bluuebunny/crossref_metadata_2025 huggingface dataset:
Fluid flow in a layered medium
Authors: Clark G. W. , Showalter R. E.
Published: December 2016

A layered medium is modeled as a continuous distribution of relatively flat cells within a spatial region. Local flow within each cell as well as the exchange with the global flow over the region is modeled by a quasilinear parabolic system of partial differential equations, and the local geometry of the individual cells is included in the model. We introduce new terms to account for the secondary flux corresponding to either transverse flow across the cells or direct cell-to-cell diffusion driven by the global density gradient. The resulting initial-boundary-value problem is shown to be well-posed and to depend continuously on the parameter defining the type of interface condition on cell boundaries.
Direct boundary integral equation method in the theory of elasticity
Authors: Kuznetsov S. V.
Published: December 2016

The Direct Boundary Integral Equation Method (BIEM), which leads to second-kind singular integral equations for all types of commonly used boundary-value problems of the theory of elasticity, is formulated. The exposition is applied to anisotropic bodies with arbitrary elastic anisotropy.
An inverse problem for the linear viscoelastic Kirchhoff plate
Authors: Cavaterra Cecilia , Grasselli Maurizio
Published: December 2016

We consider the linear Kirchhoff plate model subject to a viscoelastic damping of integral type. The damping term contains a time-dependent convolution kernel accounting for the long-range memory effects. The mechanical behavior of the plate is influenced by this memory function which is a priori unknown. That leads us to consider the question of identifying it. We study this kind of inverse problem by using the integrodifferential evolution equation governing the vertical deflection of the plate and coupling it with a set of overposed initial and boundary conditions. The problem obtained is then reduced to a nonlinear Volterra integral equation of second kind for the unknown memory kernel. Then, via the Contraction Principle, we prove local (in time) existence and uniqueness results. In addition, we show the Lipschitz continuous dependence upon the data. These results also apply to a viscoelastic beam model.
Buckling, barrelling, and surface instabilities of a finite, transversely isotropic circular cylinder
Authors: Chau K. T.
Published: December 2016

This paper investigates all possible geometric instabilities, such as buckling, barrelling, surface wrinkling, and orange-peel mode instabilities, for a finite length transversely isotropic circular cylinder under axisymmetric tension or compression. The constitutive responses of the cylinder can, in general, be transversely isotropic, compressible, and elastic-plastic; and no existence of rate potential is assumed. A general procedure is given to calculate the smallest critical ratio of radius to length,

(
a

/

L
)
)

c
r

{\left ( a/L) \right )_{cr}}

, of a circular cylinder that separates the buckling (antisymmetric) and the barrelling (axisymmetric) instabilities. Numerical results show that

(
a

/

L
)

c
r

{\left ( a/L \right )_{cr}}

is very sensitive to the constitutive model used. The stress at the maximum load point is obtained as the first term of the axisymmetric long wavelength limit; the Euler buckling formula is obtained as the first term of the antisymmetric long wavelength limit. Three different types of surface instabilities are considered. The eigenstress for the longitudinal short wavelength limit corresponds to that for the wrinkling instability of a halfspace under plane deformations; the circumferential short wavelength limit is always possible; and the longitudinal and circumferential short wavelength limit, which corresponds to the surface instability with an “orange-peel” appearance, is available along the tensile elliptic-parabolic boundary. Numerical results show that eigenmodes of

n
≥
2

n \ge 2

(where

n
n

is the circumferential wave number) can be the first bifurcation encountered under tension depending on the geometric ratio,

a

/

L

a/L

. However, such eigenmodes will not be the first possible bifurcation under compression unless the ratio of the incremental transverse shear modulus to the incremental longitudinal shear modulus drops below, approximately, 0.25.
Compactness framework and convergence of Lax-Friedrichs and Godunov schemes for a 2×2 nonstrictly hyperbolic system of conservation laws
Authors: Rubino Bruno
Published: December 2016

A compactness framework and a convergence theorem for the Lax-Friedrichs scheme and the Godunov scheme applied to the Cauchy problem for a

2
×
2

2 \times 2

nonstrictly hyperbolic system of conservation laws are established. The existence of weak solutions is proved using the theory of compensated compactness of Tartar, Murat, DiPerna, and Serre.
Nonsteady stability of the flow around the circle in the Föppl model
Authors: Tordella Daniela
Published: December 2016

The Föppl model for the incompressible flow about the circle is considered. The major feature of this model is the understanding of the convective mechanism that gives rise to the onset of the instability in the flow past the circle. Through this model the real flow is approximated by means of a potential field with two singular points, counter-rotating vortices, placed symmetrically behind the circle. This field configuration is topologically analogous to the actual steady field for Reynolds numbers below the critical value corresponding to the onset of the first instability. The intrinsic instability of the Föppl field to small perturbations may be described by a second-order linear dynamical system.
On the accuracy of the stationary forced Korteweg-de Vries equation as a model equation for flows over a bump
Authors: Shen Samuel Shan Pu
Published: December 2016

Considered are the stationary supercritical free-surface solitary waves and subcritical free-surface hydraulic falls in flows over a bump in a two-dimensional channel. The stationary forced Korteweg-de Vries equation (sfKdV) sometimes can be used to predict these solitary waves and hydraulic falls. This paper studies the deviation of the results deduced from the sfKdV from the exact solutions found by computations and experiments. The elucidation of this deviation (called the error) is by comparing the sfKdV results with the computational and experimental results since an analytic approach seems impossible. From the comparisons made in this paper and the previous studies in [6], we conclude that the sfKdV is a good model in the sense that the relative error is less than 10
Cavitation solutions to homogeneous van der Waals type fluids involving phase transitions
Authors: Yan Baisheng
Published: December 2016

In this paper, weak solutions to some special Cauchy problems involving phase transitions in

R
3

{R^3}

are constructed. These solutions exhibit the point singularity known as cavitation.
A variation of the intrinsic multiple-scale harmonic balance method
Authors: Atadan A. S.
Published: December 2016

A variation of the intrinsic multiple-scale harmonic balance method is introduced by combining the intrinsic multiple-scale harmonic balance method with the ideas introduced to modify the method of multiple-scales. The combined method has the advantage of having the desirable characteristics of both techniques. This is demonstrated by solving the Duffing equation.
Webs, iteration groups, and equivalent changes in probabilities
Authors: Aczél János , Rote Günter , Schwaiger Jens
Published: December 2016

Yew-Kwang Ng [12] listed several “reasonable properties” for equivalent changes of probabilities and other proportions. He produced a family of functions satisfying all properties and asked whether there exist essentially different ones. We show that this is the case, by constructing uncountably many families of functions satisfying all properties. We show also that there are no other solutions. Our method establishes connections with webs (nets) and iteration groups. This may be of interest both in itself and for applications.
On the forced surface waves due to a partially immersed vertical wave-maker in water of infinite depth
Authors: Rhodes-Robinson P. F.
Published: December 2016

In this paper we determine the velocity potential describing the two-dimensional antisymmetric wave motion generated by a harmonically oscillating vertical plane wave-maker partially immersed in water of infinite depth. The solution of the boundary-value problem for the velocity potential is obtained from a formulation depending on the unknown horizontal velocity on the line of antisymmetry below the wave-maker, which leads to a singular integral equation of known type for a related reduction variable. The idea was developed by Ursell to solve the corresponding transmission problem and another rigid wave-maker problem, and uses classical wave-maker theory.
Slow oscillatory Stokes flow
Authors: Smith S. H.
Published: December 2016

Two transient problems in slow viscous flow are considered where the corresponding steady-state behaviour leads to the paradoxical results of Stokes and Jeffery. First, the oscillatory flow past a circular cylinder is investigated when the frequency

λ
\lambda

tends to zero, where an outer domain of size

O

(

λ

−
1

/

2

)

O\left ( {\lambda ^{ - 1/2}} \right )

is required to ensure that the velocity conditions at infinity are satisfied. The flow close to the cylinder is quasi-steady except for a time of length

O

(
1
)

O\left ( 1 \right )

about the time of separation; most of the action takes place in the outer domain where the dominant transient behaviour extends over a time that is

O

[

{

l
n

(

λ

−
1

)

}

−
1

]

O\left [ {{{\left \{ {ln\left ( {\lambda ^{ - 1}} \right )} \right \}}^{ - 1}}} \right ]

of a complete cycle.
Boundary value problems of holomorphic vector functions and applications to anisotropic elasticity
Authors: Wang M. Z. , Yan G. P.
Published: December 2016

Using Stroh’s formalism, plane problems of anisotropic elasticity are turned into the boundary value problems of holomorphic functions. A general method is presented for solving the boundary value problems. The displacement and the stress boundary value problems of an anisotropic body in an elliptical region are solved.
Boltzmann systems for gas mixtures in 1D torus
Authors: Wu Kung-Chien
Published: December 2016

We study the 1D Boltzmann equation for a mixture of two gases on a torus with the initial condition of one gas near a vacuum and the other near a Maxwellian equilibrium state. An

L

x

∞

L

ξ
,
β

∞

L^{\infty }_{x}L^{\infty }_{\xi ,\beta }

analysis is developed to study this mass diffusion problem, which is based on the Boltzmann equation for the single species hard sphere collision in an earlier work of the author. The decay rate of the solution is algebraic for a small time region and exponential for a large time region. Moreover, the exponential rate depends on the size of the domain.
Eigenfrequencies of the non-collinearly coupled Euler-Bernoulli beam system 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 the general non-collinear Euler-Bernoulli beam equation with dissipative joints. Many different kinds of dampers are allowed, even within the same structure. This generalizes a previous result for collinear structures. Matrix techniques are used to combine asymptotic analysis with the wave propagation method. We will find that if the lengths of the beams are rational, there will be a finite number of “streams” of eigenfrequencies, and, like the collinear case, each lies asymptotically to a vertical line.
An integro-differential equation arising from an electrochemistry model
Authors: Choi Y. S. , Lui Roger
Published: December 2016

In this paper, we prove the existence and uniqueness of steady-state solutions for a system of equations arising from a model in electrochemistry. The same result was established by the authors in an earlier paper under the additional assumptions that the space-dimension

N
=
2

N = 2

and the concentrations of the charged ions satisfy an electro-neutrality condition.
On extraordinary semisimple matrix 𝑁(𝑣) for anisotropic elastic materials
Authors: Ting T. C. T.
Published: December 2016

The

6
×
6

6 \times 6

real matrix

N

(
v
)

N\left ( v \right )

for anisotropic elastic materials under a two-dimensional steady-state motion with speed

v
v

is extraordinary semisimple when

N

(
v
)

N\left ( v \right )

has three identical complex eigenvalues

p
p

and three independent associated eigenvectors. We show that such an

N

(
v
)

N\left ( v \right )

exists when

v
≠
0

v \ne 0

. The eigenvalues are purely imaginary. The material can sustain a steady-state motion such as a moving line dislocation. Explicit expressions of the Barnett-Lothe tensors for

v
≠
0

v \ne 0

are presented. However,

N

(
v
)

N\left ( v \right )

cannot be extraordinary semisimple for surface waves. When

v
=
0

v = 0

,

N

(
0
)

N\left ( 0 \right )

can be extraordinary semisimple if the strain energy of the material is allowed to be positive semidefinite. Explicit expressions of the Barnett-Lothe tensors and Green’s functions for the infinite space and half-space are presented. An unusual phenomenon for the material with positive semidefinite strain energy considered here is that it can support an edge dislocation with zero stresses everywhere. In the special case when

p
=
i

p = i

is a triple eigenvalue, this material is an un-pressurable material in the sense that it can change its (two-dimensional) volume with zero pressure. It is a counterpart of an incompressible material (whose strain energy is also positive semidefinite) that can support pressure with zero volume change.
Weak solutions to a phase-field model with non-constant thermal conductivity
Authors: Laurençot Ph.
Published: December 2016

We investigate the existence of weak solutions to a phase-field model when the thermal conductivity vanishes for some values of the order parameter. We obtain weak solutions for a general class of free energies, including non-differentiable ones. We also study the

ω
\omega

-limit set of these weak solutions, and investigate their convergence to a solution of a degenerate Cahn-Hilliard equation.
Vortices and boundaries
Authors: Chapman S. J. , Hunton B. J. , Ockendon J. R.
Published: December 2016

This paper develops a framework in which vortex/boundary interactions can be analysed for the cases of inviscid fluid dynamics and type II superconductors. Comments are made concerning the possibility of reconnection, defined as an interaction in which an initially single-component vortex eventually emerges with two components.
The calculus of variations and materials science
Authors: Ball J. M.
Published: December 2016

A review is given of the development and present state of the calculus of variations, starting from the problem of the brachistochrone, and emphasizing the current interaction with problems of materials science.
Mathematics applied in fluid motion
Authors: Stuart J. T.
Published: December 2016

Over many decades, indeed for more than a century, fluid dynamics has been the subject of many beautiful experiments and has been a proving ground for a wealth of mathematical theories, both linear and nonlinear.
Sharp stability estimates for quasi-autonomous evolution equations of hyperbolic type
Authors: Souplet Philippe
Published: December 2016

We study the energy decay of the difference of two solutions for dissipative evolution problems of the type:
\[

u
+
L
u
+
g
(

u
′

)
=
h
(
t
)
,

t
≥
0
,

u + Lu + g(u’) = h(t), \qquad t \ge 0 ,

\]
including wave and plate equations and ordinary differential equations. In the general case, when the damping term g behaves like a power of the velocity ú, the energy decreases like a negative power of time, multiplied by a constant depending on the initial energies. We provide estimates on these constants and prove their optimality. In the special case of the ordinary differential equation with periodic forcing, we establish, relying on a controllability-like technique, that the decay is in fact exponential, even under very weak damping.
Electrodiffusional free boundary problem, in a bipolar membrane (semiconductor diode), at a reverse bias for constant current
Authors: Primicerio M. , Rubinstein I. , Zaltzman B.
Published: December 2016

A singular perturbation problem, modeling one-dimensional time-dependent electrodiffusion of ions (holes and electrons) in a bipolar membrane (semi-conductor diode) at a reverse bias is analyzed for galvanostatic (fixed electric current) conditions. It is shown that, as the perturbation parameter tends to zero, the solution of the perturbed problem tends to the solution of a limiting problem which is, depending on the input data, either a conventional bipolar electrodiffusion problem or a particular electrodiffusional time-dependent free boundary problem. In both cases, the properties of the limiting solution are analyzed, along with those of the respective boundary and transition layer solutions.
Saint Venant’s principle in orthotropic planar elasticity: rates-of-diffusion for stress
Authors: Stronge W. J. , Kashtalyan M.
Published: December 2016

For plane deformations generated by an arbitrary distribution of tractions applied in a small region on the boundary of an elastic half-plane, the rates-of-decay for displacements, stresses and strain energy density are obtained as functions of complexity of the load distribution. The rates-of-decay increase in proportion to the complexity of the load distribution; i.e., they increase with the order of the smallest nonvanishing moment of the traction distribution. In orthotropic materials the elastic moduli differ in two perpendicular directions of principal stiffness; in this case as the modulus ratio

E
2

/

E
1

{E_2}/{E_1}

increases, the angular distributions of the displacement and energy density fields become channeled towards the direction of the larger elastic modulus.
Rayleigh scattering for the Kelvin-inverted ellipsoid
Authors: Dassios George , Miloh Touvia
Published: December 2016

The Kelvin-inverted ellipsoid, with the center of inversion at the center of the ellipsoid, is a nonconvex biquadratic surface that is the image of a triaxial ellipsoid under the Kelvin mapping. It is the most general nonconvex 3-D body for which the Kelvin inversion method can be used to obtain analytic solutions for low-frequency scattering problems. We consider Rayleigh scattering by such a fourth-degree surface and provide all relevant analytical calculations possible within the theory of ellipsoidal harmonics. It is shown that only ellipsoidal harmonics of even degree are needed to express the capacity of the inverted ellipsoid. Special cases of prolate or oblate spheroids and that of the sphere are recovered through appropriate limiting processes. The crucial calculations of the norm integrals, which are expressible in terms of known ellipsoidal harmonics, are outlined in Appendix B.
A note on the uniqueness of the closed orbit of the FitzHugh-Nagumo system
Authors: Hayashi Makoto
Published: December 2016

A parameter range for which the ordinary differential equations governing the FitzHugh nerve system have a unique nontrivial closed orbit is given. It is wider than those already known.
Long-time asymptotics of solutions of the third-order nonlinear evolution equation governing wave propagation in relaxing media
Authors: Varlamov Vladimir
Published: December 2016

A classical Cauchy problem for a third-order nonlinear evolution equation is considered. This equation describes the propagation of weakly nonlinear waves in relaxing media. The global existence and uniqueness of its solutions is proved and the solution is constructed in the form of a series in a small parameter present in the initial conditions. Its long-time asymptotics is calculated, which shows the presence of two solitary wave pulses traveling in opposite directions and diffusing in space. Each of them is governed by Burgers’ equation with a transfer.
Time-periodic convective patterns in a horizontal porous layer with through-flow
Authors: Dufour Françoise , Néel Marie-Christine
Published: December 2016

We investigate the time-periodic convective patterns which set on in an infinite porous layer saturated by a fluid, under the influence of a vertical temperature gradient, superimposed to a horizontal seeping through-flow. When the seepage velocity keeps moderate, it obeys the Darcy law, while the heat equation rules the evolution of the temperature. The transition towards convection is governed by the filtration Rayleigh number, and the above-mentioned system of partial differential equations has Galilean invariance: travelling waves, stationary with respect to a moving frame, solve this problem near the threshold. This follows directly from the study of natural convection. With the help of the center manifold theory, we show that other kinds of time-periodic two-dimensional structures exist: upstream to a fixed zone, their amplitude vanishes. Downstream, they resemble the travelling waves. Numerical simulation of the governing equations reproduces elements of both sets of convective structures, indexed by definite values of the time-frequency.
Spatial behaviour in plane incompressible elasticity on a half-strip
Authors: Knops R. J. , Villaggio P.
Published: December 2016

Growth and decay estimates are derived for an incompressible homogeneous isotropic elastic material occupying a plane semi-infinite strip in equilibrium under self-equilibrated loads on the base and zero traction along the lateral sides. The estimates depend upon a pair of differential inequalities for two cross-sectional line integrals related to different kinds of energy fluxes. A comparison with the exact solution shows that the estimates are somewhat conservative. The method, however, is applicable to non-rectangular plane regions.
A note on a problem in asymptotics
Authors: Shail R.
Published: December 2016

This note examines the asymptotic evaluation of two infinite series considered recently in [1] and [2], After a transformation to a more convenient form, elementary methods are used to derive extended asymptotic results (in one case the complete asymptotic expansion in the appropriate limit), and the numerical performance of the expansions is illustrated.
Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays
Authors: Ruan Shigui
Published: December 2016

The dynamics of delayed systems depend not only on the parameters describing the models but also on the time delays from the feedback. A delay system is absolutely stable if it is asymptotically stable for all values of the delays and conditionally stable if it is asymptotically stable for the delays in some intervals. In the latter case, the system could become unstable when the delays take some critical values and bifurcations may occur. We consider three classes of Kolmogorov-type predator-prey systems with discrete delays and study absolute stability, conditional stability and bifurcation of these systems from a global point of view on both the parameters and delays.
About thin film micropolar asymptotic equations
Authors: Bayada G. , Chambat M. , Gamouana S. R.
Published: December 2016

We study from a mathematical point of view the asymptotic behaviour of micropolar fluids in lubrication theory. Different assumptions are made on the dependence of the micropolar characteristic numbers and of the Dirichlet boundary conditions for the velocity and the rotation, with respect to the small gap parameter. Various generalized Reynolds equations are rigorously obtained and second-order estimates are also given.
Existence of weak solutions to the elastic string equations in three dimensions
Authors: Reiff Andrea M.
Published: December 2016

Many applied problems resulting in hyperbolic conservation laws are nonstrictly hyperbolic. As of yet, there is no comprehensive theory to describe the solutions of these systems. We examine the equations modeling an elastic string of infinite length in three-dimensional space, restricted to possess non-simple eigenvalues of constant multiplicity. We show that there exists a weak solution of the nonstrictly hyperbolic conservation law when the total variation of the initial data is sufficiently small. The proof technique is similar to Glimm’s classical existence for hyperbolic conservation laws, but necessarily departs from Glimm’s proof by not requiring strict hyperbolicity.
Sufficient condition for the existence of solutions of a free boundary problem
Authors: Hayouni Mohammed , Novruzi Arian
Published: December 2016

In this paper we study the existence of critical points of a functional depending on

Ω
⊂

R

2

\Omega \subset \mathbb {R}^{2}

through its perimeter and the solution of the Dirichlet problem in

Ω
\Omega

, under the constraint that the measure of

Ω
\Omega

is given. We give a sufficient condition for the existence of critical points using the implicit function theorem.
A modified Ginzburg-Landau model for Josephson junctions in a ring
Authors: Hill E. , Rubinstein J. , Sternberg P.
Published: December 2016

We consider the Ginzburg-Landau functional to analyze SNS junctions in a one-dimensional ring. We compare several canonical scalings. The linearized problem is solved to obtain the phase transition curves. We compute the

Γ
\Gamma

-limit of the functional in the different scalings. The interaction of several junctions is analyzed. We study the zero set of the order parameter for distinguished values of the flux. Finally, we compute the currents in the weakly nonlinear regime.
Global stability in a nonautonomous genotype selection model
Authors: Qian Chuanxi
Published: December 2016

Consider the nonautonomous difference equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Subscript n plus 1 Baseline equals StartFraction y Subscript n Baseline exp left-parenthesis beta Subscript n Baseline left-parenthesis 1 minus sigma-summation Underscript i equals 0 Overscript k Endscripts alpha Subscript i Baseline y Subscript n minus i Baseline right-parenthesis right-parenthesis Over 1 minus y Subscript n Baseline plus y Subscript n Baseline exp left-parenthesis beta Subscript n Baseline left-parenthesis 1 minus sigma-summation Underscript i equals 0 Overscript k Endscripts alpha Subscript i Baseline y Subscript n minus i Baseline right-parenthesis right-parenthesis EndFraction comma n equals 0 comma 1 comma period period period comma left-parenthesis 0.1 right-parenthesis">

y

n
+
1

=

y
n

exp
⁡

(

β
n

(
1
−

∑

i
=
0

k

α
i

y

n
−
i

)

)

1
−

y
n

+

y
n

exp
⁡

(

β
n

(
1
−

∑

i
=
0

k

α
i

y

n
−
i

)

)

,
n
=
0
,
1
,
.
.
.
,

(
0.1
)

y_{n + 1} = \frac { y_n \exp \left ( \beta _n \left ( 1 - \sum \nolimits _{i = 0}^k \alpha _i y_{n - i} \right ) \right )} {1 - y_n + y_n \exp \left ( \beta _n \left ( 1 - \sum \nolimits _{i = 0}^k \alpha _i y_{n - i} \right ) \right )}, n = 0, 1,..., \qquad \left ( 0.1 \right )

\]
where

k
k

is a nonnegative integer,

α
0

,

α
1

,
…
,

α

k
−
1

\alpha _0, \alpha _1, \dots , \alpha _{k - 1}

are nonnegative constants,

α
k

\alpha _k

is a positive constant and

{

β
n

}

\left \{ \beta _n \right \}

is a nonnegative sequence, which is used as a genotype selection model. In this paper, we first establish some criteria for the positive equilibrium of Eq. (0.1) to be globally asymptotically stable. Then some special cases of Eq. (0.1) are investigated further and more global stability results are obtained. Our results also extend and improve some known results in the literature.
Quenching for a degenerate parabolic problem due to a concentrated nonlinear source
Authors: Chan C. Y. , Jiang X. O.
Published: December 2016

Let

q
q

,

a
a

,

T
T

, and

b
b

be any real numbers such that

q
≥
0

q \ge 0

,

a
>
0

a > 0

,

T
>
0

T > 0

, and

0
>
b
>
1

0 > b > 1

. This article studies the following degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at

b
b

:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript q Baseline u Subscript t Baseline minus u Subscript x x Baseline equals a squared delta left-parenthesis x minus b right-parenthesis f left-parenthesis u left-parenthesis x comma t right-parenthesis right-parenthesis i n left-parenthesis 0 comma 1 right-parenthesis times left-parenthesis 0 comma upper T right-bracket comma">

x
q

u
t

−

u

x
x

=

a
2

δ

(
x
−
b
)

f

(
u

(
x
,
t
)

)

i
n

(
0
,
1
)

×

(
0
,
T
]

,

{x^q}{u_t} - {u_{xx}} = {a^2}\delta \left ( x - b \right )f\left ( u\left ( x, t \right ) \right ) in \left ( 0, 1 \right ) \times \left ( 0, T \right ],

\]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x comma 0 right-parenthesis equals 0 o n left-bracket 0 comma 1 right-bracket comma u left-parenthesis 0 comma t right-parenthesis equals u left-parenthesis 1 comma t right-parenthesis equals 0 f o r Baseline 0 greater-than t less-than-or-equal-to upper T comma">

u

(
x
,
0
)

=
0
o
n

[
0
,
1
]

,
u

(
0
,
t
)

=
u

(
1
,
t
)

=
0
f
o
r

0
>
t
≤
T
,

u\left ( x, 0 \right ) = 0 on \left [ 0, 1 \right ], u\left ( 0, t \right ) = u\left ( 1, t \right ) = 0 for \; 0 > t \le T,

\]
where

δ

(
x
)

\delta \left ( x \right )

is the Dirac delta function,

f
f

is a given function such that

lim

u
→

c
−

f

(
u
)

=
∞

{\lim _{u \to {c^ - }}}f\left ( u \right ) = \infty

for some positive constant

c
c

, and

f

(
u
)

f\left ( u \right )

and

f
′

(
u
)

f’\left ( u \right )

are positive for

0
≤
u
>
c

0 \le u > c

. It is shown that the problem has a unique continuous solution

u
u

before

m
a
x

{

u

(
x
,
t
)

:
0
≤
x
≤
1

}

max\left \{ {u\left ( x, t \right ) : 0 \le x \le 1} \right \}

reaches

c
−

{c^ - }

,

u
u

is a strictly increasing function of

t
t

for

0
>
x
>
1

0 > x > 1

, and if

m
a
x

{

u

(
x
,
t
)

:
0
≤
x
≤
1

}

max\left \{ {u\left ( x, t \right ):0 \le x \le 1} \right \}

reaches

c
−

{c^ - }

, then

u
u

attains the value

c
c

only at the point

b
b

. The problem is shown to have a unique

a
∗

{a^*}

such that a unique global solution

u
u

exists for

a
≤

a
∗

a \le {a^*}

, and

m
a
x

{

u

(
x
,
t
)

:
0
≤
x
≤
1

}

max\left \{ {u\left ( x, t \right ):0 \le x \le 1} \right \}

reaches

c
−

{c^ - }

in a finite time for

a
>

a
∗

a > {a^*}

; this

a
∗

{a^*}

is the same as that for

q
=
0

q = 0

. A formula for computing

a
∗

{a^*}

is given, and no quenching in infinite time is deduced.
The success of fast reaction: a discrete reaction-diffusion model
Authors: Büger Matthias
Published: December 2016

We discuss the dynamics of a system of

2
n

2n

ordinary differential equations that can be looked at as the discrete version of a system of two reaction—diffusion equations, which differ only in their sensitivity to the reaction term. Such reaction—diffusion systems occur in evolutionary models from biology. It is known that only the fastest reacting species survives in generic situations. We prove similar results for the related discrete system and give an interpretation of the results in terms of mathematical finance.
Large time behavior and global existence of solution to the bipolar defocusing nonlinear Schrödinger-Poisson system
Authors: Hao Chengchun , Hsiao Ling
Published: December 2016

In this paper, we study the large time behavior and the existence of globally defined smooth solutions to the Cauchy problem for the bipolar defocusing nonlinear Schrödinger-Poisson system in the space

R

3

{\mathbb {R}^{3}}

.
Generalisation of Hooke’s law for finite strain to include the elastic range of strain-hardening materials
Authors: Billington E. W.
Published: December 2016

The representation theorem for isotropic tensor-valued functions of symmetric second-order tensors is considered in the context of two parameters based on the Lode and Fromm parameters. A geometrical representation is established using the concept of a characteristic representation intensity function. It is shown that this geometrical representation identifies the only admissible form of the representation intensity function to be piecewise linear and continuous. This conclusion imposes a restriction on how the representation theorem can be used to formulate constitutive equations. The representation theorem is used to formulate a generalisation of Hooke’s law for finite strain that is applicable to the initial elastic range of strain-hardening materials, including the elastic conditions at initial yield.
An algorithm for finding shortest routes from all source nodes to a given destination in general networks
Authors: Yen Jin Y.
Published: December 2016

This paper presents an algorithm for finding all shortest routes from all nodes to a given destination in

N
N

-node general networks (in which the distances of arcs can be negative). If no negative loop exists, the algorithm requires

1
2

M

(

N
−
1

)

(

N
−
2

)

,
1
>
M
N
−
1

\frac {1}{2}M\left ( {N - 1} \right ) \\ \left ( {N - 2} \right ),1 > MN - 1

, additions and comparisons. The existence of a negative loop, should one exist, is detected after

1
2

N

(

N
−
1

)

(

N
−
2

)

\frac {1}{2}N\left ( {N - 1} \right )\left ( {N - 2} \right )

additions and comparisons.
Error estimates for some variational methods applicable to scattering and radiation problems
Authors: Szałek M.
Published: December 2016

We consider variational expressions helpful in calculating the approximate value of a scalar product, in Hilbert space, of an arbitrary vector

g
g

with a solution

u
∘

u^\circ

of an arbitrary inhomogeneous linear equation. Error bounds for this approximate value are given. For the case where an approximate solution of an inhomogeneous equation is sought in an arbitrary subspace of a space containing

u
∘

u^\circ

, conditions are specified for a best estimate of the error by the use of two trial vectors. A method is presented for an additional improvement of the error estimate by using four trial vectors.
Necessary conditions for applicability of Poincaré-Lighthill perturbation theory
Authors: Usher Peter D.
Published: December 2016

We establish necessary conditions for the applicability of Poincaré—Lighthill (or coordinate stretching) perturbation theory to ordinary differential equations. The criteria are simple consequences of a unique modification of the classical theory of coordinate stretching. The usefulness of the new approach and the role of the criteria of applicability are illustrated by means of simple examples.
Distortionless wave propagation in inhomogeneous media and transmission lines
Authors: Burke V. , Duffin R. J. , Hazony D.
Published: December 2016

Of concern are mechanical or electrical waves in a media which may be nonuniform and dissipative. The problem posed is to find conditions for the undistorted propagation of signals. The electrical transmission line is chosen as the general model. Along the length of the transmission line there are four functions which may be prescribed essentially arbitrarily. These are series resistance, series inductance, shunt conductance, and shunt capacitance. A differential equation is derived relating these functions which gives a necessary and sufficient requisite for distortionless transmission of a voltage wave. Various corollaries of this theorem are developed. For instance, it is shown that simultaneous voltage and current waves can be transmitted without distortion if and only if the characteristic impedance of the transmission line is positive at each point.
Asymptotic solutions of partial differential equations with damping and delay
Authors: Bojadziev G. N. , Lardner R. W.
Published: December 2016

Certain hyperbolic partial differential equations with small nonlinearities involving significant damping as well as time delay are investigated. The Krylov—Bogoliubov—Mitropolskii method is extended and applied. An application is given to longitudinal vibrations of a nonlinear elastic rod.
Gurtin-type properties associated with wave propagation in a viscous, heat-conducting gas
Authors: Murray J. C.
Published: December 2016

Variational and reciprocity principles of the Gurtin type are established for the linear initial-boundary value problems associated with small-amplitude wave motion in a viscous, heat-conducting gas.
A variational problem arising in the design of cooling fins
Authors: Barnes Earl R.
Published: December 2016

The efficiency of a cooling fin of given weight is measured by the amount of heat dissipated per unit time by the fin. It is known that the efficiency of a given fin can be altered by changing the shape of the fin. In this paper we determine the shape of the most efficient fin of given weight and length, and thickness

≤
H

\le H

and

≥
h

\ge h

.
Near-boundary expansion of Green’s function associated with clamped plates
Authors: Wu Chien Heng
Published: December 2016

The Green’s function

G

(

P
,

P
′

)

G\left ( {P,P’} \right )

associated with a clamped plate of arbitrary shape is considered, when

P
′

P’

is at a distance

O

(
ϵ
)

O\left ( \epsilon \right )

from a regular point

O
O

of the boundary. First an outer expansion of

G
G

is described, valid when

P
P

is not near

P
′

P’

. Then an inner expansion of

G
G

is constructed when both

P
P

and

P
′

P’

are near 0. The leading term of the inner expansion is just the Green’s function

G
s

{G_s}

, for the halfplane bounded by the tangent to the boundary at

O
O

, and

ϵ

−
2

G

{\epsilon ^{ - 2}}G

differs from

ϵ

−
2

G
s

{\epsilon ^{ - 2}}{G_s}

by

O

(
ϵ
)

O\left ( \epsilon \right )

. The first two terms of the inner expansion agree with the first two terms of the expansion of

G
c

{G_c}

, the Green’s function for the interior of the osculating circle of the boundary at 0, if the boundary is convex at

O
O

. If it is concave,

G
c

{G_c}

is the Green’s function for the exterior of the osculating circle. Moreover,

ϵ

−
2

G

{\epsilon ^{ - 2}}G

differs from

ϵ

−
2

G
c

{\epsilon ^{ - 2}}{G_c}

by

O

(

ϵ
2

)

O\left ( {{\epsilon ^2}} \right )

. A two-term inner expansion is explicitly given.
On the use of the interpolation polynomial for solutions of singular integral equations
Authors: Krenk Steen
Published: December 2016

On the basis of integration of singular integral equations by means of Gaussian quadrature, it is demonstrated how to obtain the corresponding approximate polynomial solution. For some special cases compact formulas are given for the strength of the singularities at the endpoints of the integration interval.
The diffraction of a plane wave by an infinite slit. I
Authors: Spahn Robert J.
Published: December 2016

The diffraction of a normally incident plane wave by an infinite slit of finite width in a perfectly conducting screen is solved using a Lebedev integral transform theorem and the Wiener-Hopf technique. This technique leads to an infinite system of equations to which the method of successive approximation is applied. In particular, an expression for the transmission coefficient (defined as the ratio of the transmitted to the incident power per unit length) is obtained for

a
≪
λ

a \ll \lambda

where

2
a

2a

is the slit width and

λ
\lambda

is the wave length of the incident wave. There is exact agreement with the known result [3] except for a constant, which has been difficult to determine because of the slow convergence of the series involved; however, a new term has been obtained.
Dynamic instability of a homogenous deformation of a thin elastic bar
Authors: Burns Timothy J.
Published: December 2016

A linear stability analysis of a homogeneous deformation at constant strain-rate of a thin elastic bar is used to show that the deformation is unstable with respect to small perturbations in the case when the stress-strain relation is concave with a single maximum.
On the steady-state continuous casting Stefan problem with nonlinear cooling
Authors: Chipot Michel , Rodrigues José-Francisco
Published: December 2016

A steady-state one-phase Stefan problem corresponding to the solidification process of an ingot of pure metal by continuous casting with nonlinear lateral cooling is considered via the weak formulation introduced in [5] for the dam problem. Two existence results are obtained, for a general nonlinear flux and for a maximal monotone flux. Comparison results and the regularity of the free boundary are discussed. An uniqueness theorem is given for the monotone case.
A note on the mathematical formulation of the problem of numerical coordinate generation
Authors: Warsi Z. U. A.
Published: December 2016

A set of second order partial differential equations for the generation of three-dimensional grids around and between arbitrary shaped bodies has been proposed. These equations basically depend on the Gauss equations for a surface and have been structured in such a way that an automatic connection is established between the succeeding generated surfaces.
A hyperbolic Stefan problem
Authors: De Socio L. M. , Gualtieri G.
Published: December 2016

Heat conduction is considered in a semi-infinite solid subjected to a high step change in surface heat flux, such that melting occurs. A time-dependent relaxation model for the energy flux is assumed, leading to a non-Fourier, non-linear equation for the thermal field, which is solved under suitable conditions on the interface displacement.
Free boundaries in one-dimensional flow
Authors: Sherman B.
Published: December 2016

Several problems are discussed regarding flow in a horizontal channel. The channel bed may be impervious to infiltration, or may allow a constant rate of infiltration, or may be impervious to the left of some point in the channel bed and allow a constant rate of infiltration to the right. The free boundary is the time history of the motion of a piston at which the water height

μ

(
x
)

\mu \left ( x \right )

has been specified. The case

μ

(
x
)

≡
0

\mu \left ( x \right ) \equiv 0

is the dam breaking problem. In the dam breaking problem in which the channel bed allows a constant rate of infiltration a more general form of a centered rarefaction wave is required, i.e., the characteristics are not straight lines. Two problems are formulated for channel beds that are partly impervious and partly porous. Shock formation may arise here. This possibility is exhibited in a single nonconservation equation.
On the completeness of the Papkovich potentials
Authors: Millar Robert F.
Published: December 2016

The Papkovich representation for the elastostatic displacement vector in a domain

D
D

is considered. The possibility of eliminating from this representation either the scalar potential

χ
\chi

or a rectangular component

ψ
\psi

of the vector potential

ψ
\psi

is examined. Earlier work is discussed and the connection is made with the oblique derivative problem of potential theory. A convexity requirement on the boundary of

D
D

is shown to be necessary in general in order that

χ
\chi

or

ψ
\psi

may be eliminated.. A result of Stippes for a domain with an internal cavity is generalized, and two new classes of domains are found for which

χ
\chi

may be eliminated.
Nonlinear dispersive instabilities in magnetic fluids
Authors: Malik S. K. , Singh M.
Published: December 2016

An asymptotic nonlinear theory of the two superposed magnetic fluids is presented taking into account the spatial as well as temporal effects. A generalized formulation of the evolution equation governing the amplitude is developed which leads to the nonlinear Klein-Gordon equation. The various stability criteria are derived from this equation. Obtained also are the bell shaped soliton and the kink solutions.
On a planar exterior problem in linear elasticity
Authors: Ramkissoon H.
Published: December 2016

In this note a representation is obtained for the solution of the planar Dirichlet boundary-value problem of linear elasticity. It consists of a double-layer potential and a linear combination of three basic functions whose form is determined by the shape of the boundary. This representation is similar to that obtained for the analogous problem in hydrodynamics.
The wavefront induced in a homogeneously shearing solid by a localized material imperfection
Authors: Toulios M.
Published: December 2016

The shape of the two dimensional wavefront induced by a line material imperfection in a large body which is being subjected to a homogeneous, time dependent antiplane shear deformation, is investigated. The body is composed of isotropic, incompressible, hyperelastic material and the constitutive relation is assumed to be such that depending on the value of one parameter, strong ellipticity fails at a strain level corresponding to the local maximum of the shear stress-strain curve. The wavefront shapes are compared when this occurs and when it does not.
On the stability of viscous flow over a stretching sheet
Authors: Bhattacharyya S. N. , Gupta A. S.
Published: December 2016

The linear stability of two-dimensional boundary layer flow of an incompresible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with an outward velocity proportional to the distance from a point on it. Using Galerkin’s method the stability equations are solved for three-dimensional disturbances periodic in a direction normal to the plane of the basic flow and it is shown that the flow is stable.
The rigid ellipsoid in the presence of a low frequency elastic wave
Authors: Dassios George , Kiriaki Kiriakie
Published: December 2016

A longitudinal, or a transverse, plane elastic wave is incident on a rigid triaxial ellipsoid. The zeroth-order and first-order low-frequency approximations are obtained explicitly at every point exterior to the ellipsoid by solving appropriate exterior boundary value problems of potential theory. The normalized scattering amplitudes are evaluated up to the

k
2

{k^2}

-order and the leading term of the scattering cross section is given explicitly.
On the solvability of a two-dimensional water-wave radiation problem
Authors: Athanassoulis G. A.
Published: December 2016

The existence of a unique weak solution for the two-dimensional water-wave radiation problem arising when a floating rigid body oscillates on the free surface is established for all but a discrete set of oscillation frequencies. The body boundary is taken to be of

C
∗

1
,
α

C_ * ^{1,\alpha }

class (see Sec. 2) and the body boundary condition is satisfied in the

L
2

{L^2}

-sense. The proof relies on an expansion theorem (Athanassoulis [1]) and on the property of the associated water-wave multipoles being a Riesz basis of

L
2

(

−
π
,
0

)

{L^2}\left ( { - \pi ,0} \right )

, a fact which is established in the present paper. Under stronger geometrical restrictions on the body boundary it is proved, using a method due to Ursell [10], that the weak solution is actually a classical one; that is, the velocity field is continuous up to ana including the body boundary.
Remarks on a paper by W. A. Day on a maximum principle under nonlocal boundary conditions
Authors: Kawohl Bernhard
Published: December 2016

In a recent paper W. A. Day proved a decay property for solutions to a linear parabolic equation with nonlocal boundary conditions. Such boundary conditions arise in thermoelasticity. We extend his result (a) from one to arbitrary space dimensions, (b) from linear to nonlinear parabolic equations, and (c) from differential equations to differential inequalities. Our tool is the classical maximum principle.