by using properties of operator coefficients. The norms of intermediate derivative operators, which are closely related to the solvability conditions, are estimated.     

Constructing C1 triangular patch of degree six by the Boolean sum of an approximation operator and an interpolation operator

A new method to construct C¹ triangular patches which satisfy the given boundary curves and cross-boundary slopes is presented. The Boolean sum of an approximation operator and an interpolation operator is employed to construct the triangular patch. The approximation operator is used to construct a polynomial patch of degree six. The polynomial of degree six affords more freedoms, which makes the approximation operator not only approximate the given boundary interpolation conditions but also have a better approximation precision in the interior of the triangle, so that the triangular patch has a better precision on both the boundary and the interior of the triangular domain. The interpolation operator is utilized to build an interpolation patch which satisfies the given boundary conditions. The Boolean sum of the approximation and interpolation patches forms the triangular patch. Comparison results of the new method with other three methods are given.     

Nonuniformly nonlinear elliptic equations of p-biharmonic type

This paper deals with the existence and multiplicity of weak solutions to nonlinear differential equations involving a general p-biharmonic operator (in particular, p-biharmonic operator) under Dirichlet boundary conditions or Navier boundary conditions. Our method is mainly based on variational arguments.     

Spectral analysis of nonselfadjoint Schrödinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schrödinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrödinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrödinger boundary value problem are given.     

Spectral analysis of nonselfadjoint Schr(o)dinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schr(o)dinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator,and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schr(o)dinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schr(o)dinger boundary value problem are given.     

On the Fredholm property of the electric field equation in the vector diffraction problem for a partially screened solid

We consider a vector problem of diffraction of an electromagnetic wave on a partially screened anisotropic inhomogeneous dielectric body. The boundary conditions and the matching conditions are posed on the boundary of the inhomogeneity domain, and under passage through it, the medium parameters have jump changes. A boundary value problem for the system of Maxwell equations in unbounded space is studied in a semiclassical statement and is reduced to a system of integro-differential equations on the body domain and the screen surfaces. We show that the quadratic form of the problem operator is coercive and the operator itself is Fredholm with zero index.     

Spectral analysis of nonselfadjoint Schrdinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schrodinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrodinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrodinger boundary value problem are given.     

Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation

An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-sided estimates for the norms of values of a nonlinear operator in terms of those of a corresponding linear operator are obtained.On this basis, two-sided estimates for the modulus of continuity of a nonlinear inverse problem in terms of that of a corresponding linear problem are obtained. A method of auxiliary boundary conditions is used to construct stable approximate solutions to the nonlinear inverse problem. An accurate (to an order) error estimate for the method of auxiliary boundary conditions is obtained on a uniform regularization class.     

Solvability of a boundary value problem for a stationary model of the magnetic hydrodynamics of a viscous heat-conducting fluid

A boundary value problem is studied for a stationary model of the magnetic hydrodynamics of a viscous heat-conducting fluid under nonhomogeneous boundary conditions on the velocity, electromagnetic field, and temperature. The model consists of the Navier-Stokes equations, the Maxwell equations, the generalized Ohm law, and the convection-diffusion equation for the temperature which are connected nonlinearly with each other. Sufficient conditions on the initial data are established that guarantee the global solvability of the problem under consideration and the local uniqueness of its solution. The properties are studied of the linear operator obtained by linearizing the operator of the original boundary value problem.     

Finite difference preconditioning cubic spline collocation method of elliptic equations

Summary. We discuss a finite difference preconditioner for the interpolatory cubic spline collocation method for a uniformly elliptic operator defined by in (the unit square) with homogeneous Dirichlet boundary conditions. Using the generalized field of values arguments, we discuss the eigenvalues of the preconditioned matrix where is the matrix of the collocation discretization operator corresponding to , and is the matrix of the finite difference operator corresponding to the uniformly elliptic operator given by in with homogeneous Dirichlet boundary conditions. Finally we mention a bound of -singular values of for a general elliptic operator in . Received December 11, 1995 / Revised version received June 20, 1996     

Necessary and sufficient conditions for the solvability of the inverse problem for the matrix Sturm-Liouville operator

The matrix Sturm-Liouville operator on a finite interval with Dirichlet boundary conditions is studied. Properties of its spectral characteristics and the inverse problem of recovering the operator from these characteristics are investigated. Necessary and sufficient conditions on the spectral data of the operator are obtained. Research is conducted in the general case, with no a priori restrictions on the spectrum. A constructive algorithm for solving the inverse problem is provided.     

Partial radiation conditions and hypersingular integral operators

The structure of the partial radiation conditions is analyzed. It is found that the principal part of the operator defining these conditions defines a hypersingular operator. The series corresponding to the operator associated with the boundary conditions is transformed to a rapidly converging series.     

Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution

We consider the spectral problem for a model second-order differential operator with an involution. The operator is given by the differential expression Lu = ?u??(?x) and boundary conditions of general form. We obtain a criterion for the basis property of the systems of eigenfunctions of this operator in terms of the coefficients in the boundary conditions.     

Estimates of Root Functions of the Adjoint of a Second-Order Differential Operator with Integral Boundary Conditions

We consider a second-order differential operator on an interval of the real line with integral boundary conditions and the adjoint of this operator. We obtain a priori estimates of the eigenfunctions and associated functions of the adjoint operator.     

Strict inequality of Robin eigenvalues for elliptic differential operators on Lipschitz domains

On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions. We prove that inequality between these functions on the boundary implies strict inequality between the eigenvalues of the two operators, provided that the inequality of the functions in the boundary conditions is strict on an arbitrarily small nonempty, open set.