Spectrum comparison for a conserved reaction-diffusion system with a variational property

We are dealing with a two-component system of reaction-diffusion equations with conservation of a mass in a bounded domain subject to the Neumann or the periodic boundary conditions. We consider the case that the conserved system is transformed into a phase-field type system. Then the stationary problem is reduced to that of a scalar reaction-diffusion equation with a nonlocal term. We study the linearized eigenvalue problem of an equilibrium solution to the system, and compare the eigenvalues with ones of the linearized problem arising from the scalar nonlocal equation in terms of the Rayleigh quotient. The main theorem tells that the number of negative eigenvalues of those problems coincide. Hence, a stability result for nonconstant solutions of the scalar nonlocal reaction-diffusion equation is applicable to the system.     

A Nonlocal Finite-Difference Boundary-Value Problem

We investigate the stability of difference schemes for the equation of heat conduction with nonlocal boundary conditions. An example is given which in a certain sense imitates the problem with variable coefficients and has an exact solution in analytical form. It is shown that the difference operator has a simple spectrum and that multiple eigenvalues appear only in the case with constant coefficients. The simple spectrum ensures that the eigenvectors of the finite-difference problem form a basis. This enables us to apply to the nonlocal problem the theory of stability of symmetrizable difference schemes.     

Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.     

Alternating direction method for the Poisson equation with variable weight coefficients in an integral condition

The Peaceman-Rachford alternating direction method is used to solve a system of difference equations approximating the Poisson equation in a rectangular domain with integral conditions with fourth-order accuracy. The convergence of the iterative method is studied on the basis of an analysis of the spectrum structure of a one-dimensional difference operator with a nonlocal condition. We study the dependence of the spectrum on the weight functions occurring in the integral conditions. In particular, we discuss the presence of complex eigenvalues with negative real parts in the spectrum of the difference operator with a nonlocal condition. The results of a numerical experiment are presented.     

Locally one-dimensional difference scheme for a pseudoparabolic equation with nonlocal conditions

We study a two-dimensional linear pseudoparabolic equation with nonlocal integral boundary conditions in one coordinate direction and use a locally one-dimensional method for solving this problem. We prove the stability of a finite-difference scheme based on the structure of spectrum of the difference operator with nonlocal conditions.     

Note on a nonlocal Sturm–Liouville problem with both right and left fractional derivatives

In this paper, we research the geometric multiplicity of eigenvalues for a nonlocal Sturm–Liouville eigenvalue problem. To this end, we study the uniqueness of solutions for a nonlocal Sturm–Liouville problem under some initial value conditions.     

Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions

In a rectangular domain, we consider the two-dimensional Poisson equation with nonlocal boundary conditions in one of the directions. For this problem, we construct a difference scheme of fourth-order approximation, study its solvability, and justify an iteration method for solving the corresponding system of difference equations. We give a detailed study of the spectrum of the matrix representing this system. In particular, we obtain a criterion for the nondegeneracy of this matrix and conditions for its eigenvalues to be positive.     

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.     

Nonlinear spectral problem for a singular system of ordinary differential equations with a nonlocal condition

We consider a nonlinear spectral problem for a system of ordinary differential equations defined on an unbounded half-line and supplemented with a nonlocal condition specified by a Stieltjes integral. We suggest a numerically stable method for finding the number of eigenvalues lying in a given bounded domain of the complex plane and for the computation of these eigenvalues and the corresponding eigenfunctions. Our approach uses a simpler (with uncoupled boundary conditions) auxiliary boundary value problem for the same equation.     

Spectral properties of a nonlocal second-order difference operator

We consider a spectral problem for a nonlocal difference operator of second derivative with variable coefficients and with a complex parameter in the boundary condition. We study the algebraic and geometric multiplicity of the eigenvalues and the sign of their real part. We obtain conditions on the parameter which ensure that the entire spectrum of the operator lies in the right complex half-plane.     

On the spectral stability in subspaces for difference schemes with nonlocal boundary conditions

We consider an initial-boundary value problem for the heat equation with nonlocal boundary conditions containing a parameter γ > 1. The spectrum of the main differential operator contains some number (depending on γ) of eigenvalues lying in the left complex half-plane, which results in the instability of the problem with respect to the initial data. For difference schemes approximating the original problem, we obtain a criterion for stability in the subspaces generated by stable harmonics.     

A fourth-order alternating-direction method for difference schemes with nonlocal condition*

We consider a finite-difference scheme of fourth-order accuracy for the two-dimensional Poisson equation in a rectangular domain with nonlocal integral conditions in one coordinate direction. The system of finite-difference equations is solved using a generalization of the Peaceman–Rachford alternating-direction implicit method. We prove the convergence of the method and estimate the rate of convergence by using the structure of the spectrum of one-dimensional difference operators with nonlocal integral conditions.     

Stability properties of autonomous homogeneous polynomial differential systems

A geometrical approach is used to derive a generalized characteristic value problem for dynamic systems described by homogeneous polynomials. It is shown that a nonlinear homogeneous polynomial system possesses eigenvectors and eigenvalues, quantities normally associated with a linear system. These quantities are then employed in studying stability properties. The necessary and sufficient conditions for all forms of stabilities characteristic of a two-dimensional system are provided. This result, together with the classical theorem of Frommer, completes a stability analysis for a two-dimensional homogeneous polynomial system.     

Eigenvalue Problem for the Laplace Operator with Nonlocal Boundary Conditions

The suggested approach to maximizing the difference between the first and second eigenvalues of the Laplace operator is based on the introduction of nonlocal boundary conditions of a special form. It is shown that the difference can be arbitrarily large.     

On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems

We deal with the stability analysis of difference schemes for a one-dimensional parabolic equation subject to integral conditions. It is based on the spectral structure of the transition matrix of a difference scheme. The stability domain is defined by using the hyperbola which is the locus of points where the transition matrix has trivial eigenvalues. The stability conditions obtained are much more general compared with those known in the literature. We analyze three separate cases of nonlocal integral conditions and solve an example illustrating the efficiency of the technique.     

The Difference Laplace Operator on a Seven-Point Nonorthogonal Pattern of a Rectangular Grid and Its Spectral Properties

We found an explicit form of eigenfunctions and eigenvalues of the difference Laplace operator on a seven-point nonorthogonal pattern of a rectangular grid for the second boundary-value problem. Estimates for maximal and minimal eigenvalues are found. Dispersion properties of an explicit difference scheme are studied for the two-dimensional wave equation. The scheme uses an approximation of the Laplace operator on a seven-point pattern.     

Nonlinear eigenvalue problem for a system of ordinary differential equations subject to a nonlocal condition

For a system of linear ordinary differential equations supplemented with a nonlocal condition specified by the Stieltjes integral, the problem of calculating the eigenvalues belonging to a given bounded domain in the complex plane is examined. It is assumed that the coefficient matrix of the system and the matrix function in the Stieltjes integral are analytic functions of the spectral parameter. A numerically stable method for solving this problem is proposed and justified. It is based on the use of an auxiliary boundary value problem and formulas of the argument principle type. The problem of calculating the corresponding eigenfunctions is also treated.     

Smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem

The implicit function theorem is applied in a nonstandard way to abstract variational inequalities depending on a (possibly infinite-dimensional) parameter. In this way, results on smooth continuation of solutions as well as of eigenvalues and eigenvectors are established under certain particular assumptions. The abstract results are applied to a linear second order elliptic eigenvalue problem with nonlocal unilateral boundary conditions (Schrödinger operator with the potential as the parameter).     

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory

In this article, the flutter vibrations of fluid-conveying thin-walled nanotubes subjected to magnetic field is investigated. For modeling fluid structure interaction, the nonlocal strain gradient thin-walled Timoshenko beam model, Knudsen number and magnetic nanoflow are assumed. The Knudsen number is considered to analyze the slip boundary conditions between the fluid-flow and the nanotube's wall, and the average velocity correction parameter is utilized to earn the modified flow velocity of nano-flow. Based on the extended Hamilton's principle, the size-dependent governing equations and associated boundary conditions are derived. The coupled equations of motion are transformed to a general eigenvalue problem by applying extended Galerkin technique under the cantilever end conditions. The influences of nonlocal parameter, strain gradient length scale, magnetic nanoflow, longitudinal magnetic field, Knudsen number on the eigenvalues and critical flutter velocity of the nanotubes are studied.