Nonselfadjoint Boundary Value Problems Research Articles

Complex Resonance As a Self-Consistent Electromagnetic Process

The phenomenon, which is classified as a complex resonance and takes place in electromagnetic structures described by nonself-adjoint boundary value problems, is considered. It is shown that the oscillation adjoint to a source is formed, when complex waves that are complex conjugate by wave numbers and amplitudes are excited in couples. This oscillation is described by the self-consistent boundary value problem for the adjoint Helmholtz equation, i.e., for the equation, whose right-hand side is the solution of the homogeneous boundary value problem. The distinctive feature of the complex resonance is in the fact that it exists in the whole range of complex waves, when the source is necessarily present. Theoretical and experimental results of the investigation of the considered phenomenon are presented.

Biorthogonal Integral Transformations in the Problems of Structural Mechanics of Shells

The paper deals with generalised vector finite integral transformation designed for the solution of initial boundary value problems described by systems of linear differential equations in partial derivatives. Described structural algorithm method provides for the construction of adjoint and invariant operators, that in contrast to the method of eigen function expansion can be considered non-selfadjoint boundary value problems with asymmetric matrix coefficients in the system of equations. An essential feature of the proposed method is the ability to produce all the components of the solution including two vectors of sound functions without any a priori information. The proposed procedure is demonstrated as a new method of analytical solution of the nonstationary problem for a shell of revolution with an arbitrary meridian.

An inverse problem for a generalized fractional diffusion

We propose a method for determining the solution and source term of a generalized time-fractional diffusion equation. The method is based on selecting a bi-orthogonal basis of L2 space corresponding to a nonself-adjoint boundary value problem. Uniqueness is proven and an existence result is obtained for smooth initial and final conditions. The asymptotic behavior of the generalized Mittag–Leffler function is used to relax the smoothness requirement on these conditions.

A class of essentially nonself-adjoint boundary value problems

A class of essentially nonself-adjoint boundary value problems

Convergence analysis of HSS-multigrid methods for second-order nonselfadjoint elliptic problems

In this paper, the multigrid methods using Hermitian/skew-Hermitian splitting (HSS) iteration as smoothers are investigated. These smoothers also include the modified additive and multiplicative smoothers which result from subspace decomposition. Without full elliptic regularity assumption, it is shown that the multigrid methods with these smoothers converge uniformly for second-order nonselfadjoint elliptic boundary value problems if the mesh size of the coarsest grid is sufficiently small (but independent of the number of the multigrid levels). Numerical results are reported to confirm the theoretical analysis.

On an A Priori Estimate for Solutions of an Elliptic Equation

In the investigation of the spectral theory of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises the problem of obtaining Lp a priori estimates for solutions about points of discontinuity of the weight function. Here we deal with this problem for the case where the weight function vanishes on a set of positive measure.

Spectral analysis for differential operators with singularities

Nonselfadjoint boundary value problems for second‐order differential equations on a finite interval with nonintegrable singularities inside the interval are considered under additional sewing conditions for solutions at the singular point. We study properties of the spectrum, prove the completeness of eigen‐ and associated functions, and investigate the inverse problem of recovering the boundary value problem from its spectral characteristics.

Asymptotic study of the elastic seadbed effects in ocean acoustics

A theoretical and asymptotic investigation of the Green' function for the system governing the propagation of time-harmonic acoustic waves in a horizontally stratified ocean with an elastic seabed is presented. Employing the surface Neumann-to-Dirichlet map for the elastic half space, we reduce the problem to an equivalent one in the layer, with a nonlocal boundarycondition at the fluid-bottom interface. The reduced problem is transformedby Hankel transform, to a non-selfadjoint boundary value problem for a second-order ordinary differential equation over the layer depth. The well posedness of this problem is investigated applying analytic Redholm theory for an equivalent Lippmann-Schwinger integral equation. An asymptotic expansionof the transformed nonlocal boundary condition is constructed in the case of a seabed with small shear modulus, and it is used to show that the Green function is a regular perturbation of that one in the case of a fluid bottom.

Vibration of rods with a concentrated mass in the presence of non-conservative forces

Separable and non-selfadjoint boundary-value problems representing the vibration of linearly elastic unidimensional systems are considered. The elastic system is modelled as a continuous distributed-parameter system where singularities in the mass distribution function can be neatly taken into account. Specifically, extending Green's function approach, the free vibration, stability and forced vibration of fixed-free rods with a tip mass and under the action of uniformly distributed non-conservative loads have been investigated analytically.

A priori bounds for solutions of an elliptic equation

SynopsisIn the general theory of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises the problem of obtaining a priori estimates for solutions about points of discontinuity of the weight function. Here we deal with this problem for the case where the weight function vanishes on a set of positive measure.

Nonselfadjoint boundary value problems at resonance with nonlinearities which may grow linearly

Nonselfadjoint boundary value problems at resonance with nonlinearities which may grow linearly

Stability of a rotating, pre-twisted, non-uniform cantilever beam with tip mass and thermal gradient subjected to a non-conservative force

The effect of thermal gradient and tangency coefficient on the stability of a pre-twisted, tapered, rotating cantilever with a tip mass and subjected to a concentrated partial follower force at the free end is investigated. The non-selfadjoint boundary value problem is formulated with the aid of a conservation law using Euler-Bernoulli theory. The associated adjoint boundary value problem is introduced and an apposite variational principle is derived. Approximate values of critical load are calculated on the basis of this variational principle and the influence of different parameters on the stability of the system is studied.

On the existence of positive solutions for a class of non–selfadjoint elliptic boundary value problems

We consider a class of non-selfadjoint el1iptic boundary value problems on a bounded domain. Combining results on weighted eigenvalue problems for non-selfadjoint operators with a general form 0f the method of sub-supersolutions we prove existence of a positive solution provided that the nonlinearity f(x,u) satisfies a suitable "crossing condition" when u varies from zero to infinity

Galerkin's method in non-selfadjoint boundary value problems of the theory of waveguides

A generalization of the incomplete Galerkin's method to complex spaces with a pseudo scalar product is given, in which the basis system of the complex eigenfunctions is orthogonal. Using this method a numerical algorithm is constructed, and its efficiency is demonstrated using the example of the solution of the non-selfadjoint boundary value problem of the theory of a plane impedance waveguide.

Existence of solutions to nonselfadjoint boundary value problems for ordinary differential equations

Existence of solutions to nonselfadjoint boundary value problems for ordinary differential equations

The discrete spectrum of natural waves of a screened open slot line

The method of operator functions is used to prove the existence of a discrete spectrum of the non-selfadjoint boundary value problem for the system of Helmholtz equations with non-linear dependence on the spectral parameter describing electromagnetic wave propagation in a screened open slot line.

Boundary value problems in transport with oblique and mixed derivative boundary conditions: More on steady state solutions

Boundary value problems involving oblique or mixed second derivative boundary conditions[1,3] arise in the description of heat (or mass) transfer in stationary media which are peripherally cooled (or leached) by a fluid in which the temperature (or concentration) is varying significantly in the direction of flow. By decomposing the energy equation into a pair of first order systems Ramkrishna and Amundson have shown[1] how the originally non-selfadjoint boundary value problems can be reduced to selfadjoint problems. The foregoing decomposition allows for an elegant analytical solution of the boundary value problems which is also numerically efficient. This paper relaxes certain redundant constraints imposed earlier on the method of solutions[1] and demonstrates the power of the method by actual numerical calculations for a constant heat source function.

Spectral analysis of a singular nonself-adjoint boundary value problem

Spectral analysis of a singular nonself-adjoint boundary value problem

On the order of convergence of finite element approximations to solutions of non-selfadjoint elliptic boundary value problems

On the order of convergence of finite element approximations to solutions of non-selfadjoint elliptic boundary value problems

Expansions associated with nonselfadjoint boundary value problems

Eigenfunction expansions are given for a class of non-self-adjoint boundary-value problems. The results apply to systems of ordinary differential equations, some of whose coefficients may be arbitrary bounded Hilbert space operators.