bluuebunny/crossref_metadata_2025 huggingface dataset:
Some exact solutions describing unsteady plane gas flows with shocks
Authors: Sachdev P. L. , Venkataswamy Reddy A.
Published: December 2016

A new class of exact solutions of plane gasdynamic equations is found which describes piston-driven shocks into non-uniform media. The governing equations of these flows are taken in the coordinate system used earlier by Ustinov, and their similarity form is determined by the method of infinitesimal transformations. The solutions give shocks with velocities which either decay or grown in a finite or infinite time depending on the density distribution in the ambient medium, although their strength remains constant. The results of the present study are related to earlier investigations describing the propagation of shocks of constant strength into non-uniform media.
Inverse scattering for an exterior Dirichlet problem
Authors: Hariharan S. I.
Published: December 2016

In this paper we consider scattering due to a metallic cylinder which is in the field of a wire carrying a periodic current. The aim of the paper is to obtain information such as the location and shape of the cylinder with a knowledge of a far-field measurement in between the wire and the cylinder. The same analysis is applicable in acoustics in the situation that the cylinder is a soft-wall body and the wire is a line source. The associated direct problem in this situation is an exterior Dirichlet problem for the Helmholtz equation in two dimensions. We present an improved low-frequency estimate for the solution of this problem using integral equation methods, and our calculations on inverse scattering are accurate to this estimate. The far-field measurements are related to the solutions of boundary integral equations in the low-frequency situation. These solutions can be expressed in terms of mapping function which maps the exterior of the unknown curve onto the exterior of a unit disk. The coefficients of the Laurent expansion of the conformal transformations can be related to the far-field coefficients. The first far-field coefficient leads to the calculation of the distance between the source and the cylinder. The other coefficients are determined by placing the source in a different location and using the corresponding new far-field measurements.
Optimization of crystal growth with diffusion
Authors: Gripenberg Gustaf
Published: December 2016

A crystal growth process involving diffusion in a half-space is studied. Both the diffusion coefficient and the interface reaction term are assumed to depend on a control parameter that is (e.g.) a function of temperature. The thickness of the deposited film after a given time or the time to reach a certain thickness is to be optimized. The full Stefan problem is not considered and the diffusion coefficient is assumed to vary slowly with the parameter.
Solution of structural optimization problems by piecewise linearization
Authors: Lorentzen R.
Published: December 2016

Structural optimization problems tend to be nonlinear and often also non-convex. In this paper it is proposed to reduce a general class of such problems to linear programming problems through piecewise linearization. They can then be solved by the highly effective linear programming computer codes currently available.
Analysis of a free boundary problem in partial lubrication
Authors: Bayada G. , Chambat M.
Published: December 2016

Hydrodynamic lubrication is concerned with a particular form of creeping flow between surfaces in relative motion where cavitation takes place. The determination of the free boundary of the cavitation area is then of fundamental importance for the computation of the characteristics of the mechanisms. Different conditions at the free boundary have been introduced. We study two of them and compare corresponding solutions with respect to film extent and pressure repartition.
New third-order bounds on the effective moduli of 𝑁-phase composites
Authors: Phan-Thien N. , Milton G. W.
Published: December 2016

We develop some new bounds on the effective moduli of

N
N

-phase composites. These new bounds are accurate up to and including terms of third order in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis StartAbsoluteValue upper K Subscript i Baseline minus upper K Subscript j Baseline EndAbsoluteValue comma StartAbsoluteValue mu Subscript i Baseline minus mu Subscript j Baseline EndAbsoluteValue right-parenthesis">

O

(

|

K
i

−

K
j

|

,

|

μ
i

−

μ
j

|

)

O\left ( {\left | {{K_i} - {K_j}} \right |,\left | {{\mu _i} - {\mu _j}} \right |} \right )

, where

K
i

{K_i}

and

μ
i

{\mu _i}

are the bulk and shear modulus, respectively, of phase

i
i

. These bounds use the same statistical information as McCoy’s and Beran-Molyneux’s bounds but are tighter than, or at worst coincident with, the latter bounds. We also present in the appendix a new perturbation solution for the effective moduli which only requires that

|

δ
μ

|

=
O

(

|

μ
i

−

μ
j

|

)

\left | {\delta \mu } \right | = O\left ( {\left | {{\mu _i} - {\mu _j}} \right |} \right )

be small.
A linear integro-differential equation for viscoelastic rods and plates
Authors: Hannsgen Kenneth B.
Published: December 2016

It is proved that the resolvent kernel of a certain integrodifferential equation in Hilbert space is absolutely integrable on

(

0
,
∞

)

\left ( {0,\infty } \right )

. The equation arises in the linear theory of isotropic viscoelastic rods and plates.
The dynamics of simple fluids in steady circular shear
Authors: Goddard J. D.
Published: December 2016

An isotropic simple fluid of constant density

ρ
\rho

is confined between two infinite horizontal planes which rotate steadily about separate vertical

(
z
)

\left ( z \right )

axes with common angular velocity

ω
\omega

. We show that at least one solution to the exact equations of motion is determined by the differential equation
\[

d

d
z

(

η

d
u

d
z

)

−
i
ρ
ω
u
=
0

\frac {d}{{dz}}\left ( {\eta \frac {{du}}{{dz}}} \right ) - i\rho \omega u = 0

\]
where

u

(
z
)

u\left ( z \right )

is a complex variable representing the horizontal velocity and

η

(

|

d
u

/

d
z

|

,
ω

)

\eta \left ( {\left | {du/dz} \right |,\omega } \right )

is a complex shear modulus. This equation represents the extension to nonlinear viscoelasticity of the previous works of Berker and of Abbott and Walters on linear viscoelastic fluids, for which

η
\eta

reduces to the usual dynamic viscosity

η
∗

(
ω
)

\eta *\left ( \omega \right )
Dynamic behavior from asymptotic expansions
Authors: Hale Jack K. , Pavlu Luiz Carlos
Published: December 2016

The purpose of this paper is to discuss stability properties of solutions of periodic and almost-periodic differential equations containing a small parameter. The existence of the solution can be obtained in the first approximation but the stability only after

k
k

approximations. We obtain the results using asymptotic expansions, higher-order averaging and the concept of exponential hyperbolicity of order

k
k

.
Explicit solutions to phase change problems
Authors: Solomon A. D. , Wilson D. G. , Alexiades V.
Published: December 2016

We examine two heat transfer and phase change problems having explicit solutions. The first involves melting of an initially cold material and clarifies the meaning of a recent result of Tarzia [5]. The second concerns a model of binary alloy solidification which, in some cases, is seen to be incorrect.
Diakoptics or tearing—a mathematical approach
Authors: Aitchison P. W.
Published: December 2016

The method of diakoptics or tearing was introduced by G. Kron in order to reduce computations in the solution of certain problems arising from large inter-connected power distribution networks. Here the method is given a purely mathematical form which can be used to solve large systems of linear equations by first solving some smaller sub-problems and then combining these solutions into a complete solution. The sub-problems are formed from sets of equations and variables which are strongly connected, within the sub-problem, but only weakly connected to those of another sub-problem.
Nonlinear breakup of an electrohydrodynamic jet
Authors: Malik S. K. , Singh M.
Published: December 2016

A weakly nonlinear theory of the breakup of a jet held together with the capillary forces in the presence of an applied electric field is presented. It is shown that the jet breaks into the main drops and their satellites whose sizes are sensitive to the wavenumbers and the electric field. The satellites always exist when the electric field is applied to the jet which, however, is not the case when the electric field is absent.
The nonlinear circular membrane under a vertical force
Authors: Dickey R. W.
Published: December 2016

The exact theory for the deformation of a plane circular membrane under a vertical force is derived. It is shown that the system of equations can be reduced to a single, nonlinear, ordinary differential equation. In addition it is shown that the Foppl approximation is the first term in an asymptotic expansion of the exact theory.
Comparison of the matched asymptotic expansions method and the two-variable technique
Authors: Bouthier M.
Published: December 2016

Either the matched asymptotic expansions method or the two-variable technique are available for treating boundary layer problems. A comparison of the two methods is achieved on dealing with elliptic boundary value problems. The two-variable technique is proved to be slightly more powerful than the matched expansions method. Nevertheless it fails to determine a closed class of approximate solutions. Such a class, which involves the results of both the asymptotic methods is set out with help of an asymptotic equivalence theorem.
Bäcklund transformations for systems of conservation laws
Authors: Kingston J. G. , Rogers C.
Published: December 2016

A class of Bäcklund transformations is introduced for systems of conservation laws. Applications are given in non-steady gasdynamics and two-phase filtration through a porous medium.
Nonlinear deflection of a thin elastic sheet on a liquid foundation
Authors: Wang Chang Yi
Published: December 2016

A thin infinite elastic sheet lies on the surface of a liquid. The nonlinear deflection due to a concentrated load is studied by both analytical and numerical methods.
Boundary value problems in microcontinuum fluid mechanics
Authors: Ramkissoon H.
Published: December 2016

This paper examines the existence and uniqueness of solutions of certain boundary-value problems associated with a seemingly complex system of coupled partial differential equations frequently encountered in the field of microcontinuum fluid mechanics. These problems are analyzed using potential theory methods and appealing to results from the theory of singular integral equations. Matrices are introduced to facilitate this analysis.
On stability and uniqueness of fluid flow through a rigid porous medium
Authors: Pericak-Spector K. A.
Published: December 2016

We study a set of equations describing the flow of an incompressible viscous fluid through a rigid porous medium. Existence, uniqueness and stability results are established for the case of a region impregnated with fluid, and uniqueness for an unsaturated region.
Conserved quantity partition for Dirac’s equation
Authors: Branson Thomas P.
Published: December 2016

Let

M
M

be the

(
n
+
1
)

(n + 1)

-dimensional Minkowski space,

n
≥
3

n \ge 3

. The energy of a solution

ψ
\psi

to Dirac’s equation in

M
M

is a sum of

n
n

terms, the

j
j

th term depending on

ψ
\psi

and the space derivative

∂
ψ

/

∂

x
j

\partial \psi /\partial {x_j}

. We show that if the Cauchy datum for

ψ
\psi

is compactly supported, then each of these terms is eventually constant. Specifically, if

ψ
\psi

is initially supported in the closed ball of radius

b
b

about the origin in space

(

R
n

)

\left ( {{R^n}} \right )

, then for times

|
t
|

≥
b

\left | t \right | \ge b

, the

j
j

th term is equal to the energy of the

j
j

th Riesz transform

(
−
Δ

)

−
1

/

2

(
∂

/

∂

x
j

)
ψ

{( - \Delta )^{ - 1/2}}(\partial /\partial {x_j})\psi

, which also solves Dirac’s equation.
On linearly coupled relaxation oscillations
Authors: Bélair Jacques , Holmes Philip
Published: December 2016

We study the dynamical behavior of a pair of linearly coupled relaxation oscillators. In such systems vastly different time scales play a crucial rôle, and solutions may be viewed as consisting of portions of slow drift linked by rapid jumps. This feature enables us to reduce the analysis from four dimensional phase space to that of a two dimensional system with discontinuous but well determined behavior at certain points on the phase plane. We determine the existence and stability of periodic motions for identical oscillators and oscillators with an uncoupled frequency ratio of

1
:
ω

1:\omega

. We give additional details on nonperiodic motions for the special case of

ω
=
2

\omega = 2

.
Generating functions for plane or axisymmetric isochoric deformations
Authors: Rooney F. J. , Carroll M. M.
Published: December 2016

Generating functions for canonical transformations of single degree of freedom Hamiltonian systems provide displacement potentials for isochoric plane or axisymmetric finite deformations.
Adiabatic shearing flow caused by time dependent inertial force
Authors: Charalambakis Nicolas
Published: December 2016

We consider the shearing adiabatic flow of an incompressible Neutonian fluid with temperature dependent viscosity caused by a time dependent “oscillatory” body force. We show that, as

t
→
∞

t \to \infty

, the flow converges with exponential rate to a rigid “oscillatory” motion at fixed temperature.
Asymptotic stability in nonlinear viscoelasticity
Authors: Beevers C. E. , Šilhavý M.
Published: December 2016

In this paper we present sufficient conditions for asymptotic stability of a homogeneous equilibrium state of a (nonlinear) elastic body with linear viscosity. The body is subject to external conditions of zero displacements on a part of the boundary, zero surface tractions on the remaining part of the boundary and zero body forces in the interior of the body. The meaning and further qualitative consequence of our conditions are also discussed.
Motion of a point vortex in a multiconnected region
Authors: Suresh Ambady
Published: December 2016

The motion of a single point vortex in a bounded region of the plane with

m
m

internal boundaries is considered. For

m
≥
1

m \ge 1

, it is shown that there exist

m
−
1

m - 1

saddle points (counted with multiplicities) where the vortex can remain stationary.
On nonuniqueness in the traction boundary-value problem for a compressible elastic solid
Authors: Ogden R. W.
Published: December 2016

For a compressible isotropic elastic solid local and global non-uniqueness of the homogeneous deformation resulting from prescribed dead-load boundary tractions is examined. In particular, for the plane-strain problem with equibiaxial in-plane tension, equations governing the paths of deformation branching from the bifurcation point on a deformation path corresponding to in-plane pure dilatation are derived. Explicit calculations are given for a specific strain-energy function and the stability of the branches is discussed. Some general results are then given for an arbitrary form of strain-energy function.
A numerical technique for the solution of singular integral equations of the second kind
Authors: Miller G. R. , Keer L. M.
Published: December 2016

A numerical technique for the solution of singular integral equations of the second kind is presented. The technique extends the piecewise quadratic method used by Gerasoulis [6], to solve singular integral equations of the first kind. Quadrature formulae are derived in terms of infinite series and are used to reduce the integral equation to a set of linear algebraic equations. Two numerical examples are given, and the results are compared with exact solutions.
Burgers flow past arbitrary ellipse
Authors: Dorrepaal J. M.
Published: December 2016

This paper examines a linearization of the Navier-Stokes equation due to Burgers in which vorticity is transported by the velocity field corresponding to continuous potential flow. The governing equations are solved exactly for the two dimensional steady flow past an ellipse of arbitrary aspect ratio. The requirement of no slip along the surface of the ellipse results in an infinite algebraic system of linear equations for coefficients appearing in the solution. The system is truncated at a point which gives reliable results for Reynolds numbers

R
R

in the range

0
>
R
≤
5

0 > R \le 5

.
A new model for thin plates with rapidly varying thickness. II. A convergence proof
Authors: Kohn Robert V. , Vogelius Michael
Published: December 2016

Our recent paper [6] presented a model for thin plates with rapidly varying thickness, distinguishing between thickness variation on a length scale longer than, on the order of, or shorter than the mean thickness. We review the model here, and identify the case of long scale thickness variation as an asymptotic limit of the intermediate case, where the scales are comparable. We then present a convergence theorem for the intermediate case, showing that the model correctly represents the solution of the equations of linear elasticity on the three-dimensional plate domain, asymptotically as the mean thickness tends to zero.
A theory for the wave-induced motion of finite monomolecular films
Authors: Dore B. D.
Published: December 2016

The fluctuating velocity field of monomolecular films of arbitrary configuration is investigated when gravity waves propagate on the air-water interface. The surface-active material is assumed to have visco-elastic properties and to be insoluble. Boundary-layer techniques are employed, and a Dirichlet boundary value problem, involving Helmholtz’ equation for the divergence of the velocity field, is obtained for the film. Circular and rectangular films are considered explicitly, whilst an approximate method is given for slender films of arbitrary orientation. Application is made to viscous wave-damping.
Note on 120 degree axial coordinates
Authors: Bleick W. E.
Published: December 2016

Recently the author introduced [1] a symmetrical system of distance coordinates for studying the planar motion of a three-body system of point masses in its internal degrees of freedom, and applied them to the small vibration problem. Subsequently he applied them [2] to the gravitational three-body problem. The purpose of this note is to give a geometrical identification of the auxiliary angles

θ

1
,

θ

2
,

{\theta _{1,}}{\theta _{2,}}

, and

θ
3

{\theta _3}

, related to the system, and to prove the identity which they satisfy.
Macroscopic global modeling of binary alloy solidification processes
Authors: Alexiades V. , Wilson D. G. , Solomon A. D.
Published: December 2016

A macroscopic mathematical model is constructed describing the evolution of the phases of a binary alloy or mixture undergoing solidification under the action of simultaneous conduction of heat and diffusion of solute. The formulation is global, in the form of a pair of conservation laws valid over the whole region occupied by the alloy in a weak (distributional) sense. Thus it is especially convenient for numerical solution since it does not require tracking of the interface, which, in fact, may develop into a “mushy zone".
Nonoscillatory differential equations with retarded and advanced arguments
Authors: Gopalsamy K.
Published: December 2016

Sufficient conditions are derived for a vector-matrix system of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d Superscript n Baseline upper X left-parenthesis t right-parenthesis Over d t Superscript n Baseline EndFraction plus left-parenthesis negative 1 right-parenthesis Superscript n minus 1 Baseline left-bracket upper P left-parenthesis t right-parenthesis upper X left-parenthesis t minus tau 1 left-parenthesis t right-parenthesis right-parenthesis plus upper Q left-parenthesis t right-parenthesis upper X left-parenthesis t plus tau 2 left-parenthesis t right-parenthesis right-parenthesis right-bracket equals 0">

d
n

X

(
t
)

d

t
n

+

(

−
1

)

n
−
1

[

P

(
t
)

X

(

t
−

τ
1

(
t
)

)

+
Q

(
t
)

X

(

t
+

τ
2

(
t
)

)

]

=
0

\frac {{{d^n}X\left ( t \right )}}{{d{t^n}}} + {\left ( { - 1} \right )^{n - 1}}\left [ {P\left ( t \right )X\left ( {t - {\tau _1}\left ( t \right )} \right ) + Q\left ( t \right )X\left ( {t + {\tau _2}\left ( t \right )} \right )} \right ] = 0

\]
to be nonoscillatory.
Stagnation flow on the surface of a quiescent fluid—an exact solution of the Navier-Stokes equations
Authors: Wang C. Y.
Published: December 2016

A lighter fluid impinges downward on a heavier, otherwise quiescent fluid. The region near the stagnation point is investigated. The Navier-Stokes equations yield similarity solutions for both upper and lower regions. The convective heat transfer is also determined.
On wave propagation in linear viscoelasticity
Authors: Hrusa W. J. , Renardy M.
Published: December 2016

We discuss the initial value problem in one-dimensional linear visco-elasticity with a step-jump in the initial data. If the memory kernel is sufficiently smooth on

[

0
,
∞

)

\left [ {0,\infty } \right )

, the solution exhibits discontinuities propagating along characteristics and a (higher order) stationary discontinuity at the position of the original step-jump. For a singular memory kernel, the propagating waves are smoothed in a manner depending on the nature of the singularity in the kernel, but the stationary discontinuity remains. We also discuss the effects of these phenomena on the regularity of solutions with arbitrary initial data.
Slow viscous flow inside a torus—the resistance of small tortuous blood vessels
Authors: Chadwick R. S.
Published: December 2016

The hydrodynamic resistance of a buckled microvessel in the form of a tightly would helix is approximated by studing the Stokes flow inside a torus. The unidirectional flow is driven by a constant tangential pressure gradient. The solution is obtained by an eigenfunction expansion in toroidal coordinates. The ratio of volume flow carried by the torus to that carried by a straight tube is computed as a function of the vessel radius: coil radius ratio. An asymptotic expansion for this flux ratio is also obtained. The results show that the resistance of a moderately curved vessel is slightly less than the resistance of a straight one, whereas the resistance of a greatly curved vessel is at most

3
3%

greater than the straight one.
Principal stress and strain trajectories in nonlinear elastostatics
Authors: Ogden R. W.
Published: December 2016

The Maxwell-Lamé equations governing the principal components of Cauchy stress for plane deformations are well known in the context of photo-elasticity, and they form a pair of coupled first-order hyperbolic partial differential equations when the deformation geometry is known. In the present paper this theme is developed for nonlinear isotropic elastic materials by supplementing the (Lagrangean form of the) equilibrium equations by a pair of compatibility equations governing the deformation. The resulting equations form a system of four first-order partial differential equations governing the principal stretches of the plane deformation and the two angles which define the orientation of the Lagrangean and Eulerian principal axes of the deformation. Coordinate curves are chosen to coincide locally with the Lagrangean (Eulerian) principal strain trajectories in the undeformed (deformed) material.
On a class of eigenvalue problems of orthotropic plates
Authors: Vendhan Chiruvai P. , Bhattacharyya Subroto Kumar
Published: December 2016

A class of eigenvalue problems for linear bending of semi-infinite orthotropic plates under the action of in-plane stress resultant in the finite direction has been developed. The structure of the eigenvalues has been investigated analytically for the case of simply supported plates and their physical basis underscored. It is shown that the treatment results in a coupled eigenvalue problem of bending as well as bifurcation buckling, thereby suggesting an alternate criterion for buckling.
Some singular, nonlinear, integral equations arising in physical problems
Authors: Lerche I.
Published: December 2016

We show that if

Q
Q

, the logarithmic decrement of plane wave intensity with position, is measured as a function of frequency, then the in-phase and quadrature components of the medium’s characteristic response function (in the form of a dielectric function for electromagnetic waves, an elastic response function for acoustic waves, etc.) satisfy a nonlinear, singular, integral equation with the in-phase and quadrature components connected through a causal Hilbert transform pair.
Finite element approximation of a reaction-diffusion equation. I. Application of topological techniques to the analysis of asymptotic behavior of the semidiscrete approximations
Authors: Khalsa Sat Nam S.
Published: December 2016

The initial-boundary value problem for a reaction-diffusion equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline equals u Subscript x x Baseline plus f left-parenthesis u right-parenthesis comma f left-parenthesis u right-parenthesis equals minus u left-parenthesis u minus b right-parenthesis left-parenthesis u minus 1 right-parenthesis comma 0 greater-than b greater-than 1 slash 2 comma left-parenthesis asterisk right-parenthesis">

u
t

=

u

x
x

+
f

(
u
)

,

f

(
u
)

=
−
u

(

u
−
b

)

(

u
−
1

)

,

0
>
b
>
1

/

2
,

(
∗
)

{u_t} = {u_{xx}} + f\left ( u \right ), \qquad f\left ( u \right ) = - u\left ( {u - b} \right )\left ( {u - 1} \right ), \qquad 0 > b > 1/2, \qquad \left ( * \right )

\]
was analyzed in [4, 17] by using the Conley index. In this paper we study the asymptotic behavior of the semidiscrete finite element approximations, with interpolation of the coefficients in the nonlinear terms. We show that for small

h
h

the spectrum of the linearized discrete steady-state problem is a “good” approximation for the spectrum of the linearized continuous steady-state problem. Using the interpretation of the Conley index as the dimension of an unstable manifold of a steady-state solution, we establish that the properties of the semidiscrete approximations are completely analogous to those of the solutions of

(
∗
)

\left ( * \right )

. The asymptotic, as

t
→
∞

t \to \infty

, optimal order convergence of the approximate solutions is proved.
On the roots of 𝑓_{𝑛}(𝑧)=𝑎+𝐾_{𝑛+1}(𝑧)/𝑧𝐾_{𝑛}(𝑧)
Authors: Conant R. J.
Published: December 2016

The function appearing in the title is examined to determine the number of roots and their general location. It is found that the roots appear in complex conjugate pairs, that the number of roots depends on

n
n

, and that, with the exception of a positive real root arising for

a
>
0

a > 0

, all roots lie in the left half-plane.
Elastodynamic crack problems in an anisotropic medium through a complex variable approach
Authors: Piva A.
Published: December 2016

The complex variable approach is used to obtain closed-form solutions to elastodynamic crack problems in a strip of anisotropic elastic medium, with one plane of symmetry, under antiplane shear stress. The two problems examined involve a finite-length crack propagating in an infinitely long strip of finite width when the edges are subjected either to constant displacements or shearing stresses. The special cases corresponding to either an orthotropic medium or an isotropic medium are recovered.
Oscillations in a delay-logistic equation
Authors: Gopalsamy K.
Published: December 2016

Sufficient conditions are derived for all nonconstant nonnegative solutions of the equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d x left-parenthesis t right-parenthesis Over d t EndFraction equals x left-parenthesis t right-parenthesis left-brace a minus sigma-summation Underscript j equals 1 Overscript n Endscripts b Subscript j Baseline x left-parenthesis t minus tau Subscript j Baseline right-parenthesis right-brace">

d
x

(
t
)

d
t

=
x

(
t
)

{

a
−

∑

j
=
1

n

b
j

x

(

t
−

τ
j

)

}

\frac {{dx\left ( t \right )}}{{dt}} = x\left ( t \right )\left \{ {a - \sum \limits _{j = 1}^n {{b_j}x\left ( {t - {\tau _j}} \right )} } \right \}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d x left-parenthesis t right-parenthesis Over d t EndFraction equals x left-parenthesis t right-parenthesis left-brace a minus b integral Subscript negative normal infinity Superscript t Baseline k left-parenthesis t minus s right-parenthesis x left-parenthesis s right-parenthesis d s right-brace">

d
x

(
t
)

d
t

=
x

(
t
)

{

a
−
b

∫

−
∞

t

k

(

t
−
s

)

x

(
s
)

d
s

}

\frac {{dx\left ( t \right )}}{{dt}} = x\left ( t \right )\left \{ {a - b\int _{ - \infty }^t {k\left ( {t - s} \right )x\left ( s \right )ds} } \right \}

\]
to be oscillatory about their respective positive steady states. The results are complementary to those in [15].
Generalized magnetohydrostatic equilibria
Authors: Shivamoggi Bhimsen K.
Published: December 2016

This note shows the existence of a generalized class of magnetohydrostatic elliptic equilibria of a perfectly conducting fluid in systems with cylindrical topology for which the surfaces of constant pressure have an elliptic transverse cross section with the eccentricity given by an arbitrary function of the axial distance. Further, it is shown that a change in the parameters characterizing the solution leads to a generalized class of hyperbolic equilibria for which the surfaces of constant pressure have a hyperbolic transverse cross section with the eccentricity given by an arbitrary function of the axial distance. Whereas the first class of equilibria is of interest in a plasma-confinement system, the second class of equilibria is of interest in a magnetic-field-line reconnection system.
Differentiation by Fourier transformation and its connection with differentiation by finite differencing
Authors: Compani-Tabrizi Behrooz , Geyer Richard G.
Published: December 2016

The relation between central-, forward-, and backward-finite differencing and differentiation by Fourier transformation is developed. The conventional rule for differentiation by Fourier transformation of a discretized function, namely, multiplication of the Fourier transform of the function by

i
k

ik

and a subsequent inverse Fourier transformation, is shown to be a first-order approximation to more complete rules. Numerical examples are provided.
Reformulation of the Fourier transformed discretized differential equations
Authors: Compani-Tabrizi Behrooz
Published: December 2016

The conventional rule for differentiation by Fourier transformation of a discretized function, namely, multiplication of the Fourier transform of the function by

i
k

ik

and a subsequent inverse Fourier transformation, was shown to be a first-order approximation to more complete rules [1]. The

i
k

ik

differentiation formula is an exact relation for a nondiscretized function with a continuum as its independent variable. However, this rule fails, particularly when applied to nonsmooth discretized functions of high-frequency content. A consequence of this result is that Fourier-transformed differential equations must be transformed into the Fourier space not by the

i
k

ik

differentiation rule, but by the appropriate correct differentiation rule if the differential equation is eventually to be solved through discretization. Helmholtz’s equation is discussed, and the solution to the simple diffusion equation is presented using the exact and the approximate differentiation formulas and a high-frequency initial field.
Examples of the use of Fourier series in structural optimization
Authors: Balasubramanyam K. , Spillers W. R.
Published: December 2016

This paper examines the use of Fourier series (finite integral transforms) in some beam-type problems of optimal structural design. In general, the series solution method reduces continuous mathematical programming problems to discrete ones. This is of particular interest when accurate results can be obtained with only a few terms of the series. Six problems of vibration and buckling are treated, most of which are not now available in the literature.
Feedback stabilization of “hybrid” bilinear systems
Authors: Slemrod M. , Rogers E. L.
Published: December 2016

This paper considers the problem of stabilizing a control system governed by a combination of partial and ordinary differential equations. The partial differential equations govern the evolution of the system in the interior of some spatial domain, and the ordinary differential equations describe the evolution of the boundary data; the control enters through the boundary ordinary differential equations in a bilinear fashion. We provide sufficient conditions for feedback stabilization of such “hybrid” systems. Two examples to wave equations with dynamic boundary conditions are provided.
On spherical inversions of polyharmonic functions
Authors: Chwang Allen T.
Published: December 2016

A general spherical inversion theorem for polyharmonic functions of order

m
m

has been obtained, and it reduces to the Kelvin transformation for

m
=
1

m = 1

. For biharmonic functions

(

m
=
2

)

\left ( {m = 2} \right )

, the present theorem has been applied to generate an explicit solution which satisfies the prescribed homogeneous boundary conditions on the surface of a sphere.
A three-dimensional stretching flow of an Oldroyd fluid
Authors: Phan-Thien N.
Published: December 2016

It is shown that a three-dimensional stretching flow of an Oldroyd-type fluid has an exact solution. As the fluid becomes Maxwellian, the solution permits a vortex sheet to propagate from the boundary into the flow domain. Furthermore, it is shown that there exists a critical Weissenberg number above which a stress component increases exponentially with time.
On the completeness of the Rayleigh-Marangoni and Graetz eigenspaces and the simplicity of their eigenvalues
Authors: Nadarajah A. , Narayanan R.
Published: December 2016

The study of convection in bounded geometries is often facilitated by consideration of model problems which require little numerical computation. Such problems typically result in a sixth-order ordinary differential system which is non-selfadjoint in the inner product associated with the space of square integrable functions in the sense of Lebesgue. Using theorems of Karlin (1971) and Naimark (1967), we prove the simplicity of the eigenvalues and completeness of the eigenspace. We illustrate some of the simplicity results with numerical calculations. The use of completeness in future problems is explained. We also consider the extended Graetz problem with homogeneous reaction and heterogeneous reaction at the wall. Similar results are shown here. Some of the above-mentioned results can be obtained by other means, but we provide this analysis as an interesting alternative.
An alternative approach to elastodynamic crack problems in an orthotropic medium
Authors: Piva A.
Published: December 2016

A similarity transformation is used to reduce the system of second-order equations, governing elastodynamic plane problems in an orthotropic medium, to a first-order elliptic system of the Cauchy-Riemann type. A complex variable notation is then introduced to derive in a straightforward way the solution of two noticeable elastodynamic crack problems.
On helicity fluctuations in turbulence
Authors: Speziale Charles G.
Published: December 2016

The role that helicity fluctuations play in relationship to small-scale intermittency and coherent structures in turbulent flows is examined from a theoretical standpoint. It has been argued recently that regions of large turbulent activity are associated with regions of small local helicity and, hence, that such a correlation exists. In this paper, it will be proven that fluctuations in the local helicity are not Galilean invariant and, consequently, it is possible to have completely different local helicity fluctuations for a given fluctuating velocity field and its associated turbulence statistics. It is thus highly doubtful that local helicity fluctuations can be of any fundamental value in correlating small-scale intermittency or coherent structures.
Pressure maximum behavior in inflation of incompressible elastic hollow spheres and cylinders
Authors: Carroll M. M.
Published: December 2016

It is shown that realistic material response allows for three qualitatively different types of behavior for inflation of incompressible elastic hollow spheres and stretched cylinders. The pressure may increase monotonically, or it may increase and then decrease, or it may increase, decrease, and then increase again. A simple condition on the material response curve for uniaxial compressive stress is used to classify materials with respect to spherical inflation and to examine which type of behavior will occur for a particular material and initial geometry. Similar results are obtained for inflation of axially stretched hollow cylinders. The results are also applicable to elastic-plastic strain hardening materials.
On oscillation of nonlinear delay differential equations
Authors: Kulenović M. R. S. , Ladas G. , Meimaridou A.
Published: December 2016

Necessary, sufficient, and necessary and sufficient conditions are obtained for all solutions of the nonlinear differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d y Over d t EndFraction plus sigma-summation Underscript j equals 1 Overscript n Endscripts q Subscript j Baseline f left-parenthesis y left-parenthesis t minus tau Subscript j Baseline right-parenthesis right-parenthesis equals 0 comma t greater-than-or-equal-to 0 comma left-parenthesis asterisk right-parenthesis">

d
y

d
t

+

∑

j
=
1

n

q
j

f

(

y

(

t
−

τ
j

)

)

=
0
,

t
≥
0
,

(
∗
)

\frac {{dy}}{{dt}} + \sum \limits _{j = 1}^n {{q_j}f\left ( {y\left ( {t - {\tau _j}} \right )} \right ) = 0,\qquad t \ge 0,} \qquad \left ( * \right )

\]
to be oscillatory. These conditions are expressed in terms of the characteristic equation of the corresponding linear “variational” equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d y Over d t EndFraction plus sigma-summation Underscript j equals 1 Overscript n Endscripts q Subscript j Baseline y left-parenthesis t minus tau Subscript j Baseline right-parenthesis equals 0 comma t greater-than-or-equal-to 0 period left-parenthesis asterisk asterisk right-parenthesis">

d
y

d
t

+

∑

j
=
1

n

q
j

y

(

t
−

τ
j

)

=
0
,

t
≥
0.

(

∗
∗

)

\frac {{dy}}{{dt}} + \sum \limits _{j = 1}^n {{q_j}y\left ( {t - {\tau _j}} \right ) = 0, \qquad t \ge 0. \qquad \left ( { * * } \right )}

\]
Our results show that for a certain class of nonlinear functions

f
,

(
∗
)

f,\left ( * \right )

oscillates if and only if

(

∗
∗

)

\left ( { * * } \right )

oscillates. As an application of our results, we obtain simple sufficient and necessary and sufficient conditions for the oscillation of several nonlinear delay differential equations which appear in applications.
A solitary group of two-dimensional deep-water waves
Authors: Yih Chia-Shun
Published: December 2016

If the magnitude of a line of concentrated force moving with a constant velocity on the surface of deep water oscillates with a constant frequency, a single group of two-dimensional gravity waves is created. The group moves with the velocity of the concentrated force, and the waves in it have a wave number and phase velocity which are determined by the velocity of the concentrated force and the frequency of oscillation of its magnitude. If these latter quantities are such that the velocity of the force is near the group velocity, in the classical sense, of the waves created, the length of the group will be very long, but in general it can take on any value. The phase velocity of the waves created is larger than that of classical deep-water waves of the same wave number, whereas the velocity of the group is less than the group velocity of these classical waves. The respective differences, as well as the magnitude of the concentrated force, tend to zero as the length of the group increases indefinitely. The solution given provides the elemental solution from which gravity wave groups caused by any travelling oscillating pressure distribution can be found.
The moving load on a string as free boundary problem
Authors: D’Acunto B.
Published: December 2016

A free boundary problem for the nonhomogeneous wave equation is studied. Such a problem arises when the motion of a load on a string is analyzed without supposing that the load velocity is known and fixed. Preliminarily the complementary equations which characterize the free boundary are determined. The mechanical problem is then solved by proving a uniqueness and existence theorem.
Temperature and moving boundary in two-phase freezing due to an axisymmetric cold spot
Authors: Gupta S. C.
Published: December 2016

Short time analytic solution of the problem of two-phase freezing due to an axisymmetric cold spot is presented. The melt could be superheated and it occupies an infinite region bounded internally by a cylinder of finite radius. Although the method of solution is valid for various other types of boundary conditions, the results in the paper are given for prescribed flux which could be time and space dependent. The method of solution is simple and straightforward and consists of assuming fictitious initial temperatures in some fictitious extensions of the region originally occupied by the melt. The spread of the solidification is much faster along the surface of the cylinder than along the interior of the cylinder and the spread along the surface always depends on material parameters. Several interesting results can be deduced as particular cases of the general results.
Supercooling and superheating effects in heterogeneous systems
Authors: Visintin A.
Published: December 2016

In the model for phase transitions in binary systems based on Fourier’s and Fick’s laws, the interface equilibrium condition

θ
=
w

\theta = w

, relating the temperature

θ
\theta

and the chemical activity

w
w

, is here replaced by a relaxation dynamics for the liquid concentration

χ
\chi

:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential chi Over partial-differential t EndFraction plus ModifyingAbove upper H With tilde Superscript negative 1 Baseline left-parenthesis chi right-parenthesis equals beta left-parenthesis theta comma w right-parenthesis">

∂
χ

∂
t

+

H
~

−
1

(
χ
)

=
β

(

θ
,
w

)

\frac {{\partial \chi }}{{\partial t}} + {\tilde H^{ - 1}}\left ( \chi \right ) = \beta \left ( {\theta ,w} \right )

\]
(

H
~

−
1

{\tilde H^{ - 1}}

: inverse of the Heaviside graph); here

β
∈

C
0

(

R
2

)

\beta \in {C^0}\left ( {{R^2}} \right )

and

s
i
g
n
β

(

θ
,
w

)

=
s
i
g
n

(

θ
−
w

)

sign\beta \left ( {\theta ,w} \right ) = sign\left ( {\theta - w} \right )

. In the case of a single dimension of space, with an interface

x
=
s

(
t
)

x = s\left ( t \right )

, a different dynamics can be considered:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s prime left-parenthesis t right-parenthesis equals beta left-parenthesis theta left-parenthesis s left-parenthesis t right-parenthesis comma t right-parenthesis comma w left-parenthesis s left-parenthesis t right-parenthesis comma t right-parenthesis right-parenthesis">

s
′

(
t
)

=
β

(

θ

(

s

(
t
)

,
t

)

,
w

(

s

(
t
)

,
t

)

)

s’\left ( t \right ) = \beta \left ( {\theta \left ( {s\left ( t \right ),t} \right ),w\left ( {s\left ( t \right ),t} \right )} \right )

\]
.
Formulation and analysis of a functional equation describing a moving one-dimensional elastic phase boundary
Authors: Pence Thomas J.
Published: December 2016

Certain elastic solids when subjected to sufficiently high loads abruptly change their mechanical properties and yet continue to respond elastically to further loading. In one dimension such mechanically induced elastic phase transitions may be due to a nonmonotonic stress-strain relationship. This appears to be particularly true for certain mineral crystals, such as calcite.
On the structure and invariance of the Barnett-Lothe tensors
Authors: Chadwick P. , Ting T. C. T.
Published: December 2016

The three real tensors introduced by Barnett and Lothe into the theory of steady plane motions of an anisotropic elastic body are shown to have algebraic representations the structure of which is largely independent of material symmetry. The allied form of the complex impedance tensor central to the analyses of surface and interfacial waves in anisotropic elastodynamics is also obtained. A detailed study of the representations yields alternative routes to known results and a variety of new relations. The paper concludes with a discussion of the invariance properties of quantities appearing in the representations under rotations of the reference frame about the direction in which the deformation is uniform.
On the solution of integral equations with a generalized Cauchy kernel
Authors: Kaya A. C. , Erdogan F.
Published: December 2016

In this paper a certain class of singular integral equations that may arise from the mixed boundary value problems in nonhomogeneous materials is considered. The distinguishing feature of these equations is that in addition to the Cauchy singularity, the kernels contain terms that are singular only at the end points. In the form of the singular integral equations adopted, the density function is a potential or a displacement and consequently the kernel has strong singularities of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis t minus x right-parenthesis Superscript negative 2 Baseline comma x Superscript n minus 2 Baseline left-parenthesis t plus x right-parenthesis Superscript n Baseline left-parenthesis n greater-than-or-equal-to 2 comma 0 greater-than x comma t greater-than b right-parenthesis">

(

t
−
x

)

−
2

,

x

n
−
2

(

t
+
x

)

n

(

n
≥
2
,
0
>
x
,
t
>
b

)

{\left ( {t - x} \right )^{ - 2}},{x^{n - 2}}{\left ( {t + x} \right )^n}\left ( {n \ge 2,0 > x,t > b} \right )

. The complex function theory is used to determine the fundamental function of the problem for the general case and a simple numerical technique is described to solve the integral equation. Two examples from the theory of elasticity are then considered to show the application of the technique.
Extremal paths and holonomic constitutive laws in elastoplasticity
Authors: Reddy B. D. , Martin J. B. , Griffin T. B.
Published: December 2016

Extremal stress paths between any two stresses are investigated for elastic-plastic materials, extending existing results which hold for the case when one of the stress points is at the origin. Assumptions about the differentiability of the various work and complementary work functions are relaxed, and it is shown that the maximum complementary work

U
^

\hat U

is a potential for the strain in the sense that the strain lies in the subdifferential of

U
^

\hat U

. In the same way the minimum work

W
^

\hat W

is a potential for stress. Parallel investigations with respect to maximum complementary plastic work and plastic work show that these quantities are potentials for stress and plastic strain increment, respectively. A holonomic constitutive law based on the relations between stress and strain, obtained when the stress history follows an extremal path, is constructed.
Existence and uniqueness in nonclassical diffusion
Authors: Kuttler K. , Aifantis Elias C.
Published: December 2016

We consider a class of diffusion models that arise in certain nonclassical physical situations and discuss existence and uniqueness of the resulting evolution equations.