bluuebunny/crossref_metadata_2025 huggingface dataset:
Finite strain solutions for a compressible elastic solid
Authors: Carroll M. M. , Horgan C. O.
Published: December 2016

Several closed form finite strain equilibrium solutions are presented for a special compressible isotropic elastic material which was proposed as a model for foam rubber by Blatz and Ko. These solutions include bending of a cylindrical sector into another sector or a rectangular block, bending of a block into a sector, expansion, compaction or eversion of cylinders or spheres, and torsion and extension of circular cylinders or tubes.
Self-induced motion of line defects
Authors: Rubinstein Jacob
Published: December 2016

The evolution of the 2-d Ginzburg-Landau functional under the Schrodinger and the diffusion dynamics is considered. We construct solutions

u

(

x
,
t

)

,
u
∈

R
2

,
x
∈

R
3

u\left ( {x, t} \right ), u \in {R^2}, x \in {R^3}

, such that the vector field

u
u

vanishes along a singular curve

γ
\gamma

. Equations of motion for

γ

(
t
)

\gamma \left ( t \right )

are derived by the method of matched asymptotic expansions.
Tchebychev nets on spheres
Authors: Samelson Sandra Lee
Published: December 2016

We show that the standard sphere on

R

3

\mathbb {R}{^3}

admits a Tchebychev net which covers an open subset of the sphere which contains a closed hemisphere.
The structure of solutions of gas dynamic equations and the formation of the vacuum state
Authors: Fan Hai Tao
Published: December 2016

The structure of the solutions of the isentropic gas dynamics equations in Eulerian coordinates given by [17] is studied. A condition under which the formation of the vacuum state occurs is obtained.
Asymptotic analysis of a rod with small bending stiffness
Authors: Wolfe Peter
Published: December 2016

In this paper we consider a model problem for the deformation of a rod with small bending stiffness. We show that this problem can be considered as a singular perturbation of the problem in which the rod is replaced by a string with no resistance to bending. We construct an approximate solution to this problem. As the bending stiffness tends to zero this solution tends to the solution of the string problem away from the ends of the rod which are assumed to be clamped. However, as one would expect, there is a boundary layer near each end of the rod. The main point of the paper is to show how to construct the boundary layer corrections.
Applications of generalized stress in elastodynamics
Authors: Meade Douglas B.
Published: December 2016

The problem under consideration is the scattering of elastic waves by inhomogeneous obstacles. The main goal is to obtain approximation techniques which are amenable to numerical implementation. For time-periodic problems a coupling procedure involving finite elements and boundary integral equations is described. For general time-dependent problems, artificial boundary methods are studied. In both cases the concept of generalized stress, as originated by Kupradze, plays a central role. The analysis is restricted to planar two-dimensional problems since these illustrate the essential ideas.
A Banach space solution of an abstract Cauchy problem which has Robin boundary data
Authors: Quinn Dennis W.
Published: December 2016

In this paper, a problem which involves the diffusion equation coupled by way of Robin boundary data with a nonlinear system of ordinary differential equations is considered. This problem has been proposed as a model to describe the absorption through the skin, the distribution throughout the body, and the metabolism of a substance in a mammal. The problem is set as an abstract Cauchy problem in a Banach space and is shown to have a unique solution. Finite dimensional approximations of the problem are obtained by replacing the spatial partial derivatives with finite differences. The approximate solutions are shown to converge to the exact solution of the original problem. Comparisons of numerical solutions with experimental data are presented.
Ignition or nonignition of a combustible solid with marginal heating
Authors: Lasseigne D. Glenn , Olmstead W. E.
Published: December 2016

The possibility of achieving ignition of a combustible solid with only marginal external heating is investigated. The heating barely raises the temperature to a level where reaction occurs. A parameter indicating the duration of time spent near the crucial level governs the ignition process. If the parameter is below a critical value, no thermal runaway can occur; if it is sufficiently large, ignition will be achieved.
Hydrodynamic stability of Rayleigh-Bénard convection with constant heat flux boundary condition
Authors: Park H. , Sirovich L.
Published: December 2016

We study the onset of thermal instability with the heat flux prescribed on the fluid boundaries. Assuming Boussinesq fluid, the Landau equation, which describes the evolution of the amplitude of the convection cells, is derived using the small amplitude expansion technique. For the case of a three-dimensional rectangular box with aspect ratio (8, 4, 1), the incipient convection cell is a two-dimensional one at

p
r
=
0.72

pr = 0.72

, which is confirmed by the numerical solution of the three-dimensional Boussinesq equation with a Chebyshev-Fourier pseudospectral code. The secondary bifurcation gives rise to an oscillatory two-dimensional roll for the same Prandtl number at

R
=
2.0

R
c

R = 2.0{R_c}

and the motion becomes three dimensional at

R
=
2.8

R
c

R = 2.8{R_c}

.
A comparison of certain elastic dissipation mechanisms via decoupling and projection techniques
Authors: Russell David L.
Published: December 2016

In this paper we study the Euler-Bernoulli elastic beam model, modified in a variety of ways to achieve an asymptotically linear relationship between damping rate and frequency. We review the so-called spatial hysteresis model and then introduce the thermoelastic/shear diffusion model, which is obtained by coupling the originally conservative elastic equations to two different diffusion processes. We then use a decoupling/triangulation process to project the coupled system onto the subspace corresponding to the lateral displacements and velocities and show that the projected system agrees in many significant respects with the spatial hysteresis model. The procedure also indicates some possibly desirable modifications in the elastic term of the spatial hysteresis model.
Spatially periodic Stokes flow stirred by a rotlet interior to a closed corrugated boundary
Authors: Pravica D. W. , Ranger K. B.
Published: December 2016

Spatially periodic solutions of the creeping flow equations are found for the stream function in which the motion is stirred by a two-dimensional rotlet in the region interior to a closed corrugated boundary. Streamlines are given for different geometrical configurations. In some cases there is separation of the streamlines in the crevices of the corrugation.
Shear instabilities for the buckling of non-linearly elastic plates
Authors: Negrón-Marrero Pablo V.
Published: December 2016

In this work we apply singularity theory and global bifurcation theory to the problem of the axisymmetric buckling of circular plates. We show that a certain shear parameter can be used as an unfolding parameter and we get detailed local and global structure of the solution set. We get conditions for a singular point to be of codimension one.
Roots of 𝐽₀(𝑧)-𝑖𝐽₁(𝑧)=0 as saddle points of the reduced logarithmic derivative of 𝐽₀(𝑧)
Authors: Abad Julio , Sesma Javier
Published: December 2016

The roots of

J
0

(
z
)

−
i

J
1

(
z
)

=
0

{J_0}\left ( z \right ) - i{J_1}\left ( z \right ) = 0

are saddle points of the function

F
0

(
z
)

≡

z

J
0
′

(
z
)

/

J
0

(
z
)

{F_0}\left ( z \right ) \equiv \\ z{J’_0}\left ( z \right )/{J_0}\left ( z \right )

. A very efficient algorithm allows one to obtain, by iteration, those roots to the desired accuracy.
Creeping flow through an annular stenosis in a pipe
Authors: Davis A. M. J.
Published: December 2016

The creeping flow disturbance of Poiseuille flow due to a disk can be determined by the use of a distribution of “ringlet” force singularities but the method does not readily adapt to the complementary problem involving an annular constriction. Here it is shown that a solvable Fredholm integral equation of the second kind with bounded kernel can be obtained for an Abel transform of the density function. The exponential decay associated with the biorthogonal eigenfunctions ensures that the flow adjusts to the presence of the constriction in at most a pipe length of half a radius on either side. Methods that depend on matching series at the plane of the constriction appear doomed to failure. The physical quantities of interest are the additional pressure drop and the maximum velocity. The lubricating effect of inlets is demonstrated by extending the analysis to a periodic array of constrictions.
Pure torsion of compressible non-linearly elastic circular cylinders
Authors: Polignone Debra A. , Horgan Cornelius O.
Published: December 2016

The large deformation torsion problem of an elastic circular cylinder, composed of homogeneous isotropic compressible nonlinearly elastic material and subjected to twisting moments at its ends, is described. The problem is formulated as a two-point boundary-value problem for a second-order nonlinear ordinary differential equation in the radial deformation field. The class of materials for which pure torsion (i.e., a deformation with zero radial displacement) is possible is described. Specific material models are used to illustrate the results.
A note on the plane vertical wavemaker in the presence of surface tension
Authors: Mandal B. N. , Bandyopadhyay S.
Published: December 2016

This paper presents another method for finding the known wave motion due to a vertical wavemaker in the presence of surface tension, either for finite constant or infinite depth, by using the Fourier integral transform technique and suitably exploiting the regularity condition of the transform.
Critical behavior of an ignition model in chemical combustion
Authors: Tonellato Peter J.
Published: December 2016

A model for the hot slab ignition problem is analyzed to determine critical conditions. The system is said to be super-critical if the solution of the reduced perturbation problem blows up in small finite time or sub-critical if the blow up time is large. Comparison principles for integral equations are used to construct upper and lower solutions of the equation. All solutions depend on two parameters

ε

−
1

{\varepsilon ^{ - 1}}

, the Zeĺdovitch number and

λ
\lambda

, the scaled hot slab size. Upper and lower bounds on a ’critical’ curve

λ
c

(
ϵ
)

{\lambda _c}\left ( \epsilon \right )

in the

(

ϵ
,
λ

)

\left ( {\epsilon , \lambda } \right )

plane, separating the super-critical from the sub-critical region, are derived based upon the lower and upper solution behavior. Numerical results confirm the parameter space analysis.
Shading without shape
Authors: Brooks Michael J. , Chojnacki Wojciech , Kozera Ryszard
Published: December 2016

A smooth object depicted in a monochrome image will often exhibit brightness variation, or shading. Of interest in computer vision is the inverse problem of how object shape may be recovered from such an image. This is referred to as the shape-from-shading problem. When the imaging conditions are such that an overhead point-source illuminates a smooth Lambertian surface, the problem may be formulated mathematically as that of finding a solution to an eikonal equation. In this paper, we seek images for which there are no corresponding object shapes. We are therefore concerned with the nonexistence of (bounded) solutions to certain eikonal equations. Specifically, we give a necessary and sufficient condition for a circularly-symmetric eikonal equation to admit exclusively unbounded solutions. In addition, we give a sufficient condition for an eikonal equation to have no solution. Examples are presented that elucidate the significance of these results for computer vision.
On the solution to a class of strongly singular linear integral equations
Authors: Gautesen A. K.
Published: December 2016

A strongly singular linear integral equation containing a small positive parameter

δ
\delta

is considered. This equation is transformed into a Fredholm integral equation of the second kind with a continuous kernel. The rate of convergence of the Neumann series for this integral equation is shown to be

o

(

δ
2

)

o\left ( {{\delta ^2}} \right )

. An example from fracture mechanics is considered in detail.
The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers’ equation
Authors: Deng Keng , Kwong Man Kam , Levine Howard A.
Published: December 2016

We study the long time behavior of solutions of Burgers’s equation with nonlocal nonlinearities:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline equals u Subscript x x Baseline plus epsilon u u Subscript x Baseline plus one half left-parenthesis a parallel-to u left-parenthesis dot comma t right-parenthesis parallel-to Superscript p minus 1 Baseline plus b right-parenthesis u comma 0 greater-than x greater-than 1 comma a comma epsilon element-of double-struck upper R comma b greater-than 0 comma p greater-than 1">

u
t

=

u

x
x

+
ε
u

u
x

+

1
2

(

a
∥
u

(

⋅
,
t

)

∥

p
−
1

+
b

)

u
,
0
>
x
>
1
,

a
,
ε
∈

R

,
b
>
0
,
p
>
1

{u_t} = {u_{xx}} + \varepsilon u{u_x} + \frac {1}{2}\left ( {a\parallel u\left ( { \cdot , t} \right ){\parallel ^{p - 1}} + b} \right )u, 0 > x > 1, \\ a, \varepsilon \in \mathbb {R}, b > 0, p > 1

, subject to

u

(

0
,
t

)

=
u

(

1
,
t

)

=
0

u\left ( {0, t} \right ) = u\left ( {1, t} \right ) = 0

. A stability-instability analysis is given in some detail, and some finite time blow up results are given.
A new method for solving dynamically accelerating crack problems. I. The case of a semi-infinite mode 𝐼𝐼𝐼 crack in elastic material revisited
Authors: Walton J. R. , Herrmann J. M.
Published: December 2016

Presented here is a new method for constructing solutions to dynamically accelerating, semi-infinite crack problems. The problem of a semi-infinite, anti-plane shear (mode III) crack accelerating dynamically in an infinite, linear, homogeneous, and isotropic elastic body has been solved previously by Freund and Kostrov. However, their methods are based upon the construction of a certain Green’s function for the ordinary two-dimensional wave equation and do not generalize to either the opening mode problem in elastic material or viscoelastic material. What is presented here is a new approach based upon integral transforms and complex variable techniques that does, in principle, generalize to both the opening modes of deformation and viscoelastic material. Moreover, the method presented here produces directly, for the mode III crack, a simple closed form expression for the crack-face profile for arbitrary applied crack-face tractions. Generalizations to opening modes of deformation and viscoelastic material produce integral equations for the crack-face displacement profile that in some cases admit closed form solutions and otherwise can be solved numerically. In contrast, the method of Freund and Kostrov yields, in mode III, an expression for the stress in front of the crack, but for opening modes provides only the stress intensity factor.
The gravity waves created by a moving source in a fluid of finite depth
Authors: Pramanik A. K. , Majumdar M.
Published: December 2016

The linear gravity waves created by a moving oscillatory source are considered in a fluid of finite uniform depth bounded on one side by a vertical cliff. The unsteady asymptotic waves are determined by Bliestien’s asymptotic expansion to the exact Fourier transform solution. Several physically interesting results are obtained. The fronts of each of the waves generated by the source are determined for all times and for all values of the parameters. The physical cause of the singularity of solution for certain values of the parameters is explained.
Stationary Binnie waves near resonance
Authors: Zhu Songping
Published: December 2016

Stationary waves in an open channel with corrugated side walls possess resonant speeds at or near which the prediction of free surface waves by the linear approximation fails. However, if the nonlinear terms in the Euler equation and boundary conditions are kept, a finite amplitude of these stationary waves can be determined. In this paper, such a nonlinear approach is presented, and the amplitude of the waves near the resonant speeds is determined to be of the order of the one-third power of the side-wall corrugation amplitude.
An explosive instability in magnetic fluids
Authors: Malik S. K. , Singh M.
Published: December 2016

The stability of a magnetic fluid subject to a normal magnetic field at the second harmonic resonance is investigated. The stability characteristics are examined on the basis of two coupled dynamical equations that are solved numerically. The solution reveals that there exists an explosive instability when the applied magnetic field strength is in the neighbourhood of the bifurcation magnetic field strength

H

m

H_{m}

.
Asymptotic solutions of a generalized Burgers equation
Authors: Vanaja V. , Sachdev P. L.
Published: December 2016

The travelling wave solutions of the generalized Burgers equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential u Over partial-differential t EndFraction equals StartFraction partial-differential Over partial-differential x EndFraction left-bracket upper D left-parenthesis u right-parenthesis StartFraction partial-differential u Over partial-differential x EndFraction right-bracket minus StartFraction partial-differential Over partial-differential x EndFraction left-bracket upper K left-parenthesis u right-parenthesis right-bracket">

∂
u

∂
t

=

∂

∂
x

[

D

(
u
)

∂
u

∂
x

]

−

∂

∂
x

[

K

(
u
)

]

\frac {{\partial u}}{{\partial t}} = \frac {\partial }{{\partial x}}\left [ {D\left ( u \right ) \frac {{\partial u}}{{\partial x}}} \right ] - \frac {\partial }{{\partial x}}\left [ {K\left ( u \right )} \right ]

\]
are related to the solution of the initial boundary value problems for the same equation, subject to initial boundary conditions relevant to the physical problem of infiltration of moisture into a homogeneous soil. The theoretical prediction of the emergence of the travelling wave solutions as intermediate asymptotics is confirmed by numerical solutions of the problem for some specific choices of the functions

D

(
u
)

D\left ( u \right )

and

K

(
u
)

K\left ( u \right )

.
Sur une classe de fluides non newtoniens: les solutions aqueuses de polymères
Authors: Amrouche C.
Published: December 2016

The aim of this paper is to study a nonlinear evolution system of third order representing an approximation of Navier-Stokes equations. This system describes the motion of a viscous fluid to which a small quantity of polymers is added. The consequently main relaxation properties of the resulting fluid are completely changed.
On phase-locked motions associated with strong resonance
Authors: Yu P. , Huseyin K.
Published: December 2016

This paper is concerned with the stability and bifurcation behaviour of a nonlinear autonomous system in the vicinity of a compound critical point characterized by two pairs of pure imaginary eigenvalues of the Jacobian. Attention is focused on the local dynamics of the system near-to-resonance. The methodology developed earlier for the bifurcation analysis into periodic and quasi-periodic motions (unification technique coupled with the intrinsic harmonic balancing) is extended to consider the stability and bifurcations of resonant cases. A set of simplified rate equations characterizing the local dynamics of the system is derived. These equations differ from those associated with nonresonant cases in that they are phase-coupled. Furthermore, the stability conditions of the phase-locked periodic bifurcation solutions are presented. All the results are expressed in explicit forms.
The thermistor problem for conductivity which vanishes at large temperature
Authors: Chen Xinfu , Friedman Avner
Published: December 2016

The thermistor problem is modeled as a coupled system of nonlinear elliptic equations. When the conductivity coefficient

σ

(
u
)

\sigma \left ( u \right )

vanishes (

u
u

= temperature) one of the equations becomes degenerate; this situation is considered in the present paper. We establish the existence of a weak solution and, under some special Dirichlet and Neumann boundary conditions, analyze the structure of the set

{

σ

(
u
)

=
0

}

\left \{ {\sigma \left ( u \right ) = 0} \right \}

and also prove uniqueness.
Nonlinear vortex trail dynamics. II. Analytic solutions
Authors: Lim Chjan C. , Sirovich Lawrence
Published: December 2016

Spatially periodic large amplitude solutions of the von Karman model are obtained in the neighborhood of singularities. These singularities correspond to vortex clusters in the physical plane. The quasi-periodic and unbounded solutions found analytically confirm earlier numerical work and show qualitative agreement with experimental observations of large-scale phenomena of vortex trails. Separatrices or heteroclinic orbits were explicitly found for an integrable approximate equation, which indicate that the von Karman model itself supports chaotic solutions.
Spectral conditions and an explicit expression for the stabilization of hybrid systems in the presence of feedback delays
Authors: Cooke K. L. , Turi J. , Turner G.
Published: December 2016

We consider the effect of feedback delays in the stabilization of linear time-invariant plants with sampled outputs. In particular, we obtain an estimate on the “minimum” sampling frequency (in terms of the spectrum of the plant and the delay in the feedback mechanism) needed for stabilization and provide an explicit expression for a stabilizing feedback control.
Unimodality and viscoelastic pulse propagation
Authors: Gripenberg Gustaf
Published: December 2016

Sufficient conditions on the stress relaxation modulus in a viscoelastic material are given in order for initially short mechanical pulses in the material to remain unimodal.
Equilibria of the circular elastica under a uniform central force field
Authors: Dickey R. W. , Roseman J. J.
Published: December 2016

A mathematical model for the problem of an inextensible circular elastica under a uniform centrally directed force field is derived and studied. It is shown analytically that stable large amplitude solutions exist at forces

P
>

P
1

P > {P_1}

, the first eigenpressure for the linearized model, and it is shown numerically that these solutions have only one axis of symmetry. These results agree with experiment. In addition, numerical solutions are calculated for states with more than one axis of symmetry which resemble those found in the literature on elastic rings under hydrostatic pressure.
Propagation and reflection of one-dimensional waves in ferroelectric ceramics
Authors: McCarthy M. F. , Moodie T. B. , Öncü T. S. , Sawatzky R. P.
Published: December 2016

Employing a linearized version of the general constitutive equations for a homogeneous deformable dielectric material body with memory, we study problems of wave propagation and reflection. We formulate these problems so as to correspond to certain well-defined experiments appearing in the literature and involving the application of time-dependent mechanical loads or prescribed voltage differences. Employing asymptotic procedures based upon geometrical optics we are able to solve problems of reflection hitherto unsolved, produce numerical results in an efficient manner, and elicit some important implications of the theory that appear not to have been noted before.
Stability of plane wave solutions of complex Ginzburg-Landau equations
Authors: Matkowsky B. J. , Volpert Vl. A.
Published: December 2016

We consider the stability of plane wave solutions of both single and coupled complex Ginzburg-Landau equations and determine stability domains in the space of coefficients of the equations.
Free-energy functions for elastic-plastic material elements
Authors: Lucchesi Massimiliano
Published: December 2016

Certain thermodynamic properties of elastic-plastic materials with work-hardening are discussed and their corresponding free-energy functions are determined.
Boundary value problems for variational integrals involving surface curvatures
Authors: Nitsche Johannes C. C.
Published: December 2016

The following investigation deals with surfaces governed by and extremal for a free energy functional which is quadratic in the principal curvatures. The associated Euler-Lagrange differential equations are derived, as are the corresponding intricate natural boundary conditions. Pertinent boundary value problems—without and with volume constraints—are formulated and discussed

1

^{1}

and existence proofs are provided for certain situations. The discussion opens the view onto an arena of rich mathematical problems which will also be of interest in engineering applications where the surfaces in question are utilized frequently as idealized models for the interfaces separating phases in real materials.
Analysis and application of a continuation method for a self-similar coupled Stefan system
Authors: Fehribach Joseph D.
Published: December 2016

This work deals with a continuation method for computing solutions to a self-similar two-component Stefan system in which the diffusion coefficients depend on the concentrations. The procedure computes a one-parameter homotopy connecting the known solution of a simplified problem (when the parameter is zero) to the solution of the problem at hand (when the parameter is one). Local convergence of the method and local existence and uniqueness of solutions for the original system are proven. Also, several examples coming from precipitant-driven protein crystal growth are discussed. The most interesting of these is a Stefan problem containing a porous media equation that corresponds to the liquid phase being in a meta-stable state near the spinodal region. The bifurcation code AUTO is used in the computations.
Nonlinear field instability and chaos in magnetic fluids
Authors: Malik S. K. , Singh M.
Published: December 2016

We study the nonlinear dynamics of normal field instability in a ferrofluid under the influence of a uniform magnetic field. In addition, a small normal sinusoidal magnetic field is superimposed on the system. An equation governing the evolution of small but finite amplitude is obtained. Applying the Melnikov method, it is shown that there exist transverse homoclinic orbits leading to chaotic motions.
Determination of the leading coefficient 𝑎(𝑥) in the heat equation 𝑢_{𝑡}=𝑎(𝑥)Δ𝑢
Authors: Hu Bei , Yin Hong-Ming
Published: December 2016

This note deals with the parabolic inverse problem of determination of the leading coefficient in the heat equation with an extra condition at the terminal. After introducing a new variable, we reformulate the problem as a nonclassical parabolic equation along with the initial and boundary conditions. The existence of a solution is established by means of the Schauder fixed-point theorem.
Complex dynamics in metal cutting
Authors: Berger B. S. , Rokni M. , Minis I.
Published: December 2016

The attractor associated with a system of nonlinear differential-delay equations, arising from the Wu-Liu metal cutting model, is shown to have a noninteger pointwise dimension and positive metric entropy. Projections of the attractor onto a two-dimensional plane substantiate the existence of complex dynamics. The result suggests that certain regenerative chatter states may be chaotic.
Global large solutions to initial-boundary value problems in one-dimensional nonlinear thermoviscoelasticity
Authors: Jiang Song
Published: December 2016

Initial boundary value problems in one-dimensional nonlinear thermo-viscoelasticity are considered, and the existence of global classical solutions is established by means of the Leray-Schauder fixed point theorem.
Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity
Authors: Racke Reinhard , Shibata Yoshihiro , Zheng Song Mu
Published: December 2016

We are mainly concerned with the Dirichlet initial boundary value problem in one-dimensional nonlinear thermoelasticity. It is proved that if the initial data are close to the equilibrium then the problem admits a unique, global, smooth solution. Moreover, as time tends to infinity, the solution is exponentially stable. As a corollary we also obtain the existence of periodic solutions for small, periodic righthand sides.
Equilibrium of permanent multivalued systems
Authors: Li Yong , Wang Huai Zhong , Lü Xian Rui
Published: December 2016

This paper is concerned with the existence problem of an equilibrium state for permanent multivalued systems. It is proved that the permanence of multivalued systems implies the existence of an equilibrium state by using the approximating technique of set-valued maps and asymptotic fixed point theory.
Integration of Prandtl’s equation with the aid of quadrature formulae of Gauss type
Authors: Dragoş Lazăr
Published: December 2016

The subject of this note is to present a method that can be used to integrate Prandtl’s equation by means of Gauss-type quadrature formulae.
On a periodic delay population model
Authors: Lalli B. S. , Zhang B. G.
Published: December 2016

The existence of a positive periodic solution for
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N prime left-parenthesis t right-parenthesis equals upper N left-parenthesis t right-parenthesis left-bracket a left-parenthesis t right-parenthesis plus b left-parenthesis t right-parenthesis upper N left-parenthesis t minus m omega right-parenthesis minus c left-parenthesis t right-parenthesis upper N squared left-parenthesis t minus m omega right-parenthesis right-bracket">

N
′

(
t
)
=
N
(
t
)

[

a
(
t
)
+
b
(
t
)
N
(
t
−
m
ω
)
−
c
(
t
)

N
2

(
t
−
m
ω
)

]

N’(t) = N(t)\left [ {a(t) + b(t)N(t - m\omega ) - c(t){N^2}(t - m\omega )} \right ]

\]
is established. Sufficient conditions are given for the periodic solution to be globally attractive.
Analytic solution of the Stefan problem in finite mediums
Authors: Kar A. , Mazumder J.
Published: December 2016

The classical Stefan problem is considered in this paper for finite mediums with Dirichlet boundary conditions. Analytic solutions for the temperature distributions and the location of the moving interface are obtained by using Lie group theory and the superposition principle. The existence of analytically exact solutions is established by proving the convergence of the solution.
Pseudo-sedimentation dialysis: an elliptic transmission problem
Authors: Nitsche Ludwig C.
Published: December 2016

In this paper we propose and analyze a novel membrane-based fractionation process that combines conventional countercurrent dialysis with a phenomenon of “pseudo-sedimentation". An elliptic transmission problem is formulated to describe the steady-state concentration field of a given chemical species. Matching of spectral expansions at the transmission boundary yields an infinite system of linear algebraic equations for the eigenfunction expansion coefficients of the solution. Existence and uniqueness of the solution, framed in terms of a Fredholm alternative, are proven within a subset of the parameter space, along with regularity at the transmission boundary. The paper also develops a perturbation solution in order to interpret the theory in physical terms. Finally, numerically generated concentration profiles indicate the degree of selectivity that could be achieved with a separation apparatus based upon our physical concept.
The energy release rate for transient dynamic mode 𝐼 crack propagation in a general linearly viscoelastic body
Authors: Herrmann J. M. , Walton J. R.
Published: December 2016

A mathematical model of a semi-infinite mode I crack that suddenly begins to propagate at constant speed is constructed for a general linear viscoelastic body. Expressions for the Laplace transform of the stress, displacement, and stress intensity factor are derived for general loadings. A Barenblatt type process zone is incorporated into the model and used to determine the total energy flux into the crack tip. This energy release rate,

G

(
t
)

G\left ( t \right )

, is constructed for two specific loadings: one following the advancing crack tip, the second remaining fixed as the crack tip advances. In each case

G

(
t
)

G\left ( t \right )

is analyzed by asymptotic and numerical methods to determine its qualitative form and, in particular its rate of decay to its steady-state value. The effect of such simplifying assumptions as quasi-static propagation or an elastic material is also illustrated. The second loading is intended as an idealized model of the dynamic fracture experiments of Ravi-Chandar and Knauss [13–16].
Lagrangian mechanics, Gauss’ principle, quadratic programming, and generalized inverses: new equations for non-holonomically constrained discrete mechanical systems
Authors: Kalaba Robert E. , Udwadia Firdaus E.
Published: December 2016

In this paper we formulate Lagrangian mechanics as a constrained quadratic minimization problem. This quadratic minimization problem is then solved using the theory of generalized inverses of matrices thereby obtaining the explicit equations of motion of constrained, discrete mechanical systems. The approach extends the boundaries of Lagrangian mechanics in that we provide a general formulation for describing the constrained motion of such systems without either the use of Lagrange multipliers or the use of quasi-coordinates. An important feature of the approach is that we do not require prior knowledge of the specific set of constraints to accomplish this formulation. This makes the equations presented here more generally useful, and perhaps more aesthetic, than the Gibbs-Appell equations which require a felicitous choice of problem-specific quasi-coordinates. The new equations of motion presented here are applicable to both the holonomic and nonholonomic constraints that Lagrangian mechanics deals with. They are obtained in terms of the usual generalized coordinates used to describe the constrained system. Furthermore, they can be integrated by any of the currently available numerical integration methods, thus yielding analytical and/or computational descriptions of the motions of constrained mechanical systems.
A principal stress contour for nonlinear elastic deformations
Authors: Shi Jingyu
Published: December 2016

We seek special principal stress contours, for a nonlinear elastic plane deformation, on which both the principal stresses are constant and the principal axes do not change directions. It is found that they must be straight lines and that the lines must be parallel to one another or intersect at one point. It is also shown that the principal stress contours in the deformation for the straightening of a circular arc are such straight lines.
Three-dimensional steady-state indentation problem for a general viscoelastic material
Authors: Golden J. M. , Graham G. A. C. , Lan Q.
Published: December 2016

The steady-state problem is considered for periodic normal loading by a smooth, rigid indentor on a half-space exhibiting general viscoelastic behaviour. A technique is developed for summing the infinite series that arise. The method is applicable to the case in which the viscoelastic behaviour is described by a discrete spectrum model, in other words, by a finite sum of decaying exponentials. Numerical results are presented for two decay times. The extension to any number of decay times is straightforward.
A note on Onsager’s relations
Authors: Šilhavý M.
Published: December 2016

A nonlinear viscoelastic material with the heat flux obeying a generalization of Cattaneo’s law is considered. It is shown that for slow processes with small gradients of temperature the exact constitutive equations can be approximated by those of a linear viscous material with Fourier heat conduction. As a consequence of the thermodynamic restrictions on the original constitutive equations, the approximate constitutive equations are shown to satisfy the principle of local equilibrium for energy and entropy, and the kinetic coefficients giving the viscous stress and heat flux vector satisfy Onsager’s relations.
Oscillation and stability in a simple genotype selection model
Authors: Grove E. A. , Kocić V. Lj. , Ladas G. , Levins R.
Published: December 2016

We study the oscillation, the stability, and the global attractivity of the simple genotype selection model
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Subscript n plus 1 Baseline equals StartFraction y Subscript n Baseline e Superscript beta left-parenthesis 1 minus 2 y Super Subscript n minus k Superscript right-parenthesis Baseline Over 1 minus y Subscript n Baseline plus y Subscript n Baseline e Superscript beta left-parenthesis 1 minus 2 y Super Subscript n minus k Superscript right-parenthesis Baseline EndFraction comma n equals 0 comma 1 comma period period period comma">

y

n
+
1

=

y
n

e

β

(
1
−
2

y

n
−
k

)

1
−

y
n

+

y
n

e

β

(
1
−
2

y

n
−
k

)

,

n
=
0
,
1
,
.
.
.
,

{y_{n + 1}} = \frac {{{y_n}{e^{\beta \left ( 1 - 2{y_{n - k}}\right )}}}}{{1 - {y_n} + {y_n}{e^{\beta \left (1 - 2{y_{n - k}} \right )}}}}, \qquad n = 0, 1,...,

\]
where

β
∈

(
0
,
∞
)

\beta \in \left ( 0, \infty \right )

and

k
∈
{
0
,
1
,
2
,
.
.
.
}

k \in \{ 0, 1, 2,...\}

.
A parabolic system modeling the thermoelastic contact of two rods
Authors: Andrews Kevin T. , Shi Peter , Shillor Meir , Wright Steve
Published: December 2016

We consider an initial-boundary value problem for a nonlinear parabolic system that arises naturally in modeling behavior of two (possibly dissimilar) thin homogeneous rods. Each rod is held fixed at one end and is free to expand or contract at the other, as a result of the evolution of its temperature and stress fields. The two rods may also come into contact at their free ends. We establish the existence and uniqueness of a strong solution to the system, assuming that the thermal expansion coefficients are small. The proofs rely on a priori estimates, the functional method of Ladyzhenskaya, and Schauder’s fixed-point theorem.
Study of an inclined, segmented cantilever with tip load
Authors: Wang C. Y. , Du Q.
Published: December 2016

The problem of a tip-loaded

N
N

-segment cantilever with torsion springs at the joints is formulated through nonlinear difference equations. The solutions are obtained by both numerical and perturbation methods. As the base is rotated the cantilever shows nonlinear behavior such as nonuniqueness, hysteresis, and catastrophic changes. A stability criterion is proposed.
Steady time-harmonic oscillations in a linear thermoelastic plate model
Authors: Schiavone Peter , Tait R. J.
Published: December 2016

We examine the bending of a Mindlin-type thermoelastic plate when the source terms are time-harmonic with angular frequency

ω
\omega

, and sufficient time has elapsed for the system to have reached a steady-state. We show that in an infinite plate the solution can be represented as the sum of five waves all but one of which exhibit damping. By formulating appropriate radiation conditions we prove uniqueness results for exterior boundary value problems subject to certain regularity assumptions and a condition on the angular frequency of oscillation.
A Galerkin solution to a regularized Cauchy singular integro-differential equation
Authors: Frankel Jay I.
Published: December 2016

This paper presents a Galerkin approach for solving a regularized version of the Cauchy singular, linear integro-differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d normal upper Theta Over d x EndFraction left-parenthesis x right-parenthesis minus f left-parenthesis x right-parenthesis equals lamda integral Subscript y equals 0 Superscript 1 Baseline StartFraction normal upper Theta left-parenthesis y right-parenthesis Over x minus y EndFraction d y comma x element-of left-parenthesis 0 comma 1 right-parenthesis">

d
Θ

d
x

(
x
)

−
f

(
x
)

=
λ

∫

y
=
0

1

Θ

(
y
)

x
−
y

d
y
,

x
∈

(
0
,
1
)

\frac {{d\Theta }}{{dx}}\left ( x \right ) - f\left ( x \right ) = \lambda \smallint _{y = 0}^1\frac {{\Theta \left ( y \right )}}{{x - y}}dy, \qquad x \in \left ( 0, 1 \right )

\]
, subject to

Θ

(
0
)

=
Θ

(
1
)

=
0

\Theta \left ( 0 \right ) = \Theta \left ( 1 \right ) = 0

. This equation has appeared in both combined infrared gaseous radiation and molecular conduction, and elastic contact studies. A regularized formulation is produced which suggests the use of an expansion technique where the orthogonal basis functions are chosen as the Chebychev polynomials of the first kind. Accurate results, requiring a minimal computational cost, are formally documented and compared to a purely numerical solution.
Monotonic, completely monotonic, and exponential relaxation functions in linear viscoelasticity
Authors: Del Piero Gianpietro , Deseri Luca
Published: December 2016

A priori restrictions on the relaxation function of linear viscoelasticity are studied under regularity assumptions weaker than those usually made in the literature. The new set of assumptions is sufficient to define, by a limit procedure, the work done in deformation processes in which some parts are subject either to extreme retardations or to extreme accelerations. The use of such processes results in a considerable simplification of the proofs of some classical results. Under the same assumptions, we give a characterization of the monotonicity of the relaxation function in terms of work. We also extend an earlier one-dimensional characterization of complete monotonicity due to Day, and prove that the work done in every closed path in stress-strain space is nonnegative if and only if the relaxation function is of exponential type.
A novel method for simulating the complex Ginzburg-Landau equation
Authors: Goldman Daniel , Sirovich Lawrence
Published: December 2016

We present a split-step method for integration of the complex Ginzburg-Landau equation in any number of spatial dimensions. The novel aspect of the method lies in the fact that each portion of the splitting is explicitly integrable. This leads to an extremely fast, stable, and efficient procedure. A comparison is made with spectral and pseudospectral procedures which have appeared in the literature.
On unstable and neutral spectra of incompressible inviscid and viscid fluids on the 2D torus
Authors: Liu Vincent Xiaosong
Published: December 2016

We completely determine all the unstable modes and neutral modes for the Kolmogorov flow

(
sin
⁡

(
2

x
2

)

,
0
)

\left ( \sin \left ( 2{x_2} \right ), 0 \right )

on the two-dimensional torus

[
0
,
2
π
]

×

[
0
,
2
π
]

\left [ 0, 2\pi \right ] \times \left [ 0, 2\pi \right ]

for both the Navier-Stokes equations and the Euler equations. We also give comments on “vorticity generation".
Steady state solution for electrochemical processes with multiple reacting species
Authors: Yu Xun
Published: December 2016

At steady state, the dissociation-association electrochemical reactions in a dilute solution are governed by the nondimensional equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential Over partial-differential x EndFraction left-parenthesis d Subscript i Baseline StartFraction partial-differential u Subscript i Baseline Over partial-differential x EndFraction plus d Subscript i Baseline e Subscript i Baseline StartFraction partial-differential phi Over partial-differential x EndFraction u Subscript i Baseline right-parenthesis plus upper R Subscript i Baseline equals 0 comma i equals 1 comma period period period comma m comma 0 less-than-or-equal-to x less-than-or-equal-to 1 comma sigma-summation Underscript i equals 1 Overscript m Endscripts e Subscript i Baseline u Subscript i Baseline equals 0">

∂

∂
x

(

d
i

∂

u
i

∂
x

+

d
i

e
i

∂
ϕ

∂
x

u
i

)

+

R
i

=
0
,

i
=
1
,
.
.
.
,
m
,
0
≤
x
≤
1
,

∑

i
=
1

m

e
i

u
i

=
0

\frac {\partial }{{\partial x}}\left ( {{d_i}\frac {{\partial {u_i}}}{{\partial x}} + {d_i}{e_i}\frac {{\partial \phi }}{{\partial x}}{u_i}} \right ) + {R_i} = 0, \qquad i = 1,...,m, 0 \le x \le 1, \\ \sum \limits _{i = 1}^m {{e_i}{u_i} = 0}

\]
where

d
i

,

e
i

{d_i}, {e_i}

, and

u
i

{u_i}

are the diffusion coefficient, charge, and concentration of the

i
i

th species, respectively.

ϕ
\phi

is the electric potential of the solution.

R
i

{R_i}

is the net source of the

i
i

th species in the solution which includes the external input or production due to the reactions in the solution. The extra electroneutrality condition

∑

i
=
1

m

e
i

u
i

=
0

\sum _{i = 1}^m {e_i}{u_i} = 0

determines the electric potential

ϕ
\phi

. This system of nonlinear differential equations is subject to the nonlinear boundary conditions modeling the actual electrode kinetics. We prove the existence of the solution to this system and construct an iterative numerical algorithm to compute the solution. Numerical results are also presented in the paper.
On the well-posedness of the initial value problem for elastic-plastic oscillators with isotropic work-hardening
Authors: Wang Keming
Published: December 2016

The system of differential equations of elastic-plastic oscillators with isotropic work-hardening is converted to a system of differential inclusions and well-posedness is established using maximal monotone operator theory when the external force

f
∈

W

1
,
1

(
0
,
T
;
R
)

f \in {W^{1, 1}}\left ( 0, T; R \right )

. By a more delicate analysis, well-posedness is also established for

f
∈

L
1

(
0
,
T
;
R
)

f \in {L^1}\left ( 0, T; R \right )

.
A generalisation of Prüfer’s transformation
Authors: Anderson N. , Arthurs A. M.
Published: December 2016

A canonical version of the Prüfer system for Sturm-Liouville equations is presented. The partial uncoupling of the first-order equations remains, providing a basis for their solution in succession.
A model for the effect of roughness on the load capacity of journal bearings
Authors: Pravica D. W.
Published: December 2016

The flow generated by a rotating circular cylinder inside a corrugated outer cylinder is considered. The corrugations are small and are used to model roughness. The force on the inner cylinder, as a linear function of displacement, is found. This requires a Stokes expansion to determine a first-order correction in the Reynolds number

R
R

. The result implies as expected that transverse roughness acts to increase the load capacity of the bearing at low

R
R

, which is confirmed by previous numerical experiments. In addition, roughness can induce reversed flow at much smaller displacements than for smooth bearings.
A Riemann problem for an elastic bar that changes phase
Authors: Lin Yier
Published: December 2016

This paper is concerned with the dynamics of an elastic bar that can undergo reversible stress-induced phase transformations. We consider a Riemann problem in which the initial strains belong to a single metastable phase and prove uniqueness of solution that satisfies a nucleation criterion and a kinetic law at all subsonic and sonic phase boundaries. This paper generalizes the results of [3]; the authors of [3] considered a piecewise-linear material for which no wave fans exist, shock waves always travel at the acoustic speed, and shock waves are dissipation-free. The material model of the present paper does not suffer from these degeneracies.