Linear and nonlinear nonlocal boundary value problems for differential-operator equations

This study focuses on nonlocal boundary value problems (BVPs) for linear and nonlinear elliptic differential-operator equations (DOEs) that are defined in Banach-valued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal L_p -regularity and Fredholmness of linear BVPs, uniformly with respect to this parameter, are presented. This fact implies that the appropriate differential operator is a generator of an analytic semigroup. Then, by using these results, the existence, uniqueness and maximal smoothness of solutions of nonlocal BVPs for nonlinear DOEs are shown. These results are applied to nonlocal BVPs for regular elliptic partial differential equations, finite and infinite systems of differential equations on cylindrical domains, in order to obtain the algebraic conditions that guarantee the same properties.     

Exact modes for post-buckling characteristics of nonlocal nanobeams in a longitudinal magnetic field

An exact mode solution that investigates the prebuckling and postbuckling characteristics of nonlocal nanobeams with fixed–fixed, hinged–hinged, and fixed–hinged boundary conditions in a longitudinal magnetic field is determined. The geometric nonlinearity arising from mid-plane stretching is considered to obtain the nonlinear governing equation of motion by virtue of Hamilton's principle. The influences of the nonlocal and magnetic parameters on the prebuckling and postbuckling dynamics of nanobeams with various boundary conditions are evaluated, indicating that the critical buckling force can be decreased with the increase of the nonlocal parameter while can be increased with increasing the magnetic parameter. It is demonstrated that the first natural frequency of the nanobeam with fixed–fixed and fixed–hinged conditions in postbuckling configuration is increased from zero to a constant value for higher values of the nonlocal parameter with increasing the axial force. The second natural frequency of the buckled nanobeam is always decreased with an increase of the nonlocal parameter. The results show that the internal resonance between the first and second modes of the postbuckling nanobeams can be quickly and easily activated by increasing the nonlocal parameters, especially for fixed–fixed and hinged–hinged boundary conditions. In addition, the results obtained by exact mode solution are compared those obtained by classical mode solution. It is found that the classical mode is valid only for nonlocal nanobeams with the hinged–hinged boundary conditions.     

On the stability of the index of unbounded nonlocal operators in Sobolev spaces

Unbounded operators corresponding to nonlocal elliptic problems on a bounded region G ⊂ ℝ² are considered. The domain of these operators consists of functions in the Sobolev space W ₂ ^m (G) that are generalized solutions of the corresponding elliptic equation of order 2m with the right-hand side in L ₂(G) and satisfy homogeneous nonlocal boundary conditions. It is known that such unbounded operators have the Fredholm property. It is proved that lower order terms in the differential equation do not affect the index of the operator. Conditions under which nonlocal perturbations on the boundary do not change the index are also formulated. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 116–135.     

Solution of the Bitsadze–Samarskii problem for a parabolic system in a semibounded domain

We consider a nonlocal initial–boundary value Bitsadze–Samarskii problem for a spatially one-dimensional parabolic second-order system in a semibounded domain with nonsmooth lateral boundary. The boundary integral equation method is used to construct a classical solution of this problem under the condition that the vector function on the right-hand side in the nonlocal boundary condition only has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

Decay solution for the renewal equation with diffusion

In this paper we consider age structured equation with diffusion under nonlocal boundary condition and nonnegative initial data. We prove existence, uniqueness and the positivity of the solution to the above problem. Our main result is to get an exponential decay of the solution for large times toward such a study state. To this end we prove a weighted Poincaré–Wirtinger’s type inequality in unbounded domain.     

On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators

We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domain H ⊂ ℝ _σ ^m . A criterion for the existence and uniqueness of solutions and for the continuous dependence of the solution on the boundary function is established. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1131 – 1136, August, 2005.     

On the existence of a Feller semigroup with atomic measure in a nonlocal boundary condition

The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. An elliptic operator of second order is considered on a plane bounded region G. Its domain of definition consists of continuous functions satisfying a nonlocal condition on the boundary of the region. In general, the nonlocal term is an integral of a function over the closure of the region G with respect to a nonnegative Borel measure μ(y, dη) ∈ ∂G. It is proved that the operator is a generator of a Feller semigroup in the case where the measure is atomic. The smallness of the measure is not assumed. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 164–179.     

Stability of a family of weighted finite-difference schemes

Existence and uniqueness theorems are proved for a weighted finite-difference scheme approximating the heat equation with a nonlocal boundary condition containing a parameter. Bounds are derived that guarantee stability of the solution in the initial values in the mean-square grid norm. Translated from Prikladnaya Matematika i Informatika, No. 29, 2008, pp. 64–87.     

Effects of boundary conditions on pattern formation in a nonlocal prey–predator model

Effects of periodic and Neumann boundary conditions on a nonlocal prey–predator model are investigated. Two types of kernel functions with finite supports are used to characterize the nonlocal interactions. These kernel functions are modified to handle the Neumann boundary condition. Numerical techniques to find the Turing and spatial-Hopf thresholds for Neumann boundary condition are also described. For a fixed range of nonlocal interaction with a given kernel function, Turing bifurcation curves corresponding to both the boundary conditions are close to each other. The same is true for the spatial-Hopf bifurcation curves too. However, the nonlinear solutions inside the Turing domain as well as spatial-Hopf domain depend on the boundary condition. Thus, boundary conditions play important roles in a nonlocal model of prey-predator interaction.     

Nonlinear localized waves in a medium with nonlocal interaction

Soliton solutions are found for nonlinear integro-differential equations with a type λ/(τ-τ′) kernel used to describe particle tunneling and magnetic and superconducting vortices in a medium with nonlocal interaction. The Fourier transform method is applied to derive asymptotic formulas for even and odd localized solutions. Analytical solutions are found for particular parameter values. A complete pattern is constructed for the behavior of soliton solutions in an arbitrary range of the interaction parameter λ by means of numerical calculations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 3, pp. 366–379, March, 1998.     

Stability of a nonlocal difference problem with a complex parameter

We consider a difference scheme for the heat equation with nonlocal boundary conditions containing a complex parameter γ. We single out stability domains in the complex γ-plane and derive a criterion for stability with respect to the initial data in the form of a condition on γ and the grid increments.     

Green’s functions for stationary problems with nonlocal boundary conditions

In this paper, we investigate Green’s functions for various stationary problems with nonlocal boundary conditions. We express the Green’s function per Green’s function for a problem with classical boundary conditions. This property is illustrated by various examples. Properties of Green’s functions with nonlocal boundary conditions are compared with those for classical problems. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-73/09.     

Automatic Control via Thermostats of a Hyperbolic Stefan Problem with Memory

A hyperbolic Stefan problem based on the linearized Gurtin—Pipkin heat conduction law is considered. The temperature and free boundary are controlled by a thermostat acting on the boundary. This feedback control is based on temperature measurements performed by real thermal sensors located within the domain containing the two-phase system and/or at its boundary. Three different types of thermostats are analyzed: simple switch, relay switch, and a Preisach hysteresis operator. The resulting models lead to integrodifferential hyperbolic Stefan problems with nonlinear and nonlocal boundary conditions. Existence results are proved in all the cases. Uniqueness is also shown, except in the situation corresponding to the ideal switch. Accepted 27 May 1997     

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models

The free vibration response of single-walled carbon nanotubes (SWCNTs) is investigated in this work using various nonlocal beam theories. To this end, the nonlocal elasticity equations of Eringen are incorporated into the various classical beam theories namely as Euler-Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Reddy beam theory (RBT) to consider the size-effects on the vibration analysis of SWCNTs. The generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations of each nonlocal beam theory corresponding to four commonly used boundary conditions. Then molecular dynamics (MD) simulation is implemented to obtain fundamental frequencies of nanotubes with different chiralities and values of aspect ratio to compare them with the results obtained by the nonlocal beam models. Through the fitting of the two series of numerical results, appropriate values of nonlocal parameter are derived relevant to each type of chirality, nonlocal beam model, and boundary conditions. It is found that in contrast to the chirality, the type of nonlocal beam model and boundary conditions make difference between the calibrated values of nonlocal parameter corresponding to each one.     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

On a Time-Optimal Control Problem Associated with the Heat Exchange Process

The initial-boundary problem for the heat conduction equation inside a bounded domain is considered. It is supposed that on the boundary of this domain the heat exchange according to Newton’s law takes place. The control parameter is equal to the magnitude of output of hot or cold air and is defined on a given part of the boundary. An estimate of the minimal time for achieving the given average temperature is found.     

A fractional nonlocal time-space viscoelasticity theory and its applications in structural dynamics

To overcome the long wavelength and time limits of classical elastic theory, this paper presents a fractional nonlocal time-space viscoelasticity theory to incorporate the non-locality of both time and spatial location. The stress (strain) at a reference point and a specified time is assumed to depend on the past time history and the stress (strain) of all the points in the reference domain through nonlocal kernel operators. Based on an assumption of weak non-locality, the fractional Taylor expansion series is used to derive a fractional nonlocal time-space model. A fractional nonlocal Kevin–Voigt model is considered as the simplest fractional nonlocal time-space model and chosen to be applied for structural dynamics. The correlation between the intrinsic length and time parameters is discussed. The effective viscoelastic modulus is derived and, based on which, the tension and vibration of rods and the bending, buckling and vibration of beams are studied. Furthermore, in the context of Hamilton’s principle, the governing equation and the boundary condition are derived for longitudinal dynamics of the rod in a more rigorous manner. It is found that when the external excitation frequency and the wavenumber interact with the intrinsic microstructures of materials and the intrinsic time parameter, the nonlocal space-time effect will become substantial, and therefore the viscoelastic structures are sensitive to both microstructures and time.