bluuebunny/crossref_metadata_2025 huggingface dataset:
The time-dependent Stokes paradox
Authors: Smith S. H.
Published: December 2016

When a uniform stream starts to flow impulsively at time

t
=
0

t = 0

past a two-dimensional body, the solution of the Stokes equation indicates that the velocity grows without bound as

t
→
∞

t \to \infty

. It is seen that this is a natural extension of the Stokes paradox to unsteady flows. Two resolutions of this result are presented: firstly, through an analysis of the Oseen equation on the basis of singular perturbation theory and, secondly, through considering the two-dimensional body as the limit of a three-dimensional body when the length increases without bound.
Lyapunov stability via differential moments
Authors: Charron R. J.
Published: December 2016

We show how differential moments can be used to construct Lyapunov functions for general, autonomous and nonautonomous, second- and third-degree ordinary differential equations. In certain instances, one can extend classical results to sequences of Lyapunov functions.
A 3-dimensional hyperbolic Stefan problem with discontinuous temperature
Authors: Li De Ning
Published: December 2016

The hyperbolic heat transfer model is obtained by replacing the classical Fourier’s law with the relaxation relation

τ

q
t

+
q
=
−
k
∇
T

\tau {q_t} + q = - k\nabla T

. The conditions are derived for the local existence and uniqueness of classical solutions for a 3-dimensional Stefan problem of hyperbolic heat transfer model where the temperature may sustain a jump across the phase change interface.
Voltage-current characteristics of multidimensional semiconductor devices
Authors: Schmeiser Christian
Published: December 2016

The steady state drift-diffusion model for the flow of electrons and holes in semiconductors is simplified by perturbation techniques. The simplifications amount to assuming zero space charge and low injection. The limiting problems are solved and explicit formulas for the voltage-current characteristics of bipolar devices can be obtained. As examples, the

p
n

pn

-diode, the bipolar transistor and the thyristor are discussed. While the classical results of a one-dimensional analysis are confirmed in the case of the diode, some important effects of the higher dimensionality appear for the bipolar transistor.
Unsteady diffusion from a source in uniform stream
Authors: Wang C. Y.
Published: December 2016

Basic exact solutions for an unsteady point heat source in a uniform stream are presented. The governing equation is solved using changes of variables and Laplace transform. Both diffusion and convection are important. The periodic source sheds a trail of thermal energy packets, alternating and rapidly decreasing in strength. The suddenly started source reaches steady state in a time of 4(diffusivity)/(velocity)

2

^{2}

.
A nonstationary problem in the theory of electrolytes
Authors: Krzywicki A. , Nadzieja T.
Published: December 2016

The equations describing the density of ions which appear in the theory of electrolytes take the form

f
t

=

f

x
x

+

(

f

u
x

)

x

,

u

x
x

=
−
f

{f_t} = {f_{xx}} + {\left ( {f{u_x}} \right )_x}, {u_{xx}} = - f

, in the one-dimensional case. In the paper the existence of solutions and their behaviour as time goes to infinity is discussed.
On the propagation of maximally dissipative phase boundaries in solids
Authors: Abeyaratne Rohan , Knowles James K.
Published: December 2016

This paper is concerned with the kinetics of propagating phase boundaries in a bar made of a special nonlinearly elastic material. First, it is shown that there is a kinetic law of the form

f
=
φ

(

s
˙

)

f = \varphi \left ( {\dot s} \right )

relating the driving traction

f
f

at a phase boundary to the phase boundary velocity

s
˙

\dot s

that corresponds to a notion of maximum dissipation analogous to the concept of maximum plastic work. Second, it is shown that a modified version of the entropy rate admissibility criterion can be described by a kinetic relation of the above form, but with a different

φ
\varphi

. Both kinetic relations are applied to the Riemann problem for longitudinal waves in the bar.
Oscillations and global attractivity in a discrete delay logistic model
Authors: Kuruklis S. A. , Ladas G.
Published: December 2016

Consider the discrete delay logistic model
\[

N

t
+
1

=

α

N
t

1
+
β

N

t
−
k

,

(
1
)

{N_{t + 1}} = \frac {{\alpha {N_t}}}{{1 + \beta {N_{t - k}}}}, \qquad \left ( 1 \right )

\]
where

α
∈

(

1
,
∞

)

,
β
∈

(

0
,
∞

)

\alpha \in \left ( {1, \infty } \right ), \beta \in \left ( {0, \infty } \right )

, and

k
∈

N

=

{

0
,
1
,
2
,
.
.
.

}

k \in \mathbb {N} = \left \{{0, 1, 2,...} \right \}

. We obtain conditions for the oscillation and asymptotic stability of all positive solutions of Eq. (1) about its positive equilibrium

(

α
−
1

)

/

β

\left ( {\alpha - 1} \right )/\beta

. We prove that all positive solutions of Eq. (1) are bounded and that for

k
=
0

k = 0

and

k
=
1

k = 1

the positive equilibrium

(

α
−
1

)

/

β

\left ( {\alpha - 1} \right )/\beta

is a global attractor.
Local energy decay for the damped plate equation in exterior domains
Authors: Jiang Song
Published: December 2016

We establish a rate of local energy decay for solutions of the damped plate equation with variable coefficients in exterior domains by using the spectral analysis to the corresponding stationary problem.
Some remarks on the Stefan problem with surface structure
Authors: Gurtin Morton E. , Soner H. Mete
Published: December 2016

This paper discusses a generalized Stefan problem which allows for supercooling and superheating and for capillarity in the interface between phases. Simple solutions are obtained indicating the chief differences between this problem and the classical Stefan problem. A weak formulation of the general problem is given.
On two circular inclusions in harmonic problems
Authors: Honein E. , Honein T. , Herrmann G.
Published: December 2016

In this paper, we derive the solution for two circular cylindrical elastic inclusions perfectly bonded to an elastic matrix of infinite extent, under anti-plane deformation. The two inclusions have different radii and possess different elastic properties. The matrix is subjected to arbitrary loading. The solution is obtained, via iterations of Möbius transformations, as a rapidly convergent series with an explicit general term involving the complex potential of the corresponding homogeneous problem, i.e., when the inclusions are absent and the matrix material occupies the entire space and is subjected to the same loading. This procedure has been termed “heterogenization."
Comparison principle for some nonlocal problems
Authors: Deng Keng
Published: December 2016

In this paper, for the parabolic equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline equals normal upper Delta u plus g left-parenthesis x comma u right-parenthesis comma left-parenthesis x comma t right-parenthesis element-of normal upper Omega times left-parenthesis 0 comma upper T right-parenthesis">

u
t

=
Δ
u
+
g

(

x
,
u

)

,

(

x
,
t

)

∈

Ω
×

(

0
,
T

)

{u_t} = \Delta u + g\left ( {x, u} \right ), \left ( {x, t} \right ) \in \\ \Omega \times \left ( {0, T} \right )

, with nonlocal boundary conditions

u

|

∂
Ω

=

∫

Ω

f

(

x
,
y

)

u

(

y
,
t

)

d
y

u\left | {_{\partial \Omega }} \right . = \int _{\Omega } f\left ( {x, y} \right )u\left ( {y, t} \right )dy

, we establish the comparison theorem and local existence of the solution. We also discuss its long time behavior.
Stokes flow in a two-dimensional spatially periodic boundary driven by a combination of line singularities
Authors: Ma K. , Pravica D. W.
Published: December 2016

The analytical solution is obtained for two-dimensional creeping flow generated by a source-sink combination located on a spatially periodic boundary. Separation is found to occur when the source and sink are located on concave regions of the boundary. The velocity profile of the two-dimensional Poiseuille flow for a channel is compared to that of the circle problem and the rough channel obtained from the analytical solution. It is found that the mid-channel velocity (or pressure gradient) of the rough case is greater (less) than that of the circle problem.
Parameter identification for an abstract Cauchy problem by quasilinearization
Authors: Brewer Dennis W. , Burns John A. , Cliff Eugene M.
Published: December 2016

A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasilinearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations.
Propagation of a combustion front in a striated solid medium: a homogenization analysis
Authors: Brauner C.-M. , Fife P. C. , Namah G. , Schmidt-Lainé C.
Published: December 2016

A model problem for the propagation of a combustion front through a periodically inhomogeneous medium is posed. The existence of a steady state solution is proved, in which the front’s velocity is periodic in time. Computer simulations are carried out. Finally, through rigorous homogenization techniques, it is shown that when the wavelength of the inhomogeneity is small, the solution may be approximated by a travelling wave solution of the corresponding problem for a medium with certain constant properties.
The use of Delaunay curves for the wetting of axisymmetric bodies
Authors: Basa P. , Schön J. C. , Salamon P.
Published: December 2016

The wetting of a given solid figure is of importance in many fields of science ranging from physics and materials science to geology and medicine. An important special case that is generic to many situations is the wetting of an axisymmetric solid by a liquid of the same material. This problem is equivalent to minimizing the total surface area of the condensed phase (liquid + solid). Its solution is a mosaic of wet and dry regions on the solid. The shape of the wet regions is described by Delaunay curves. The analytic properties of these curves are discussed, and the wetting of several interesting solid configurations is presented.
Asymptotic solutions for double pendant and extended sessile drops
Authors: O’Brien S. B. G. M.
Published: December 2016

Asymptotic solutions are obtained for small double pendant drops which are shown to consist of two (or more) nearly spherical drops joined together by a thin neck region, the upper drop being slightly larger. Equivalent solutions for so-called extended sessile drops represent the curling of the drop profile in on itself beyond the point where the inclination of the drop becomes greater than

π
\pi

radians.
Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates
Authors: Dassios G. , Hadjinicolaou M. , Payatakes A. C.
Published: December 2016

The stream function

ψ
\psi

for axisymmetric Stokes flow satisfies the well-known equation

E
4

ψ
=
0

{E^4}\psi = 0

. In spheroidal coordinates the equation

E
2

ψ
=
0

{E^2}\psi = 0

admits separable solutions in the form of products of Gegenbauer functions of the first and second kind, and the general solution is then represented as a series expansion in terms of these eigenfunctions. Unfortunately, this property of separability is not preserved when one seeks solutions of the equation

E
4

ψ
=
0

{E^4}\psi = 0

. The nonseparability of

E
4

ψ
=
0

{E^4}\psi = 0

in spheroidal coordinates has impeded considerably the development of theoretical models involving particle-fluid interactions around spheroidal objects. In the present work the complete solution for

ψ
\psi

in spheroidal coordinates is obtained as follows. First, the generalized 0-eigenspace of the operator

E
2

{E^2}

is investigated and a complete set of generalized eigenfunctions is given in closed form, in terms of products of Gegenbauer functions with mixed order. The general Stokes stream function is then represented as the sum of two functions: one from the 0-eigenspace and one from the generalized 0-eigenspace of the operator

E
2

{E^2}

. A rearrangement of the complete expansion, in such a way that the angular-type dependence enters through the Gegenbauer functions of successive order, leads to some kind of semiseparable solutions, which are given in terms of full series expansions. The proper solution subspace that provides velocity and vorticity fields, which are regular on the axis, is given explicitly. Finally, it is shown how these simple and generalized eigenfunctions reduce to the corresponding spherical eigenfunctions as the focal distance of the spheroidal system tends to zero, in which case the separability is regained. The usefulness of the method is demonstrated by solving the problem of the flow in a fluid cell contained between two confocal spheroidal surfaces with Kuwabara-type boundary conditions.
Numerical computations for antiplane shear in a granular flow model
Authors: Garaizar Xabier
Published: December 2016

We describe an algorithm for the numerical resolution of elasto-plastic deformations in the context of antiplane shear models. The algorithm is a second-order Godunov method. For these models the eigenvalues associated to the hyperbolic system are discontinuous. We test the algorithm on several examples.
Kinematical approach to the shakedown analysis of some structures
Authors: Pham D. C. , Stumpf H.
Published: December 2016

From Koiter’s kinematical shakedown theorem, a new variational problem is deduced, which does not contain integrals over a time parameter and gives an upper bound on the safety factor. For a broad class of practical problems, including planar bar systems subjected to combined axial and bending loads, it leads to the exact value of the shakedown factor. The possible inadaptation modes (incremental, alternating, or mixed) on the shakedown boundary are determined.
Existence of classical solutions for singular parabolic problems
Authors: Chan C. Y. , Wong Benedict M.
Published: December 2016

Let

L
u
≡

u

x
x

+
b

u
x

/

x
−

u
t

Lu \equiv {u_{xx}} + b{u_x}/x - {u_t}

with

b
b

a constant less than 1. Its Green’s function corresponding to first boundary conditions is constructed by eigenfunction expansion. With this, a representation formula is established to obtain existence of a classical solution for the linear first initial-boundary value problem. Uniqueness of a solution follows from the strong maximum principle. Properties of Green’s function and of the solution are also investigated.
Eigenfrequencies of curved Euler-Bernoulli beam structures with dissipative joints
Authors: Paulsen William H.
Published: December 2016

In this paper, we will compute asymptotically the eigenfrequencies for the in-plane vibrations of an Euler-Bernoulli beam system with dissipative joints, which allow the beams to be curved into an arc of a circle. This enhances the author’s previous result for structures involving straight beams, given in his preprint “Eigenfrequencies of the non-collinearly coupled Euler-Bernoulli beam system with dissipative joints". Matrix techniques are used to combine asymptotic analysis with a form of the wave propagation method.
Equilibrium of multivalued permanent systems with nonconvex right-hand sides
Authors: Li Yong , Wang Huai Zhong , Lü Xian Rui
Published: December 2016

The existence of periodic solutions for differential inclusions with nonconvex right-hand sides is proved. As an application, an affirmative answer to the existence of an equilibrium for permanent multivalued systems with nonconvex righthand sides is given.
Unstable simple modes of the nonlinear string
Authors: Cazenave Thierry , Weissler Fred B.
Published: December 2016

We prove instability of high-energy simple modes for the nonlinear vibrating string equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace Subscript u vertical-bar Sub Subscript partial-differential normal upper Omega Subscript equals 0 comma Superscript u Super Subscript t t Superscript minus left-parenthesis a plus b integral Subscript 0 Superscript pi Baseline u Super Subscript x Super Superscript 2 Superscript left-parenthesis t comma x right-parenthesis d x right-parenthesis u Super Subscript x x Superscript equals 0 comma">

{

u

|

∂
Ω

=
0
,

u

t
t

−

(

a
+
b

∫
0
π

u
x
2

(

t
,
x

)

d
x

)

u

x
x

=
0
,

\left \{ {_{u\left | {_{\partial \Omega } = 0,} \right .}^{{u_{tt}} - \left ( {a + b \int _0^\pi {u_x^2 \left ( {t, x} \right ) dx} } \right ) {u_{xx}} = 0,}} \right .

\]
in

Ω
=

(

0
,
π

)

\Omega = \left ( {0, \pi } \right )

, where

a
≥
0
,
b
>
0

a \ge 0, b > 0

.
Love’s circular patch problem revisited: closed form solutions for transverse isotropy and shear loading
Authors: Hanson Mark T. , Puja Igusti W.
Published: December 2016

This paper considers the title problem of uniform pressure or shear traction applied over a circular area on the surface of an elastic half space. The half space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is adopted where the elastic field is written in terms of three harmonic functions. The known point force potential functions are used to find the solution for uniform pressure or shear traction over a circular area by quadrature. Using methods developed by Love (1929) and Fabrikant (1988), the elastic displacement and stress fields for normal and shear loading are evaluated in terms of closed form expressions containing complete elliptic integrals of the first, second, and third kinds. The solution for uniform normal pressure on an isotropic half space was previously found by Love (1929). The present results for transverse isotropy including shear loading are new. During the course of this research, a new relation has been discovered between different forms of the complete elliptic integral of the third kind. This has allowed the present solution to be put in a more convenient form than that used by Love. Following a limiting procedure allows the isotropic solution to be obtained. It is shown that for normal loading the present results agree with Love’s solution, while the results for shear loading of an isotropic half space are also apparently new. Special consideration is also given to derive the limiting form of the elastic field on the

z
z

-axis and the surface

(

z
=
0

)

\left ( {z = 0} \right )

.
Non-classical diffusion equations related to birth-death processes with two boundaries
Authors: Mansourati Z. G. , Campbell L. L.
Published: December 2016

A pair of forward and backward diffusion equations is considered. In the forward equation, boundary values appear in the differential equation, and in the backward equation, boundary values are related to average values of the solution in the interior of the domain. The forward equation can be regarded as a diffusion approximation to a type of birth-death process with returns to the interior, or as a heat equation in one dimension where heat flowing out from the boundaries is returned to the interior. Existence and uniqueness theorems are proved, and some properties of the associated eigenvalues and eigenfunctions are deduced. An expression for the steady-state solution is obtained. Some information on the goodness of the diffusion approximation is also obtained.
Inertial manifolds of incompressible, nonlinear bipolar viscous fluids
Authors: Bloom Frederick , Hao Wenge
Published: December 2016

The existence of an inertial manifold is established for the nonlinear system of equations describing the motion of a bipolar incompressible viscous fluid. In this paper we consider only the case of a spatially periodic velocity field. Existence of an inertial manifold for the model complements earlier work on the existence of compact global attractors for bipolar viscous fluids and serves to further highlight the differences between the bipolar model and the usual model based on the linear Stokes constitutive relation.
An inverse method for determining parameters for folded structures
Authors: Lerche I.
Published: December 2016

General analytic solutions are presented here to an equation relating two angles describing fault shape, and two angles describing fold shape. The analytic solutions provide any of the four angles when the other three are given.
Instability and stability of numerical approximations to discrete velocity models of the Boltzmann equation
Authors: Peszek Robert
Published: December 2016

We study a standard, explicit finite difference approximation of the 2-D Broadwell model and construct a numerical solution with the sum-norm growing in time faster than any polynomial. Our construction is based on a structure of a self-similar fractal!
Periodic solutions and equilibrium states for functional-differential inclusions with nonconvex right-hand side
Authors: Li Yong , Zhou Qinde , Lu Xianrui
Published: December 2016

In this paper, we present a general existence theorem for periodic solutions for functional differential inclusions with nonconvex right-hand side. As an application, we prove that for a multivalued system with nonconvex right-hand side, permanence implies the existence of a pre-equilibrium state, which answers an open problem proposed by Hutson.
Conditions of compatibility for the solid-liquid interface
Authors: Baldoni F. , Rajagopal K. R.
Published: December 2016

The seminal theory of singular surfaces propounded by Hadamard and Thomas is examined within the context of the dynamics of a solid-liquid interface. It is shown that most of the hypotheses upon which Clapeyron’s equation is based can be weakened and two generalized versions of it are derived: with and without curvature effects. The remaining part of the paper is mainly focused on the interface conditions for the classical Stefan problem. The counterpart of Clapeyron’s equation for such a problem will give an explicit expression for the supercooling temperature without recourse to linearization procedures. Furthermore, a decay law for the latent heat of melting is given which shows, in an explicit way, its complex dependence upon the curvature and the normal speed of the interface. Finally, a transport equation for the interface temperature is derived and a qualitative solution of a simplified version of it is given for the particular case in which the jump in the Helmoltz free energies of the bulk phases is a conserved quantity throughout the field.
Voltage-current characteristics of a 𝑝𝑛-diode from a drift-diffusion model with nonlinear diffusion
Authors: Jüngel Ansgar , Schmeiser Christian
Published: December 2016

A drift-diffusion model with density-dependent diffusion coefficients for the flow of electrons and holes in a semiconductor crystal is considered. It contains a new class of models for recombination-generation effects as well as boundary conditions modelling Ohmic contacts. Existence of steady-state solutions is proven. For a planar

p
n

pn

-diode the qualitative properties of steady-state solutions in dependence on the applied voltage is examined and, in particular, voltage-current characteristics are discussed.
On an integrodifferential equation arising in a theory of phase transitions in solids
Authors: Abeyaratne Rohan , Knowles James K.
Published: December 2016

This note is concerned with some properties of an integrodifferential equation arising in a continuum model of solid-solid phase transitions.
Continuous dependence results for a problem in penetrative convection
Authors: Ames Karen A. , Payne Lawrence E.
Published: December 2016

Continuous dependence inequalities are derived for a system of equations that models penetrative convection in a thermally conducting viscous fluid with a linear buoyancy law. Both the forward-in-time problem and the improperly posed backward-in-time problem are analyzed. These results indicate that solutions depend continuously on a parameter in the boundary data.
On the summation of a particular non-uniformly-convergent series
Authors: Dhondt Guido
Published: December 2016

In a previous article the force between a dielectric cylinder and a conducting half-space for the cylinder approaching the half-space was obtained as the limit of a non-uniformly-convergent series. A special asymptotic analysis was performed to get a closed-form solution for this limit. The present article focuses on a delicate and crucial step in this analysis and gives a justification by a mathematically rigorous proof.
On Constanda’s matrix of nonuniqueness in a theory of plates in asymmetric elasticity
Authors: Ru Chong-Qing , Schiavone Peter
Published: December 2016

We apply the direct method to the solution of the Dirichlet problem in the bending of micropolar plates and show that a necessary and sufficient condition for solubility is the nonsingularity of a special constant matrix constructed for any smooth closed boundary curve.
Kinetic and fluid aspects of gas discharges
Authors: Nouri A.
Published: December 2016

Gas discharges are investigated in two levels: the kinetic one, where a stationary electron distribution is determined, and the fluid one, where use of the small mobility of ions with respect to the mobility of electrons leads to a simplified model and a boundary layer analysis.
Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping
Authors: Hsiao L. , Luo Tao
Published: December 2016

We consider a system of isentropic flow through porous media and show that nonlinear diffusive phenomena occur even if there are shocks in the flow.
Influence of natural convection on stability of reaction fronts in liquids
Authors: Garbey M. , Taïk A. , Volpert V.
Published: December 2016

We study the influence of natural convection on stability of reaction fronts in liquids. In our previous article [6] we considered the case where the reactants were in a liquid phase and the product of the reaction was solid. In this paper we study the case where both of them are liquid. We carry out a linear stability analysis and show that the results are essentially different compared to the case of a solid product.
Computational anatomy: an emerging discipline
Authors: Grenander Ulf , Miller Michael I.
Published: December 2016

This paper studies mathematical methods in the emerging new discipline of Computational Anatomy. Herein we formalize the Brown/Washington University model of anatomy following the global pattern theory introduced in [1, 2], in which anatomies are represented as deformable templates, collections of 0, 1, 2, 3-dimensional manifolds. Typical structure is carried by the template with the variabilities accommodated via the application of random transformations to the background manifolds. The anatomical model is a quadruple

(
Ω
,
H
,
I
,
P
)

\left ( \Omega , H, I, P \right )

, the background space

Ω

=
˙

U
α

M
α

\Omega \dot = {U_\alpha }{M_\alpha }

of 0, 1, 2, 3-dimensional manifolds, the set of diffeomorphic transformations on the background space

H

:
Ω
↔
Ω

{H} : \Omega \leftrightarrow \Omega

, the space of idealized medical imagery

I
I

, and

P
P

the family of probability measures on

H
H

. The group of diffeomorphic transformations

H
H

is chosen to be rich enough so that a large family of shapes may be generated with the topologies of the template maintained. For normal anatomy one deformable template is studied, with

(
Ω
,
H
,
I
)

\left ( \Omega , H, I \right )

corresponding to a homogeneous space [3], in that it can be completely generated from one of its elements,

I
=

H

I

t
e
m
p

,

I

t
e
m
p

∈
I

I = {HI_{temp}}, {I_{temp}} \in I

. For disease, a family of templates

U
α

I

t
e
m
p

α

{U_\alpha }I_{temp}^\alpha

are introduced of perhaps varying dimensional transformation classes. The complete anatomy is a collection of homogeneous spaces

I

t
o
t
a
l

=

U
α

(

I
α

,

H
α

)

{I_{total}} = {U_\alpha }\left ( {I^\alpha }, {H^\alpha } \right )

.
Stability analysis for a mathematical model of the lac operon
Authors: Mahaffy Joseph M. , Savev Emil Simeonov
Published: December 2016

A mathematical model for induction of the lac operon is derived using biochemical kinetics and includes delays for transcription and translation. Local analysis of the unique equilibrium of this nonlinear model provides conditions for stability. Techniques are developed to determine Hopf bifurcations, and stability switching is found for the delayed system. Near a double bifurcation point a hysteresis of solutions to two stable periodic orbits is studied. Global analysis provides conditions on the model for asymptotic stability. The biological significance of our results is discussed.
A finite difference approximation for a nonlinear size-structured phytoplankton aggregation model
Authors: Ackleh Azmy S. , Ferdinand Robert R.
Published: December 2016

We consider a nonlinear model that describes the dynamics of a phytoplankton population with aggregation and competition between individual cells. A finite difference method is developed for approximating the solution of this partial differential equation. The convergence of this approximation to a unique bounded variation solution of the model is proved. Numerical results showing the accuracy of this scheme are presented.
A parameter-expansion method for the scattering of plane waves by an elliptic cylindrical and a hyperboloidal scatterer
Authors: Acho Thomas M.
Published: December 2016

The scattered wave components of a plane polarized electromagnetic wave by an elliptic cylindrical scatterer are obtained as a parameter expansion of the more exact circular cylindrical obstacle. The scattering potential of a scalar wave by a hyperboloidal scatterer as a parameter expansion of a circular cone is also obtained. The parameter of expansion in both cases is the semi-interfocal distance.
Major simplifications in a current linear model for the motion of a thermoelastic plate
Authors: Simmonds J. G.
Published: December 2016

A dynamic model for a thin thermoelastic plate proposed by Lagnese and Lions in 1988 [1] has been used recently by several authors (e.g., [2]–[5]) to study existence and stability of solutions to initial/boundary-value problems. Simple, systematic order-of-magnitude arguments show that it is consistent to neglect several terms appearing in the governing differential equations that couple a temperature moment to the average vertical displacement. Further, because the time scale on which the temperature adjusts itself to the strain rate contribution to the energy equation is quite small compared with the longest (isothermal) period of free vibration of the plate, the energy equation can be solved for the temperature in terms of derivatives of the vertical displacement and hence the system reduced to a single equation, only slightly more complicated than the classical (Kirchhoff) equation of motion. Among other things, it is shown that the temperature has a cubic rather than a linear variation through the thickness. Finally, another order-of-magnitude estimate for a clamped aluminum plate of one meter radius and 1mm thickness shows that thermal damping acting alone takes on the order of 200 cycles of vibration to halve the initial amplitude.
On second sound at the critical temperature
Authors: Saxton K. , Saxton R. , Kosinski W.
Published: December 2016

Based on low-temperature experimental data in solid dielectric crystals, we derive a model of heat conduction for rigid materials using the theory of thermo-dynamic internal state variables. The model is intended to admit wavelike propagation of heat below—and diffusive conduction above—a particular temperature value

ϑ

λ

{\vartheta {_\lambda }}

. A rapid decay of the speed of thermal waves occurs just below this temperature, coincident with the conductivity of the material reaching a peak. An analysis of weak and strong discontinuity waves is given in order to exhibit several main features of the proposed model.
On the oblique wave penetration in elastic solids with a doubly periodic array of cracks
Authors: Scarpetta Edoardo , Sumbatyan Mezhlum A.
Published: December 2016

In the context of wave propagation in damaged (elastic) solids, an analytical approach for oblique penetration of a plane wave through a doubly periodic array of cracks is developed. By using a uniform approximation in one-mode range previously obtained, we give explicit representations for the wave field throughout the structure and the relevant parameters. Two figures show the peculiarity of such results.
Singular perturbation approach to an elastic dry friction problem with non-monotone coefficient
Authors: Renard Yves
Published: December 2016

In this paper we study a one-dimensional dynamic model of dry friction with slip velocity dependent coefficient. In many cases, this model has more than one solution. We introduce a perturbed friction condition which allows us to regain the uniqueness of the solution. We show that the perturbed problem’s solutions pointwise converge to a particular solution of the initial problem when the perturbation parameter tends to zero. The singular perturbation approach provides the analysis of a criterion used to select a solution of the problem, and suggests a method to study more elaborated dry friction problems.
Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers with generalized polytrope laws
Authors: Brown Kevin G. , Choudhury S. Roy
Published: December 2016

The linear stability of zero and finite width, arbitrarily compressible, and magnetized velocity shear layers with isotropic or anisotropic pressure is investigated. Such flows, modeled by the magnetohydrodynamic equations with generalized polytrope laws for the pressure parallel and perpendicular to the magnetic field, are relevant in various astrophysical, geophysical and space plasma configurations. The conditions for instability of shear layers of zero width are derived. For layers of finite width, shooting numerical schemes are employed to satisfy the Sommerfeld radiation conditions of outgoing, spatially damping modes in a frame comoving with the plasma flow. The resulting eigenvalues for the angular frequency and linear growth rate are mapped out for different regions of the wave number/Mach number space. For polytrope indices corresponding to the double adiabatic and magnetohydrodynamic equations, the results reduce to those obtained earlier using these models.
Existence and uniqueness for a quasistatic frictional bilateral contact problem in thermoviscoelasticity
Authors: Rochdi M. , Shillor M.
Published: December 2016

We prove the existence and uniqueness of the weak solution for a quasistatic thermoviscoelastic problem which describes bilateral frictional contact between a deformable body and a moving rigid foundation. The model consists of the heat equation for the temperature, the elliptic viscoelasticity system for the displacements, the SJK-Coulomb law of friction and frictional heat generation condition. The proof is accomplished in two steps. First, the existence of solutions for a regularized problem is established and a priori estimates obtained. Then the limit function, which is the weak solution of the original problem, is shown to be the unique fixed point of the solution operator when the friction coefficient is small.
On a 𝑝-Laplacian type of evolution system and applications to the Bean model in the type-II superconductivity theory
Authors: Yin Hong-Ming
Published: December 2016

In this paper we study the Cauchy problem for a

p
p

-Laplacian type of evolution system

H
t

+
∇
×

[

|

∇
×
H

|

p
−
2

∇
×
H

]

=
F

{H_t} + \nabla \times \left [ {{{\left | {\nabla \times H} \right |}^{p - 2}}\nabla \times H} \right ] = F

. This system governs the evolution of a magnetic field H, where the displacement currently is neglected and the electrical resistivity is assumed to be some power of the current density. The existence, uniqueness, and regularity of solutions to the system are established. Furthermore, it is shown that the limit solution as the power

p
→
∞

p \to \infty

solves the problem of Bean’s model in the type-II superconductivity theory. The result provides us information about how the superconductor material under the external force becomes the normal conductor and vice versa. It also provides an effective method for finding numerical solutions to Bean’s model.
Linear stability for a thermoelectromagnetic material with memory
Authors: Amendola Giovambattista
Published: December 2016

In this paper we study the behaviour of a three-dimensional linear thermoelectromagnetic material, which has constitutive equations with memory effects for both the heat flux and the electric current density. We develop a linearized theory of thermodynamics, in which context we are able to introduce a maximal free energy defined in the frequency domain. Using this free energy, a domain of dependence is obtained. Moreover, we prove a theorem of uniqueness, existence, and asymptotic stability.
Existence and nonexistence of global solutions of the wave equation with a nonlinear boundary condition
Authors: Ackleh Azmy S. , Deng Keng
Published: December 2016

We study the initial-boundary value problem
\[

u

t
t

=

u

x
x

,

0
>
x
>
∞
,

t
>
0
,

{u_{tt}} = {u_{xx}}, \qquad 0 > x > \infty , \qquad t > 0,

\]

\[

−

u
x

(
0
,
t
)

=
h

(
u

(
0
,
t
)

)

,

t
>
0
,

- {u_x}\left ( 0, t \right ) = h\left ( u\left ( 0, t \right ) \right ), \qquad t > 0,

\]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x comma 0 right-parenthesis equals f left-parenthesis x right-parenthesis comma u Subscript t Baseline left-parenthesis x comma 0 right-parenthesis equals g left-parenthesis x right-parenthesis comma 0 greater-than x greater-than normal infinity period">

u

(
x
,
0
)

=
f

(
x
)

,

u
t

(
x
,
0
)

=
g

(
x
)

,

0
>
x
>
∞
.

u\left ( x, 0 \right ) = f\left ( x \right ), \qquad {u_t}\left ( x, 0 \right ) = g\left ( x \right ), \qquad 0 > x > \infty .

\]
We establish criteria for existence and nonexistence of global solutions, and we present the growth rate at blow-up.
On nonlinearly detuned third harmonic ripples between two stratified fluids
Authors: Jones Mark C. W.
Published: December 2016

The problem of two semi-infinite fluids in uniform horizontal motion parallel to their interface is studied. Attention is focused on the interfacial disturbances that are caused by the interaction between a fundamental mode and its third harmonic. A series expansion for the disturbance profile is obtained in which the leading-order amplitudes are assumed to be slowly varying functions in time and space. By use of this expression we are able to derive a pair of coupled nonlinear Schrödinger-type equations which model the evolution of the interface. Solutions to this system are found and thus we are able to describe the possible wave profiles, which turn out to be tripleor quintuple-dimpled. We also find that at perfect resonance three profiles are always possible but that at near-resonance there may be one or three profiles depending on the values of the parameters present in the problem.
Conical quantum billiard revisited
Authors: Liboff Richard L.
Published: December 2016

Eigenstates of a particle confined to a cone of finite length capped by a spherical surface element are derived. A countable infinite set of solutions is obtained corresponding to integer azimuthal and orbital quantum numbers

(
m
,
l
)

\left ( m, l \right )

. These solutions apply to a discrete subset of the domain of half vertex angles,

0
≤

θ
0

≤
π

/

2

0 \le {\theta _0} \le \pi /2

. For arbitrary real orbital quantum numbers,

l
→
ν

l \to \nu

, solutions are given in terms of the hypergeometric function, with

ν
=
ν

(

θ
0

)

\nu = \nu \left ( {\theta _0} \right )

, and are valid in the

θ
0

{\theta _0}

domain,

0
≤

θ
0

>
π

/

2

0 \le {\theta _0} > \pi /2

. Eigenstates are either nondegenerate or two-fold degenerate. Numerical examples of both classes of solutions are included. For the case

μ
=
cos
⁡
π

/

4

\mu = \cos \pi /4

, the ground-state wavefunction and eigenenergy are
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Subscript upper G Baseline equals upper P Subscript nu Baseline left-parenthesis mu right-parenthesis j Subscript nu Baseline left-parenthesis x Subscript nu Baseline 1 Baseline r slash a right-parenthesis comma upper E Subscript upper G Baseline equals italic h over two pi squared left-parenthesis 6.4387 right-parenthesis squared slash left-parenthesis 2 upper M a squared right-parenthesis">

φ
G

=

P
ν

(
μ
)

j
ν

(

x

ν
1

r

/

a
)

,

E
G

=

ℏ
2

(
6.4387
)

2

/

(
2
M

a
2

)

{\varphi _G} = {P_\nu }\left ( \mu \right ){j_\nu }\left ( {x_{\nu 1}}r/a \right ), \qquad {E_G} = {\hbar ^2}{\left ( 6.4387 \right )^2}/\left ( 2M{a^2} \right )

\]
where

ν
=
2.54791
,

P
ν

(
μ
)

\nu = 2.54791, {P_\nu }\left ( \mu \right )

are Legendre functions,

x

ν
1

{x_{\nu 1}}

is the first finite zero of the spherical Bessel function

j
ν

(
x
)

{j_\nu }\left ( x \right )

,

M
M

is the mass of the confined particle and

a
a

is the edgelength of the cone. Solutions constructed also represent the scalar

r
^

⋅
E

\hat r \cdot E

electric field, where

r
^

\hat r

is the unit radius from the vertex of the cone. The first excited state of the conical quantum billiard has the nodal surface

μ
=
1

\mu = 1

for all

0
≤

μ
0

≤
1

0 \le {\mu _0} \le 1

.
Reinforcement of elastic structures in the presence of imperfect bonding
Authors: Lipton Robert
Published: December 2016

The two-dimensional problem of plane strain is considered in the presence of imperfectly bonded elastic reinforcements. A geometric criterion on the shape and size of the elastic reinforcement is found that determines when the effects of imperfect bonding overcome the benefits of the reinforcement. The criterion is given in terms of an eigenvalue problem posed on the surface of the reinforcement.
On numerical entropy inequalities for a class of relaxed schemes
Authors: Tang Huazhong , Tang Tao , Wang Jinghua
Published: December 2016

In [4], Jin and Xin developed a class of first- and second-order relaxing schemes for nonlinear conservation laws. They also obtained the relaxed schemes for conservation laws by using a Hilbert expansion for the relaxing schemes. The relaxed schemes were proved to be total variational diminishing (TVD) in the zero relaxation limit for scalar equations. In this paper, by properly choosing the numerical entropy flux, we show that the relaxed schemes also satisfy the entropy inequalities. As a consequence, the

L

1

{L^{1}}

convergence rate of

O

(

Δ
t

)

O\left ( \sqrt {\Delta t} \right )

for the relaxed schemes can be established.
A simplified, nonlinear thermodynamic theory of beamshells
Authors: Simmonds J. G.
Published: December 2016

This paper develops a nonlinear thermodynamical theory for arbitrary elastic beamshells (infinite cylindrical shells in plane strain) in which approximations are made only in the First Law of Thermodynamics (Conservation of Energy) and in the associated constitutive relations. The basic approach is straightforward: the three-dimensional equations of motion and the Second Law of Thermodynamics (Clausius-Duhem Inequality) for an infinite cylindrical body subject to external loads and heating are written in integral-impulse form and then specialized to beamshells. This requires neither formal expansions in a thickness coordinate nor a priori kinematic hypotheses such as those associated with the names of Kirchhoff or Cosserat. The resulting one-dimensional, time-dependent equations involve a vector stress resultant N, a scalar stress couple

M
M

, a vector translational momentum L, a scalar rotational momentum

R
R

, an entropy resultant

S
S

, an average reciprocal temperature

T
T

, and an average transverse temperature gradient

G
G

. The unknowns N,

M
M

, L,

R
R

, and

S
S

are defined in terms of thickness-weighted integrals, but

T
T

and

G
G

are defined in terms of the surface values of the three-dimensional absolute temperature. A power identity yields, automatically, definitions of a strain vector e and a scalar bending strain

k
k

whose local rates are conjugate, respectively, to N and

M
M

. Once an elastodynamic (kinetic plus strain) energy of the beamshell is defined, the introduction of a free energy introduces an additional unknown

F
F

, an entropy couple conjugate to

G
G

. Enforcement of the Second Law for all possible thermodynamic processes, à la Coleman and Noll [1], plus the key assumption that the time derivative of

F
F

is a function of the state variables only, leads to a complete and consistent set of simplified constitutive relations. In the present approach there is just one entropy inequality and just one energy equation, in contrast to that of Green and Naghdi [2] who introduce a hierarchy of such equations, essentially one for each director they introduce.
The generalized quasilinearization method for parabolic integro-differential equations
Authors: Vatsala A. S. , Wang Liwen
Published: December 2016

In this paper we consider the nonlinear parabolic integro-differential equation with initial and boundary conditions. We develop the method of generalized quasilinearization to generate linear iterates that converge quadratically to the unique solution of the nonlinear parabolic integro-differential equation. For this purpose, we establish comparison results for the parabolic integro-differential equation. These comparison results are used to develop monotone sequences and to establish quadratic convergence.
Asymptotic behaviour of the energy in partially viscoelastic materials
Authors: Muñoz Rivera Jaime E. , Peres Salvatierra Alfonso
Published: December 2016

In this paper we study models of materials consisting of an elastic part (without memory) and a viscoelastic part, where the dissipation given by the memory is effective. We show that the solutions of the corresponding partial viscoelastic model decay exponentially to zero, provided the relaxation function also decays exponentially, no matter how small is the viscoelastic part of the material
Linear stability of rimming flow
Authors: O’Brien S. B. G.
Published: December 2016

We consider the case of rimming flow where a thin film of viscous liquid coats the walls of a cylinder whose axis is horizontal and which is rotating with constant angular velocity. It has been experimentally established that such flows are often unstable and that the liquid often segregates into “rings” along the length of the tube. Using a leading-order lubrication theory, we utilise recently established steady solutions [10], which in some instances contain shocks, to examine the linear stability of the flow when subjected to two-dimensional disturbances. All solutions are shown to be at least neutrally stable. We suggest that further investigations should include higher-order (small) effects and that the origin of the observed instabilities lies in these terms.
A mechanism for linear instability in two-dimensional rimming flow
Authors: O’Brien S. B. G.
Published: December 2016

In rimming flow, a thin film of viscous liquid coats the inside of a cylinder whose axis is horizontal and which is rotating with constant angular velocity. It has been established experimentally that such flows are often unstable with a variety of secondary flow regimes having been observed experimentally [15]. We use a lubrication approximation extended to the first order in the dimensionless film thickness (including the small effects of the variation of the film pressure across its thickness and the surface tension) and study the stability of the steady solutions to two-dimensional disturbances. The modified evolution equation is found to have both asymptotically stable and unstable solutions arising from the pressure terms. Surface tension effects place a restriction on the critical wave number when instability occurs: in many cases, surface tension prevents instability.