bluuebunny/crossref_metadata_2025 huggingface dataset:
Anharmonic analysis of a time-dependent packed bed thermocline
Authors: Margolis Stephen B.
Published: December 2016

A vectorized separation of variables approach is applied to a coupled pair of parabolic partial differential equations describing the degradation of a thermocline in a packed bed thermal storage tank. The time-dependent quasi-one-dimensional model includes the effects of finite tank length, thermal conduction in the direction parallel to the tank walls, and heat transfer between the fluid and solid components of the bed. For certain classes of boundary conditions, the analysis leads to an eigenvalue problem for the spatial dependence of the fluid and solid temperatures in the bed. The eigenvalues and corresponding eigenfunctions are readily calculated, and completeness of the eigenfunctions follows from a transformation to an integral equation by the construction of a Green’s tensor function. The method is illustrated by an example which arises in the analysis of the thermal storage subsystem of a central solar receiver power plant.
Isoperimetric inequalities for the Dirichlet eigenvalue problem
Authors: Horgan Cornelius O. , Wheeler Lewis T.
Published: December 2016

Isoperimetric inequalities for the first Dirichlet eigenvalue are discussed, with application to the development of an approximate formula appropriate for thin strip domains.
On the dynamics of a second-order thin rod
Authors: Pastrone Franco
Published: December 2016

A second-order theory of hyperelastic thin rods is developed from a three-dimensional point of view. The basic topics of geometry and kinematics are given and strain variables are introduced. By means of the virtual work theorem we obtain the equations of motion; the internal work theorem determines the choice of stress variables and, under the hypothesis of hyperelasticity, stress-strain relations are shown. Finally a suitable choice of a second-degree potential, in the Signorini sense, is given.
Generalization of dual structural optimization problems in terms of fractional programming
Authors: Morris A. J.
Published: December 2016

Duality now plays an important role in the theory of optimum structures but has not been given adequate detailed consideration within this context. The paper makes a limited attempt to satisfy this requirement through a generalization of the associated duality theory by formulating the structural optimization as a fractional program. This provides some new forms for the dual objective function and crystalizes some of the intrinsic problems associated with dual structural programs.
Vibrations of long narrow plates. I
Authors: Chadwick R. S.
Published: December 2016

An asymptotic theory for the determination of the natural flexural modes and eigenvalues of a long narrow plate having a quite general planform shape is presented. A local transition layer exists in the vicinity of the widest portion of the plate, which reveals the essential structure of the flexural modes. Mode shapes are computed for trapezoidal and semi-elliptical planforms. The theory is relevant to an understanding of frequency discrimination in the cochlea (inner ear).
Vibrations of long narrow plates. II
Authors: Chadwick R. S.
Published: December 2016

An asymptotic theory for the natural bending modes of a long narrow plate having variable width and thickness is constructed. A two-variable expansion and a transition layer expansion are matched to find the mode shapes and eigenvalues.
Curves along which plane waves can interfere
Authors: Karp S. N. , Machover M.
Published: December 2016

Partial results are given on a conjecture in inverse scattering theory concerning the interference of two-dimensional plane waves. The conjecture states that an odd number of plane waves of the same frequency can only cancel each other at isolated points and not along a simple continuous curve. It is partially confirmed here for curves which are nearly flat at some point. An analysis is also made for various possible nodes for an even number of plane waves.
Energy-like Liapunov functionals for linear elastic systems on a Hilbert space
Authors: Walker J. A.
Published: December 2016

An approach is presented for generating energy-like functionals for linear elastic dynamic systems on a Hilbert space. The objective is to obtain a family of functionals which may be used for stability analysis of the equilibrium; i.e., Liapunov functionals. Although the energy functional, when one exists, is always a member of this family, the family is shown to exist even when an energy functional does not. Several discrete and distributed-parameter examples are presented, as are certain specific techniques for utilizing this approach.
A procedure for determination of the exponential stability of certain differential-difference equations
Authors: Datko R.
Published: December 2016

Certain continuity properties of the spectra of linear autonomous differential-difference equations which depend on a parameter are developed. These results are used to obtain a practical criterion for determination of the exponential stability of these systems.
Monotone methods for the Thomas-Fermi equation
Authors: Mooney J. W.
Published: December 2016

The boundary-value problem for the ionized atom case of the Thomas-Fermi equation is transformed to a certain convex nonlinear boundary-value problem. Two iterative procedures, previously developed for such problems, are constructed for the ionized atom problem. A comparative analysis of the efficiency of the iteration schemes is presented. The existence and uniqueness of a solution is established and the solution is shown to have monotonic dependence on the boundary conditions. Numerical bounds are obtained for a specific problem.
Nearly isochoric elastic deformations: volume changes in plane strain
Authors: Ogden R. W.
Published: December 2016

Plane deformations of nearly incompressible elastic solids are examined with a view to calculating the volume changes accompanying an arbitrary (plane) deformation. The dilatation is shown to be determined from a knowledge of the deformation appropriate to the corresponding incompressible material under the same boundary conditions. This work parallels that given in a previous paper for three-dimensional deformations and relies on the decomposition of the deformation gradient into its dilatational and isochoric parts.
Fluctuating boundary layers in a rotating fluid with variable suction
Authors: Purushothaman R.
Published: December 2016

The viscous flow in a semi-infinite fluid above a porous oscillating disk with unsteady normal velocity in a rotating frame is studied. Expressions for velocity and skin friction are obtained. For the case of suction, the oscillations are contained in boundary layers of finite thickness. In the case of injection, resonance occurs when the imposed frequency is twice the angular velocity of rotation. It is found that the resonance can be avoided under certain conditions.
Free boundary problems with radiation boundary conditions
Authors: Tao L. N.
Published: December 2016

The paper is concerned with the free boundary problem of a semi-infinite body of arbitrarily prescribed initial temperature, subject to a mixed or radiation boundary condition at its face. The analytically exact solutions of temperature of both phases and the interfacial position are established in series of time and functions of the error integral family. Convergence of these series solutions is studied and proved. A few remarks on the solutions and their simplifications are then offered. A discussion on the analyticity of the solutions is also given. The paper concludes with an illustrative example, the so-called one-phase problem.
On secondary vorticity in internal waves
Authors: Dore B. D. , Al-Zanaidi M. A.
Published: December 2016

The generation of secondary vorticity in maintained and temporally-decaying wave motions is investigated for internal waves in fluid of great depth. In the case of two semi-infinite, homogeneous fluids of different density, the interfacial boundary layers generate a second-order, mean vorticity which diffuses inwards into the interior of both fluids, and the net vorticity produced is zero. For a continuously-stratified fluid, the free surface layer plays an indirect role and secondary vorticity, initially generated only within stratified regions by the action of a Reynolds stress, diffuses, in general, over the whole fluid, and no steady-state vorticity field is established. In finite depths, a steady state ultimately exists for maintained waves, and the mass transport velocity field is investigated.
Resonant frequencies in an electromagnetic eccentric spherical cavity
Authors: Kanellopoulos John D. , Fikioris John G.
Published: December 2016

The interior boundary-value electromagnetic (vector) problem in the region between two perfectly conducting spheres of radii

R
1

{R_1}

,

R
2

{R_2}

and distance

d
d

between their centers is considered. Surface singular integral equations are used to formulate the problem. Use of spherical vector wave functions and related addition theorems reduces the solution of the integral equations to the problem of solving an infinite set of linear equations. Their determinant is evaluated in powers of

k
d
=
2
π
d

/

λ

kd = 2\pi d/\lambda

to a few terms. It is then specialized to the axially symmetric case and set equal to zero. This yields closed-form expressions for the coefficients

g

n
s

{g_{ns}}

in the resulting relations
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Subscript n s Baseline left-parenthesis k d right-parenthesis equals omega Subscript n s Baseline left-parenthesis 0 right-parenthesis left-bracket 1 plus g Subscript n s Baseline left-parenthesis k d right-parenthesis squared plus dot dot dot right-bracket">

ω

n
s

(

k
d

)

=

ω

n
s

(
0
)

[

1
+

g

n
s

(

k
d

)

2

+
⋅
⋅
⋅

]

{\omega _{ns}}\left ( {kd} \right ) = {\omega _{ns}}\left ( 0 \right )\left [ {1 + {g_{ns}}{{\left ( {kd} \right )}^2} \\ + \cdot \cdot \cdot } \right ]

for the natural frequencies of the cavity. Numerical results, comparisons and possible generalizations are also included.
Small-amplitude internal waves due to an oscillatory pressure
Authors: Mahanti N. C.
Published: December 2016

The initial-value problem of waves in superposed fluids which are otherwise unlimited due to an axisymmetrical oscillatory pressure is solved by the method of integral transforms. The wave integral representing interfacial displacement is evaluated in an asymptotic form that remains uniformly valid through the transition zone.
Exponential asymptotic expansions for nonlinear differential equations
Authors: Day William B.
Published: December 2016

An exponential method for obtaining asymptotic expansions of the solution of a linear differential equation was presented by Brull and Soler in [3], We extend this method to the nonlinear differential equation of the type

L

(
u
)

+

Σ
1

N

f

(

x
,
t

)

u
n

=
0

L\left ( u \right ) + {\Sigma _1}^Nf\left ( {x, t} \right ){u^n} = 0

, where

t
t

is the small parameter. Three examples are used to illustrate the technique and to explain how uniformly valid expansions may be obtained.
Lifting-line theory for a swept wing at transonic speeds
Authors: Cook L. Pamela
Published: December 2016

The boundary-value problems describing the first-order corrections
\[

O

(

1

A
R

,

1

A
R

ln
⁡
A
R

)

O\left ( {\frac {1}{{AR}},\frac {1}{{AR}}\ln AR} \right )

\]
to two-dimensional flow about a lifting swept wing at transonic speeds

(

M
∞

>
1

)

\left ( {{M_\infty } > 1} \right )

are derived. The corrections are found by the use of the method of matched asymptotic expansions on the transonic small disturbance equations. The wing is at a sweep angle of

O

(

(

1
−

M
∞
2

)

1

/

2

)

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

in the physical plane, hence of

O

(
1
)

O\left ( 1 \right )

in the transonic small disturbance plane. As has been noted for subsonic flow, the finite sweep angle necessitates the introduction of terms

O

(

(

A
R

)

−
1

ln
⁡
A
R

)

O\left ( {{{\left ( {AR} \right )}^{ - 1}}\ln AR} \right )

. These terms arise naturally in the matching process. Of particular interest is the derivation of the near field of a skewed lifting-line which is found by Mellin transform techniques. Also of interest is the fact that the influence of the nonzero sweep angle can be completely separated from the unswept solution.
Global analysis of magnetic domains
Authors: Miranker W. L. , Willner B. E.
Published: December 2016

We study the phase transition problem for ferromagnetic materials. By means of examples we show that a mathematical bifurcation process may be used to describe the evolution of one configuration of magnetic domains into another.
Similarity solutions of the nonlinear diffusion equation
Authors: Grundy R. E.
Published: December 2016

The paper considers similarity solutions of the nonlinear diffusion equation of the form

t
α

f

(
η
)

{t^\alpha }f\left ( \eta \right )

where

η
=
r

t

−
δ

\eta = r{t^{ - \delta }}

or

exp
⁡

(

α
t

)

f

(
η
)

\exp \left ( {\alpha t} \right )f\left ( \eta \right )

where

η
=
r
exp
⁡

(

−
δ
t

)

\eta = r\exp \left ( { - \delta t} \right )

. The novel feature of the paper is that the second-order differential equation for

f
f

is reduced to a system of first-order equations and a phase plane analysis of one member of the system can be made. In this way we may discuss the existence and uniqueness of all the solutions for

f

(
n
)

f\left ( n \right )

. Restricting the discussion to plane geometry, we list all the continuous solutions to the basic problem on

0
≤
η
≤
∞

0 \le \eta \le \infty

with

f

(
0
)

=
U
≥
0

f\left ( 0 \right ) = U \ge 0

and

f

(
∞
)

=
0

f\left ( \infty \right ) = 0

. Solutions of previous authors are identified as special cases.
Plane harmonic functions in the presence of a surface layer of arbitrary stiffness
Authors: Zahalak George Ireneus
Published: December 2016

Complex variable techniques are employed to characterize two-dimensional solutions

u

(

x
,
y

)

u\left ( {x, y} \right )

of Laplace’s equation which satisfy the boundary condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket beta left-parenthesis partial-differential squared u slash partial-differential y squared right-parenthesis plus left-parenthesis partial-differential u slash partial-differential x right-parenthesis right-bracket Subscript x equals 0 Baseline equals 0">

[

β

(

∂
2

u

/

∂

y
2

)

+

(

∂
u

/

∂
x

)

]

x
=
0

=
0

{\left [ {\beta \left ( {{\partial ^2}u/ \\ \partial {y^2}} \right ) + \left ( {\partial u/\partial x} \right )} \right ]_{x = 0}} = 0

, where

β
\beta

is referred to as the surface-stiffness parameter. Simple closed-form singular solutions are derived which satisfy this boundary condition and represent source and dislocation singularities. The former is used to synthesize the field generated by a small inclusion of arbitrary shape on which

u
=
1

u = 1

, in the presence of a boundary at

y
=
0

y = 0

on which

u
=
0

u = 0

. At points not near the inclusion the field has the form of a function of position and surface stiffness multiplied by a strength factor which depends on the size and shape of the inclusion and the surface stiffness. Detailed calculations are presented for two extreme shapes of inclusions—a shallow, wide inclusion on the surface and a deep, narrow inclusion penetrating below the surface—which exhibit the relation between the field near the inclusion and the distant field, and show explicitly the dependence of the strength factor on surface stiffness and inclusion size and shape. The nature and strength of the singularities at the tips of the inclusions are also examined and it is found that a tip singularity at the surface changes character as the surface stiffness varies.
Generic bifurcation in the obstacle problem
Authors: Mallet-Paret John
Published: December 2016

We consider a class of singularities, locally of the form

y
2

=
p

(
x
)

{y^2} = p\left ( x \right )

near the origin in

R
2

{R^2}

, describing the shape of a free boundary curve arising from an elliptic free boundary value problem. The point of view taken is that of generic bifurcation, in particular with more than one parameter present. Of prime interest is a description of the unfoldings of such singularities, their normal forms, and generic conditions for one- and two-parameter unfoldings. The two simplest cases corresponding to perturbations of singularities

y
2

=

x
n

+
O

(

x

n
+
1

)

,
n
=
4
,
5

{y^2} = {x^n} + O\left ( {{x^{n + 1}}} \right ),n = 4, 5

are treated in greater detail and the bifurcation diagram for a generic two-parameter unfolding is given. Our results do not rigorously concern the free boundary problem itself, but rather set down a formal framework, or model, for studying this problem in terms of bifurcation theory. We prove theorems describing this model. Nevertheless, our results have a bearing on any rigorous analysis of this problem since they form the necessary first step to such an analysis. The theory for computing the normal forms of solutions up to first order, for example, is given here.
Power series solutions for the 𝑚th-order-matrix ordinary differential equation
Authors: Ruiz-Claeyssen Julio , Zevallos Gutierrez Mauro
Published: December 2016

A description of the fundamental solution of the mth-order linear ordinary differential equation with matrix coefficients is given in terms of power series and the Green function. The second-order equation is discussed.
Generalized thermoelasticity for anisotropic media
Authors: Dhaliwal Ranjit S. , Sherief Hani H.
Published: December 2016

The equations of generalized thermoelasticity for an anisotropic medium are derived. Also, a uniqueness theorem for these equations is proved. A variational principle for the equations of motion is obtained.
Relaxation oscillations of a van der Pol equation with large critical forcing term
Authors: Grasman J.
Published: December 2016

A van der Pol equation with sinusoidal forcing term is analyzed with singular perturbation methods for large values of the parameter. Asymptotic approximations of (sub)harmonic solutions with period

T
=
2
π

(

2
n
−
1

)

,
n
=
1
,
2
,
.
.
.

T = 2\pi \left ( {2n - 1} \right ),n = 1, 2, ...

are constructed under certain restricting conditions for the amplitude of the forcing term. These conditions are such that always two solutions with period

T
=
2
π

(

2
n
±
1

)

T = 2\pi \left ( {2n \pm 1} \right )

coexist.
Diffusion systems reacting at the boundary
Authors: Olmstead W. E.
Published: December 2016

The problem considered is that of two species or chemical concentrations which independently diffuse within the same or adjacent regions. The coupling interaction takes place only along a common boundary. This boundary reaction is allowed to be either totally dissipative wherein both species are removed by the interaction, or semi-dissipative wherein one species is stimulated at the expense of the other. This physical situation is modeled by two independent, linear heat equations, each defined over a one-dimensional, semi-infinite domain. Associated with each heat equation is a boundary flux condition containing a nonlinear interactive term which couples the solutions of the two heat equations. With only boundary interaction, the problem can be reduced to the study of two coupled Volterra integral equations. By using monotone operator methods these integral equations are shown to have positive solutions. Uniqueness is also established. The large-time asymptotic behavior of the solutions is examined for the cases of both fast and slow decay of data.
On continuous dependence in finite elasticity
Authors: Spector Scott J.
Published: December 2016

We investigate the relation between stability and continuous dependence for a nonlinearly elastic body at equilibrium. We show that solutions of the governing equations that lie in a convex, stable set of deformations depend continuously on the body forces and the surface tractions. The definition of stability used is essentially due to Hadamard.
Translational addition theorems for spheroidal scalar and vector wave functions
Authors: Sinha Bateshwar P. , Macphie Robert H.
Published: December 2016

The translational addition theorems for spheroidal scalar wave functions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript m n Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis h comma xi right-parenthesis upper S Subscript m n Baseline left-parenthesis h comma eta right-parenthesis exp left-parenthesis j m phi right-parenthesis semicolon i equals 1 comma 3 comma 4">

R

m
n

(
i
)

(

h
,
ξ

)

S

m
n

(

h
,
η

)

exp
⁡

(

j
m
ϕ

)

;
i
=
1
,
3
,
4

R_{mn}^{\left ( i \right )}\left ( {h, \xi } \right ){S_{mn}}\left ( {h, \eta } \right )\exp \left ( {jm\phi } \right ); i = 1, 3, 4

and spheroidal vector wave functions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript m n Superscript x comma y comma z left-parenthesis i right-parenthesis Baseline left-parenthesis h semicolon xi comma eta comma phi right-parenthesis comma upper N Subscript m n Superscript x comma y comma z left-parenthesis i right-parenthesis Baseline left-parenthesis h semicolon xi comma eta comma phi right-parenthesis semicolon i equals 1 comma 3 comma 4">

M

m
n

x
,
y
,
z

(
i
)

(

h
;
ξ
,

η
,
ϕ

)

,

N

m
n

x
,
y
,
z

(
i
)

(

h
;
ξ
,
η
,
ϕ

)

;
i
=
1
,
3
,
4

M_{mn}^{x, y, z\left ( i \right )}\left ( {h; \xi , \\ \eta , \phi } \right ), N_{mn}^{x, y, z\left ( i \right )}\left ( {h; \xi , \eta , \phi } \right ); i = 1, 3, 4

, with reference to the spheroidal coordinate system at the origin

O
O

, have been obtained in terms of spheroidal scalar and vector wave functions with reference to the translated spheroidal coordinate system at the origin

O
′

O’

, where

O
′

O’

has the spherical coordinates (

r
0

,

θ
0

,

ϕ
0

{r_0},{\theta _0},{\phi _0}

) with respect to

O
O

. These addition theorems are useful in acoustics and electromagnetics in those cases involving spheroidal radiators and scatterers.
Asymptotic expansions of eigenvalues and eigenfunctions of random boundary value problems
Authors: Day William B.
Published: December 2016

An asymptotic procedure is developed for calculating the eigenvalues and eigenfunctions of linear boundary-value problems which may contain random coefficients in the operator. The corresponding asymptotic series for the solution of a second-order initial-value problem is shown to be convergent.
On a uniform asymptotic expansion of a Fourier-type integral
Authors: Wong R.
Published: December 2016

An alternative derivation is given for a uniform asymptotic expansion of the integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis h right-parenthesis equals integral Subscript q Superscript z Baseline g left-parenthesis StartRoot y squared minus q squared EndRoot right-parenthesis sine y h d y left-parenthesis h right-arrow plus normal infinity right-parenthesis">

I

(
h
)

=

∫
q
z

g

(

y
2

−

q
2

)

sin
⁡
y
h
d
y

(

h
→
+
∞

)

I\left ( h \right ) = \int _q^z {g\left ( {\sqrt {{y^2} - {q^2}} } \right )\sin yh dy \qquad \left ( {h \to + \infty } \right )}

\]
, which has been obtained recently by Schmidt. Here

q
q

varies in the interval

[

0
,

q
0

]

\left [ {0,{q_0}} \right ]

,

q
0

{q_0}

is some fixed number less than

z
z

, and

z
z

is finite. A similar result is obtained for integrals having an infinite range of integration. Realistic bounds are also provided for the error terms associated with the expansions. Our approach is based on a summability method introduced by Olver.
On a conjecture concerning the means of the eigenvalues of random Sturm-Liouville boundary value problems
Authors: Boyce William E.
Published: December 2016

It has been known for several years that the expected value

⟨

λ
1

⟩

\left \langle {{\lambda _1}} \right \rangle

of the smallest eigenvalue of a self-adjoint positive definite random Sturm-Liouville boundary value problem satisfies the relation

⟨

λ
1

⟩

≤

μ
1

\left \langle {{\lambda _1}} \right \rangle \le {\mu _1}

where

μ
1

{\mu _1}

is the smallest eigenvalue of the corresponding deterministic problem obtained by replacing each random coefficient by its mean. It has been an open question whether similar inequalities are valid for the higher eigenvalues. The answer is negative, as shown by the counterexample given in this note.
The two-variable technique for singular partial differential problems and its justification
Authors: Bouthier M.
Published: December 2016

Dirichlet problems with a small parameter in factor of the highest derivative are considered for bounded domains. A two-variable technique is formalized in order to carry out the study of the main boundary layer. A “secular “ hypothesis is made, and a unique and uniformly valid asymptotic expansion is obtained. However, it is shown that the “secular” hypothesis may be weakened and this yields a whole set of expansions. Then the asymptotic validity of each expansion in the set can be proven for second-order operators by means of an extension of a theorem due to Eckhaus and Jager.
On three-dimensional generalizations of the Boussinesq and Korteweg-de Vries equations
Authors: Infeld E.
Published: December 2016

Three-dimensional generalizations of two different forms of the Boussinesq equation are derived. They are investigated for stability of slowly varying nonlinear wavetrains. The results obtained are then compared with the stability properties following from the full water wave equations. Agreement is found to be good for

h
0

k
0

{h_0}{k_0}

(depth times wavenumber) of order one. This is very satisfactory, as the Boussinesq equations are only supposed to be valid for small

h
0

k
0

{h_0}{k_0}

. In particular, one version of the Boussinesq equation is found to yield instability with respect to one-dimensional perturbations for

h
0

k
0

>
1.5

{h_0}{k_0} > 1.5

(as against 1.36 for the full equations). Finally, a similar comparison is performed for the three-dimensional Korteweg—de Vries equation.
Stretching the circular cylindrical elastic sheet
Authors: Bowman B. F.
Published: December 2016

The theory of thin elastic sheets, neglecting bending, is applied to many thin surface problems. The classical theory for these surfaces is inadequate when the strains are small, the surface is unstressed before deformation, and boundary displacements are prescribed. The equations in this case are degenerate. Attempts to solve the exact equations for these elastic membranes have failed for small strains. The purpose of this paper is to determine the behavior of the solution to the exact equations for the elastic surface which is initially a circular cylinder and which is deformed by a uniform pull at the ends keeping the end radii fixed. A resolution of the small-strain behavior both analytically and numerically is of particular importance.
A note on the method of multiple scales
Authors: Veronis George
Published: December 2016

A modification of the method of multiple scales makes use of the expansion of parameters of the system in order to remove undesirable characteristics of the solution obtained by the usual multiple-scale method. When applied to the damped harmonic oscillator, the modification leads to the exact solution. For the Duffing equation it leads to an approximation which can be reduced to the solution reported by Nayfeh [1], However, the solution derived here appears to be more accurate and the frequency takes on a form without nonuniformities.
On monotonicity of the optimal strain path in linear viscoelasticity
Authors: Spector Scott J.
Published: December 2016

We consider the problem of finding the strain path

e

(
t
)

,
0
≤
t
≤
T

e\left ( t \right ), 0 \le t \le T

which minimizes the work done by a one-dimensional linear viscoelastic material. The material is assumed to be initially unstrained; the time interval is fixed; and, the final strain is specified. It has previously been shown

1

^{1}

that this problem has a unique solution which is referred to as the optimal strain path. We prove (i) The optimal strain path is monotone. (ii) An estimate of the work done on the optimal strain path. The first proves a conjecture of Gurtin, MacCamy and Murphy, while the second is an alternative proof to a result of Day.
Stable small-amplitude solutions in reaction-diffusion systems
Authors: Ermentrout G. Bard
Published: December 2016

Bifurcation and perturbation techniques are used to construct small-amplitude periodic wave-trains for general systems of reaction and diffusion. All solutions are characterized by the amplitude

a
a

and the wavenumber

k
k

. For scalar diffusion,

k
∼
a

k \sim a

, while for certain types of nonscalar diffusion,

k
k

is bounded away from zero as

a
↘
0

a \searrow 0

. For certain ranges of

a
a

and

k
k

, linear stability of waves is demonstrated.
Small-amplitude waves on the surface of a layer of a viscous liquid
Authors: Cortelezzi L. , Prosperetti A.
Published: December 2016

We study the initial-value problem posed by the small-amplitude waves on the surface of a layer of a viscous fluid of infinite lateral extent. The problem of the motion of the interface is reduced to an integro-differential equation which is solved by means of the Laplace transform. Explicit numerical results for illustrative cases are presented.
Khachiyan’s algorithm for linear inequalities: optimization and implementation
Authors: Bisshopp Frederic
Published: December 2016

An optimized version of Khachiyan’s algorithm is developed here, and an APL implementation of it is provided. The operation count for

M
M

linear constraints on

N
N

variables is brought from the

O

(

N
4

M
2

)

O\left ( {{N^4}{M^2}} \right )

of the original algorithm to

O

(

N
4

M
l
n
N

)

O\left ( {{N^4}M ln N} \right )

with a smaller coefficient. There is no significant change of the storage requirement, which remains at

O

(

N
M

)

O\left ( {NM} \right )

locations.
Planar premixed-flame/end-wall interaction: the jump conditions across the thin flame
Authors: Bush W. B. , Fink S. F.
Published: December 2016

Within the context of time-dependent interaction of a planar premixed laminar flame with a cold parallel end wall, the jump conditions for the first (spatial) derivatives of the dependent variables across the thin flame are obtained through the solution of the nonlinear diffusive-reactive boundary-value problem that describes the structure of the flame zone. Recently developed numerical techniques are employed to yield solutions of this boundary-value problem.
On the eigentheory of operators which exhibit a slow variation
Authors: Sirovich Lawrence , Knight B. W.
Published: December 2016

A class of linear operators which exhibit slow variation is considered. If the kernel of the operator is

K

(

x
−
y
,

1
2

ε

(

x
+
y

)

)

K\left ( {x - y,\frac {1}{2}\varepsilon \left ( {x + y} \right )} \right )

,

ε
\varepsilon

the parameter of slowness, then its Wigner transform is defined to be

K
~

(

p
,
q

)

=
∫
K

(

u
,
p

)

exp
⁡

(

−
i
u
p

)

d
u

\tilde K\left ( {p, q} \right ) = \smallint K\left ( {u, p} \right )\exp \left ( { - iup} \right )du

. The eigenvalues of such operators are shown to follow an area rule: if the curve

λ
n

=

K
~

(

p
,
q

)

{\lambda _n} = \tilde K\left ( {p, q} \right )

contains the area

A

(
λ
)

=

(

2
n
+
1

)

π
ε

A\left ( \lambda \right ) = \left ( {2n + 1} \right )\pi \varepsilon

then

λ
n

{\lambda _n}

is an eigenvalue. Forms for the corresponding eigenfunctions are also obtained. Classical WKB theory is shown to be a special case and other examples are given.
A note on the stability of traveling-wave solutions to a class of reaction-diffusion systems
Authors: Bell Jonathan
Published: December 2016

Many classes of reaction-diffusion systems have been shown to have traveling-wave solutions. For a class of such systems for which a comparison theorem can be used, we establish a wave stability result which roughly states that these wave solutions are asymptotically stable to perturbations which lie in some weighted

L
p

{L_p}

-space if their speeds are sufficiently large. We then apply this result to some excitable systems, namely a model of the Belousov-Zhabotinskii reaction, a substrate-inhibition biochemical model, and a class of models recently studied by Fife.
The axisymmetric branching behavior of complete spherical shells
Authors: Lange Charles G. , Kriegsmann Gregory A.
Published: December 2016

The purpose of this paper is to describe the axisymmetric branching behavior of complete spherical shells subjected to external pressure. By means of an asymptotic integration technique (based on the smallness of the ratio of the shell thickness to the shell radius) applied directly to a differential equation formulation, we are able to continue the solution branches from the immediate vicinity of the bifurcation points, where the solution has the functional form predicted by the linear buckling theory, to the region where the solution consists of either one or two “dimples” with the remainder of the shell remaining nearly spherical. The analysis deals with a novel aspect of bifurcation theory involving “closely spaced” eigenvalues.
The parabolic umbilic catastrophe and its application in the theory of elastic stability
Authors: Hui David , Hansen Jorn S.
Published: December 2016

The implications of the parabolic umbilic catastrophe in the theory of elastic stability are investigated. In particular, the influence of terms in the potential energy which are deemed necessary for a complete analysis and the isolation of primary critical surfaces are considered. The results are demonstrated for the example of the buckling and initial post-buckling of a spherical shell under the influence of a constant as well as a spatially variable pressure.
Oseen flow of a compressible fluid past a flat plate
Authors: Homentcovschi Dorel
Published: December 2016

The steady two-dimensional flow of a compressible fluid past a flat plate inclined to the uniform stream is studied on the basis of the Oseen approximation. The use of the momentum equations written in distributions allows us to obtain in a straight-forward fashion the integral equations of the skin friction and of the lift. These equations are integrated for various values of the Reynolds and Mach numbers. The values obtained for the lift in the limit case of the inviscid fluid coincide with those known in conventional aerodynamics for the subsonic and for supersonic flow.
Solutions of the electromagnetic wave equations for point dipole sources and spherical boundaries
Authors: Hirvonen M. T.
Published: December 2016

Solutions of the electromagnetic wave equation are derived in systems containing spherical interfaces when the source field is that of a magnetic or electric point dipole. Piecewise constant electromagnetic parameters are assumed, but their values as well as the frequency of the source field are arbitrary. The solutions are obtained in terms of scalar and vector spherical harmonics. A sphere embedded in full space with a radial or transverse source dipole is considered explicitly.
On some epidemic models
Authors: Gripenberg Gustaf
Published: December 2016

The qualitative behavior of the solution

x
x

of the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t right-parenthesis equals k left-parenthesis p left-parenthesis t right-parenthesis minus integral Subscript 0 Superscript t Baseline upper A left-parenthesis t minus s right-parenthesis x left-parenthesis s right-parenthesis d s right-parenthesis left-parenthesis f left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline a left-parenthesis t minus s right-parenthesis x left-parenthesis s right-parenthesis d s right-parenthesis comma t greater-than-or-equal-to 0">

x

(
t
)

=
k

(

p

(
t
)

−

∫
0
t

A

(

t
−
s

)

x

(
s
)

d
s

)

(

f

(
t
)

+

∫
0
t

a
(
t
−
s
)
x
(
s
)
d
s

)

,
t
≥
0

x\left ( t \right ) = k\left ( {p\left ( t \right ) - \smallint _0^tA\left ( {t - s} \right )x\left ( s \right )ds} \right )\left ( {f\left ( t \right ) + \smallint _0^ta(t - s)x(s)ds} \right ),t \ge 0

\]
is studied. This equation arises in the study of the spread of an infectious disease that does not induce permanent immunity.
Precessional advances of a Lock-Fowler missile
Authors: Rath P. C. , Pal J.
Published: December 2016

It is proved that for apses-to-apses motions which are purely direct (retrograde), a Lock-Fowler missile satisfies the Halphen inequality. The result is independent of all launching conditions.
The potential distribution in a constricted cylinder: an exact solution
Authors: Rosenfeld A. M. , Timsit R. S.
Published: December 2016

An exact solution to the Laplace equation is derived for the distribution of electric potential within a long cylinder carrying a circular constriction along its axis. The expression obtained for the potential distribution is reduced to a form which may be readily evaluated and is highly accurate for a ratio of constriction radius to cylinder radius approaching unity. Exact expressions both for the electric current density within the constriction and for the spreading resistance (i.e., the increase in resistance of the cylinder due to constriction of current flow lines) are also obtained.
On a problem in the dynamic theory of cracks
Authors: Tait R. J. , Moodie T. Bryant
Published: December 2016

We show that it is possible, for a certain case of a traveling crack problem, to obtain an explicit solution, in the entire region of interest, in terms of elementary functions. This affords a simple way of constructing level stress curves in the entire region, in contrast with the general case when simple expressions are obtainable at best along a particular axis.
A mathematical model for linear elastic systems with structural damping
Authors: Chen G. , Russell D. L.
Published: December 2016

We present a mathematical model exhibiting the empirically observed damping rates in elastic systems. The models studied are of the form (

A
A

the relevant elasticity operator)
\[

x
¨

+
B

x
˙

+
A
x
=
0

\ddot x + B\dot x + Ax = 0

\]
with

B
B

related in various ways to the positive square root,

A

1

/

2

{A^{1/2}}

, of

A
A

. Comparison with existing “ad hoc” models is made.
Lower bounds for eigenvalues of Sturm-Liouville problems with discontinuous coefficients: integral equation methods
Authors: Horgan C. O. , Spence J. P. , Andry A. N.
Published: December 2016

This paper is concerned with application of the theory of Fredholm integral equations to Sturm-Liouville problems with discontinuous coefficients. Such problems occur naturally in many areas of application involving mechanics of heterogeneous media. Due to the nonsmoothness of the coefficients, the eigenvalue spectrum may exhibit severe irregularities. Lower bounds for eigenvalues are obtained which reflect this behavior. Numerical results are presented for example problems previously treated using other methods.
Stability and bifurcation in a modulated Burgers system
Authors: Olmstead W. E. , Davis S. H.
Published: December 2016

The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude

A

(
τ
)

A\left ( \tau \right )

of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate

θ

(
τ
)

\theta \left ( \tau \right )

. Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in

θ

(
τ
)

\theta \left ( \tau \right )

. For each of these oscillatory-type modulations, it is found that

A
2

(
τ
)

{A^2}\left ( \tau \right )

has the same long-time mean value as the unmodulated case, implying no alteration of the final mean kinetic energy. Applications to various fluid-dynamical phenomena are discussed.
An inequality for the coefficient 𝜎 of the free boundary 𝑠(𝑡)=2𝜎√𝑡 of the Neumann solution for the two-phase Stefan problem
Authors: Tarzia Domingo Alberto
Published: December 2016

We consider a semi-infinite body (e.g. ice), represented by

(

0
,
+
∞

)

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

, with an initial temperature

−
c
>
0

- c > 0

having a heat flux

h

(
t
)

=
−

h
0

/

t

(

h
0

>
0

)

h\left ( t \right ) = - {h_0}/\sqrt t \left ( {{h_0} > 0} \right )

in the fixed face

x
=
0

x = 0

. If

h
0

>
c

k
1

/

π

a
1

{h_0} > c{k_1}/\sqrt {\pi {a_1}}

there exists a solution, of Neumann type, for the resulting two-phase Stefan problem. If we connect it with the Neumann problem (on

x
=
0

x = 0

the body has a temperature

b
>
0

b > 0

we obtain the inequality erf

(

σ

/

a
2

)

>

(

k
2

b

a
1

/

k
1

c

a
2

)

\left ( {\sigma /{a_2}} \right ) > \left ( {{k_2}b{a_1}/{k_1}c{a_2}} \right )

for the coefficient

σ
\sigma

of the free boundary

s

(
t
)

=
2
σ

t

s\left ( t \right ) = 2\sigma \sqrt t

, where

k
i

{k_i}

, and

a
i
2

a_i^2

are respectively the thermal conductivity and thermal diffusivity coefficients of the corresponding

i
i

phase

i
=
1
:

i = 1:

solid phase,

i
=
2
:

i = 2:

liquid phase). If

h
0

>
c

k
1

/

π

a
1

{h_0} > c{k_1}/\sqrt {\pi {a_1}}

there is no solution of the initial problem and if

h
0

=
c

k
1

/

π

a
1

{h_0} = c{k_1}/\sqrt {\pi {a_1}}

the problem has no physical meaning and corresponds to the case where the latent heat of fusion

L
L

tends to infinity.
Thermodynamic principle of virtual dissipation and the dynamics of physical-chemical fluid mixtures including radiation pressure
Authors: Biot M. A.
Published: December 2016

Equations of motion are obtained for a viscous fluid mixture including thermal and intermolecular diffusion as well as chemical reactions and radiation pressure. They are derived by applying the thermodynamic principle of virtual dissipation. The method also incorporates a new approach to the chemical thermodynamics of open systems which leads to new concepts and formulas for the heat of reaction and the affinity. They are simpler and more general than classical values. Instead of chemical potentials, new “ convective potentials “ are used which involve physical properties restricted to the system. They do not require extrapolations to absolute zero or the use of undetermined constants. No statistical theory is involved. A noncalorimetric evaluation of the heat of mixing is obtained from the concept of injection pressure of each substance in the mixture. Field equations are derived and a coupling between viscous stress gradients and diffusion is brought out. The convective potentials lead to a new evaluation of the thermodynamic functions of mixtures as well as a new generalized formulation of the Gibbs–Duhem theorem. Translational invariance of the dissipation is discussed and related to total momentum balance. Lagrangian equations of motion with generalized collective coordinates are derived directly from the variational principle and should provide a powerful approach to problems of stellar dynamics with radiation pressure.
On the steady-state propagation of an anti-plane shear crack in an infinite general linearly viscoelastic body
Authors: Walton Jay R.
Published: December 2016

The steady-state propagation of a semi-infinite anti-plane shear crack is considered for a general infinite homogeneous and isotropic linearly viscoelastic body. Inertial terms are retained and the only restrictions placed on the shear modulus are that it be positive, continuous, decreasing and convex. For a given integrable distribution of shearing tractions travelling with the crack, a simple closed-form solution is obtained for the stress intensity factor and for the entire stress field ahead of and in the plane of the advancing crack. As was observed previously for the standard linear solid, the separate considerations of two distinct cases, defined by parameters

c
c

and

c
∗

c*

, arises naturally in the analysis. Specifically,

c
c

and

c
∗

c*

denote the elastic shear wave speeds corresponding to zero and infinite time, and the two cases are (1)

0
>
υ
>
c
∗

0 > \upsilon > c*

and (2)

c
∗
>
υ
>
c

c* > \upsilon > c

, where

υ
\upsilon

is the speed of propagation of the crack. For case (1) it is shown that the stress field is the same as in the corresponding elastic problem and is hence independent of

υ
\upsilon

and all material properties, whereas, for case (2) the stress field depends on both

υ
\upsilon

and material properties. This dependence is shown to be of a very elementary form even for a general viscoelastic shear modulus.
A geometric-optical series and a WKB paradox
Authors: Gray Samuel H.
Published: December 2016

We discuss a solution of the one-dimensional reduced wave equation with non-constant velocity. We show that, for sufficiently small total velocity variations, this solution is exact. Furthermore, it lends itself to (high-frequency) asymptotic analysis and to elementary numerical analysis in a natural way. For reflected waves, we show that asymptotically small reflection implies numerically small reflection, thus resolving a paradox of classical WKB theory.
Stress fields in plane irrotational flows of viscous and perfectly plastic continua
Authors: Yin Wan Lee
Published: December 2016

The viscous and viscoplastic stress fields in three families of plane irrotational flows are determined by using the properties of complex analytic functions. In these flows the slip line fields are defined by the continuation of an orthogonal pair of characteristics that are straight lines, logarithmic spirals or Cornu’s spirals, respectively. The three families of solutions comprise all plane flows which, when subjected to changes in the time scale, continue to satisfy the dynamical equations of a viscoplastic continuum in the sense of Bingham.
A test for stability of linear differential delay equations
Authors: Mahaffy J. M.
Published: December 2016

The changes in the stability of a system of linear differential delay equations resulting from the delay are studied by analyzing the associated eigenvalues of the characteristic equation. A specific contour is mapped by the characteristic equation into the complex plane to give an easy test for stability from an application of the argument principle. When the real part of an eigenvalue is positive, the contour gives bounds on the imaginary part which are important in certain applications to nonlinear problems.
A multi-phase Stefan problem describing the swelling and the dissolution of glassy polymer
Authors: Tu Yih-o
Published: December 2016

In the swelling and the dissolution of certain glassy polymers, three distinctive regimes are present. They are (1) the liquid solution wherein the disassociated polymer molecules are carried away by diffusion, (2) the gel layer of rubbery polymer containing large solvent concentration, and (3) the glassy phase of the polymer where there is very little solvent penetration. The gel/liquid interface that separates the diffusion of the disassociated polymer in the liquid solution from that of the solvent in the polymer is characterized by a constant disassociation concentration. The position of this gel/liquid interface is described explicitly either by a relationship between diffusion processes, or by the rate of disassociation at the interface in addition to the diffusion processes, depending on whether the disassociation rate exceeds the diffusion capability in removing the disassociated polymer molecules at the interface.