bluuebunny/crossref_metadata_2025 huggingface dataset:
Global solutions to the lake equations with isolated vortex regions
Authors: Huang Chaocheng
Published: December 2016

The vorticity formulation for the lake equations in

R
2

{R^2}

is studied. We assume that the initial vorticity has the form

ω

(
x
,
0
)

=

ω
0

(
x
)

χ

Ω
¯

0

\omega \left ( x, 0 \right ) = {\omega _0}\left ( x \right ){\chi _{{\bar \Omega }_0}}

, where the initial vortex region

Ω
0

{\Omega _0}

is a

C

1
+
α

{C^{1 + \alpha }}

domain and

ω
0

∈

C
α

(

Ω
¯

0

)

{\omega _0} \in {C^\alpha }\left ( {\bar \Omega _0} \right )

. It is shown that the Cauchy problem can be formulated as an integral system. Global existence and uniqueness of the

C

1
+
α

{C^{1 + \alpha }}

solution to the integral system are established. Consequently, the lake equation admits a unique weak solution, global in time, in the form of

ω

(
x
,
t
)

=

ω
t

(
x
)

χ

Ω
¯

t

\omega \left ( x, t \right ) = {\omega _t}\left ( x \right ){\chi _{{{\bar \Omega }_t}}}

, where

ω
t

(
x
)

∈

C
x
α

(

Ω
¯

t

)

{\omega _t}\left ( x \right ) \in C_x^\alpha \left ( {\bar \Omega _t} \right )

and

∂

Ω
t

∈

C
α

\partial {\Omega _t} \in {C^\alpha }

.
Uniform asymptotic solutions for lamellar inhomogeneities in piezoelectric solids
Authors: Dascalu Cristian , Homentcovschi Dorel
Published: December 2016

We study the problem of a lamellar inhomogeneity of arbitrary shape embedded in a piezoelectric matrix of infinite extent. Uniform asymptotic solutions for the equations of elastostatics and electrostatics on this configuration are obtained. The first order terms, in the inhomogeneity thickness, are explicitly determined for piezoelectric inclusions, rigid inclusions of electric conductor, impermeable cracks, and cracks with inside electric field. We give real-form expressions of mechanical and electric fields at the interface and on the inhomogeneity axis. Detailed first order solutions are obtained for elliptic and lemon-shaped inhomogeneities. It is found that, while for elliptic piezoelectric inclusions the perturbation stresses and electric displacements at the inclusion ends have the same order as those given at infinity, for a lemon-shaped inclusion they are an order-of-magnitude smaller. Intensity factors are calculated for lemon-shaped cavities. It is shown that, when inside electric fields are considered, the stress intensity coefficients are influenced by the material anisotropy.
Cooperative systems with any number of species
Authors: Delgado Manuel , Suárez Antonio
Published: December 2016

In this paper we study the positive solutions of a cooperative system of any number of equations which consider the case of the slow diffusion and include the Lotka-Volterra model. We determine conditions of existence of global solution and blow-up in finite time in terms of the value of the spectral radius of a certain nonnegative matrix associated to the system. The results generalize the ones known for the particular case of two equations and we justify them by using the specific properties of nonnegative matrices which translate the cooperative character of the system.
Front motion in multi-dimensional viscous conservation laws with stiff source terms driven by mean curvature and variation of front thickness
Authors: Fan Haitao , Jin Shi
Published: December 2016

The bistable reaction-diffusion-convection equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript t Baseline u plus nabla dot f left-parenthesis u right-parenthesis equals minus StartFraction 1 Over epsilon EndFraction g left-parenthesis u right-parenthesis plus epsilon normal upper Delta u comma x element-of double-struck upper R Superscript n Baseline comma u element-of double-struck upper R left-parenthesis 1 right-parenthesis">

∂
t

u
+
∇
⋅
f

(
u
)

=

−

1
ϵ

g

(
u
)

+
ϵ
Δ
u

,

x
∈

R

n

,

u
∈

R

(
1
)

{\partial _t}u + \nabla \cdot f\left ( u \right ) = {- \frac {1}{\epsilon }g\left ( u \right ) + \epsilon \Delta u}, \qquad {x \in \mathbb {R}{^n}}, {u \in \mathbb {R}} \qquad \left ( 1 \right )

\]
is considered. Stationary traveling waves of the above equation are proved to exist when

f

(
u
)

f\left ( u \right )

is symmetric and

g

(
u
)

g\left ( u \right )

is antisymmetric about

u
=
0

u = 0

. Solutions of initial value problems tend to almost piecewise constant functions within

O

(
1
)

ϵ

O\left ( 1 \right )\epsilon

time. The almost constant pieces are separated by sharp interior layers, called fronts. The motion of these fronts is studied by asymptotic expansion. The equation for the motion of the front is obtained. In the case of

f
=
b

u
2

f = b{u^2}

and

g

(
u
)

=
a
u

(
1
−

u
2

)

g\left ( u \right ) = au\left ( 1 - {u^2} \right )

, where

b
∈

R

n

b \in {\mathbb {R}^{n}}

and

0
>
a
∈

R

0 > a \in \mathbb {R}

are constants, the front motion equation takes a more explicit form, showing that the front’s speed is
\[

ϵ

(

k
+

∇
μ

μ

⋅
T

)

\epsilon \left ( {k + \frac {{\nabla \mu }}{\mu } \cdot T} \right )

\]
, where

κ
\kappa

is the mean curvature of the front,

μ
\mu

is the width of the planar traveling of (1) in the normal direction n of the front, and T is a vector tangential to the front. Both

κ
\kappa

and

∇
μ

/

μ
⋅
T

\nabla \mu /\mu \cdot T

T are elliptic operators, contributing to the shrinkage of closed curves. An ellipse in

R

2

{\mathbb {R}^{2}}

is found to preserve its shape while shrinking.
Global solvability of a dissipative Frémond model for shape memory alloys. I. Mathematical formulation and uniqueness
Authors: Bonetti Elena
Published: December 2016

The mathematical formulation of a dissipative Frémond model for shape memory alloys is given in terms of an initial and boundary values problem. Uniqueness of sufficiently regular solutions is proved by use of a contracting estimates procedure in the case when quadratic dissipative contributions are neglected in the energy balance. The related existence result is only established while its proof will be detailed by the author in a subsequent paper.
Spatial behaviour in a Mindlin-type thermoelastic plate
Authors: D’Apice Ciro , Chiriţă Stan
Published: December 2016

Spatial behaviour is studied for the transient solutions in the bending of a Mindlin-type thermoelastic plate. Some appropriate time-weighted line-integral measures are associated with the transient solutions and the spatial estimates are established for these measures describing spatial behaviour results of the Saint-Venant and Phragmén-Lindelöf type. A complete description of the spatial behaviour is obtained by combining the spatial estimates with time-independent and time-dependent decay and growth rates. For a thermoelastic plate whose middle surface is like a semi-infinite strip, it is shown, by means of the maximum principle, that at infinity a sharper spatial decay holds and it is dominated by the thermal characteristics only. Uniqueness results are also established.
Global solvability of a dissipative Frémond model for shape memory alloys. II. Existence
Authors: Bonetti Elena
Published: December 2016

The paper investigates an initial and boundary values problem which is derived from a dissipative Frémond model for shape memory alloys. Existence of a global solution for the abstract version of the evolution problem is proved by use of a semi-implicit time discretization scheme combined with an a priori estimates-passage to the limit procedure.
“Concentrated capacity” model of ion-exchange funneling in a modified (thin film coated) heterogeneous electrodialysis membrane
Authors: Antontsev Stanislav , Meirmanov Anvarbek M. , Rubinstein Isaak , Zaltzman Boris
Published: December 2016

Inexpensive highly permselective heterogeneous ion exchange membranes are prohibitively polarizable by a direct electric current for use in electrodialysis. According to recent experiments, polarizability of these membranes may be considerably reduced by casting on their surface a thin layer of cross-linked polyelectrolyte, weakly charged with the same sign as the membrane’s charge. The present paper is concerned with this effect. In order to explain this feature, a simple limiting ion-exchange ’funnel’ model of a modified membrane is derived from the original two-layer model. In this model, asymptotically valid for a thin coating, solution of the ionic transport equations in it is replaced, via a suitable averaging procedure, by a single nonlinear boundary condition for the membrane/solution interface, which itself has the same order as the bulk equation. Rigorous analysis of the ’funnel’ model shows that the value of the limiting current through a modified membrane, which is the main quantitative characteristic of its polarizability, is equal to that through a homogeneous membrane for any positive value of the funneling parameter. The limiting problem for a vanishing funneling parameter is analyzed.
Approximate solutions to slightly viscous conservation laws
Authors: Kreiss Gunilla , Efraimsson Gunilla
Published: December 2016

We study an approximate solution of a slightly viscous conservation law in one dimension, constructed by two asymptotic expansions that are cut off after the third order terms. In the shock layer, an inner solution is valid and an outer solution is valid elsewhere.
Stress constrained 𝐺 closure and relaxation of structural design problems
Authors: Lipton Robert
Published: December 2016

A generic relaxation for stress constrained optimal design problems is presented. It is accomplished by introducing the stress constrained G closure. For a finite number of stress constraints, an explicit characterization of the stress constrained G closure is given. It is shown that the stress constrained G closure is characterized by all G limits together with their derivatives. A local representation of the set of all G limits and their derivatives is developed.
Global existence and asymptotic behavior to the solutions of 1-D Lyumkis energy transport model for semiconductors
Authors: Chen Li , Hsiao Ling , Li Yong
Published: December 2016

The global existence and asymptotic behavior of smooth solutions to the initial-boundary value problem for the 1-D Lyumkis energy transport model in semiconductor science is studied. When the boundary is insulated, the smooth solution of the problem converges to a stationary solution of the drift diffusion equations, exponentially fast as

t
→
∞

t \to \infty

.
Smooth domain method for crack problems
Authors: Khludnev Alexander M. , Sokołowski Jan
Published: December 2016

Equilibrium problems for elastic bodies in domains with cracks are considered. Inequality type boundary conditions are imposed at the crack describing a mutual nonpenetration between the crack faces. A new formulation for such problems is proposed in smooth geometrical domains for two-dimensional elasticity and Kirchhoff plates.
Asymptotics for the time-dependent thermistor problem
Authors: Cimatti Giovanni
Published: December 2016

Using the Liapunoff method, the global asymptotic stability of the stationary solution of the time-dependent thermistor problem is proved.
Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives
Authors: Van Buren Arnie L. , Boisvert Jeffrey E.
Published: December 2016

Alternative expressions for calculating the prolate spheroidal radial functions of the second kind

R

m
l

(
2
)

(
c
,
ξ
)

R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right )

and their first derivatives with respect to

ξ
\xi

are shown to provide accurate values over wide parameter ranges where the traditional expressions fail to do so. The first alternative expression is obtained from the expansion of the product of

R

m
l

(
2
)

(
c
,
ξ
)

R_{ml}^{\left ( 2 \right )}(c, \xi )

and the prolate spheroidal angular function of the first kind

S

m
l

(
1
)

(
c
,
η
)

S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right )

in a series of products of the corresponding spherical functions. A similar expression for the radial functions of the first kind was shown previously to provide accurate values for the prolate spheroidal radial functions of the first kind and their first derivatives over all parameter ranges. The second alternative expression for

R

m
l

(
2
)

(
c
,
ξ
)

R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right )

involves an integral of the product of

S

m
l

(
1
)

(
c
,
η
)

S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right )

and a spherical Neumann function kernel. It provides accurate values when

ξ
\xi

is near unity and

l
−
m

l - m

is not too large, even when

c
c

becomes large and traditional expressions fail. The improvement in accuracy using the alternative expressions is quantified and discussed.
Quantum Euler-Poisson systems: global existence and exponential decay
Authors: Jüngel Ansgar , Li Hailiang
Published: December 2016

A one-dimensional transient quantum Euler-Poisson system for the electron density, the current density, and the electrostatic potential in bounded intervals is considered. The equations include the Bohm potential accounting for quantum mechanical effects and are of dispersive type. They are used, for instance, for the modelling of quantum semiconductor devices.
𝐿¹-convergence rate of viscosity methods for scalar conservation laws with the interaction of elementary waves and the boundary
Authors: Liu Hongxia , Pan Tao
Published: December 2016

This paper is concerned with global error estimates for viscosity methods to initial-boundary problems for scalar conservation laws

u
t

+
f

(
u
)

x

=
0

{u_t} + f{\left ( u \right )_x} = 0

on

[

0
,
∞

)

×

[

0
,
∞

)

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

, with the initial data

u

(

x
,
0

)

=

u
0

(
x
)

u\left ( {x, 0} \right ) = {u_0}\left ( x \right )

and the boundary data

u

(

0
,
t

)

=

u
−

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

, where

u
−

{u_ - }

is a constant,

u
0

(
x
)

{u_0}\left ( x \right )

is a step function with a discontinuous point, and

f
∈

C
2

f \in {C^2}

satisfies

f
>
0
,
f

(
0
)

=

f
′

(
0
)

=
0

f > 0, f\left ( 0 \right ) = f’\left ( 0 \right ) = 0

. The structure of global weak entropy solution of the inviscid problem in the sense of Bardos-Leroux-Nedelec [11] is clarified. If the inviscid solution includes the interaction that the central rarefaction wave collides with the boundary

x
=
0

x = 0

and the boundary reflects a shock wave, then the error of the viscosity solution to the inviscid solution is bounded by

O

(

ϵ

1

/

2

+
ϵ

|

I
n
ϵ

|

+
ϵ
)

O\left ( {\epsilon ^{1/2}} + \epsilon \left | {In\epsilon } \right | + \epsilon \right )

in

L
1

{L^1}

-norm. If the inviscid solution includes no interaction of the central rarefaction wave and the boundary or the interaction that the rarefaction wave collides with the boundary and is absorbed completely or partially by the boundary, then the error bound is

O

(
ϵ

|

I
n
ϵ

|

+
ϵ
)

O\left ( \epsilon \left | {In\epsilon } \right | + \epsilon \right )

. In particular, if there is no central rarefaction wave included in the inviscid solution, the error bound is improved to

O

(
ϵ
)

O\left ( \epsilon \right )

. The proof is given by a matching method and the traveling wave solutions.
Computation of Fokker-Planck equation
Authors: Yau Stephen S.-T.
Published: December 2016

In plasma physics, the interaction of radio-frequency waves with a plasma is described by a Fokker-Planck equation with an added quasilinear term. In nonlinear filtering with conditional probability density of the state

x
t

{x_t}

given the observation

{

y

(
s
)

:
0
≤
s
≤
t

}

\left \{ {y\left ( s \right ):0 \le s \le t} \right \}

is also described by a Fokker-Planck equation with an added first order term. Method for solving Fokker-Planck equation by means of ordinary differential equations is discussed.
Spatially non-homogeneous coagulation equations with source terms
Authors: Shirvani M. , van Roessel H. J.
Published: December 2016

The physical process of coagulation or coalescence of particles is often modelled by Smoluchowski’s coagulation equation, an infinite system of nonlinear differential equations governing the binary interactions of particles of different sizes. One of the physical assumptions underlying the coagulation equation is spatial homogeneity, in particular, the assumption that coagulation of particles is governed only by particle type or size. The physical shape of the cloud of particles, as well as the relative position of particles of various type, are ignored.
Fluid velocity fields derived from vorticity singularities
Authors: Ranger K. B.
Published: December 2016

The present article addresses the existence by construction of random velocity fields produced by prescribed flow singularities which are consistent with the equations governing the motion of a two-dimensional viscous incompressible liquid. By showing first that the vorticity can be arbitrary and consistent with the flow equations it is then possible to determine the velocity field explicitly in terms of the vorticity and its derivatives.
Minimal entropy conditions for Burgers equation
Authors: De Lellis Camillo , Otto Felix , Westdickenberg Michael
Published: December 2016

We consider uniformly convex,

1
−
d

1 - d

scalar conservation laws. We show that a single uniformly convex entropy is sufficient to characterize a Kruzhkov solution. The proof uses the concept of viscosity solution for the related Hamilton-Jacobi equation.
An 𝐿²(Ω)-based algebraic approach to boundary stabilization for linear parabolic systems
Authors: Nambu Takao
Published: December 2016

We study the stabilization problem of linear parabolic boundary control systems. While the control system is described by a pair of standard linear differential operators

(

L
,
τ

)

\left ( {L, \tau } \right )

, the corresponding semigroup generator generally admits no Riesz basis of eigenvectors. Very little information on the fractional powers of this generator is needed. In this sense our approach has enough generality as a prototype to be used for other types of parabolic systems. We propose in this paper a unified algebraic approach to the stabilization of a variety of parabolic boundary control systems.
Love waves in stratified monoclinic media
Authors: Kuznetsov Sergey V.
Published: December 2016

A mathematical model for analysis of Love waves propagating in stratified anisotropic (monoclinic) media is presented; this model is based on a newly developed Modified Transfer Matrix (MTM) method. Closed form dispersed relations are obtained for media consisting of one or two orthotropic layers lying on orthotropic substrate. Conditions for existence of Love waves are analyzed. Horizontally polarized shear surface waves of non-Love type are constructed. A numerical algorithm is worked out for obtaining dispersion relations for Love waves propagating in stratified media containing a large number of layers.
Optimal elimination for sparse symmetric systems as a graph problem.
Authors: Spillers W. R. , Hickerson Norris
Published: December 2016

The optimal (requiring the minimum number of multiplications) ordering of a sparse symmetric system of linear algebraic equations to be used with Gaussian elimination is first developed as a graph problem which is then treated using the functional equation techniques of dynamic programming. A simple algorithm is proposed as an alternative to the more lengthy procedures of dynamic programming and this algorithm is shown to be effective for systems whose graphs are “grids".
Classical analytic representations
Authors: Curme G. L. , Hille Einar
Published: December 2016

Mittag-Leffler-like fractional decompositions are constructed for the gamma, Jacobian elliptic and for the quotient of Bessel functions. These results illustrate a technique herein developed which permits a complete ML decomposition of a large class of meromorphic functions without resorting to any other previously derived information or knowledge of the particular function. Typically, the point of departure of this note stands in contrast to a statement in Knopp [1, p. 44], during his development of the ML decomposition of

π
cot
⁡
π
z

\pi \cot \pi z

, which reads as follows: “The still undetermined entire function,

G

(
z
)

G\left ( z \right )

, cannot be ascertained solely from the nature and position of the poles."
Theory of irreducible operators of linear systems
Authors: Chiu H. H.
Published: December 2016

The complementary differential operator is devised to decouple two coupled partial differential equations with variable coefficients. This complementary operator transforms two coupled partial differential equations into a pair of uncoupled equations either of (i) fourth order, (ii) third order, or (iii) second order. The conditions for the existence of linearly dependent solutions for two coupled equations with constant coefficient is also discussed. The method is applied to the propagation of two sound waves in liquid helium at finite temperatures.
Further properties of the nonseparable solutions of the Helmholtz wave equation
Authors: Moseley Donald S.
Published: December 2016

Nonseparable solutions

W

(
n
)

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

of

(

∇
2

+

k
2

)

W

(
n
)

=
0

\left ( {{\nabla ^2} + {k^2}} \right ){W^{\left ( n \right )}} = 0

are linearly independent, but inter-related through a generative differential operator. The nonseparable of order

n
=
0

n = 0

is the familiar separable solution. In two cartesian coordinates, a sum of zero and second order solutions can describe transverse motion of a membrane of unique boundary contour. In three coordinates the same sum can describe acoustic pressure in a uniquely shaped cavity with pressure-release walls.
A Riccati solution for Burgers’ equation
Authors: Rodin Ervin Y.
Published: December 2016

In the last few years a one-dimensional, time-dependent and nonlinear approximation to the Navier-Stokes equations, (1) below, has found applications in fields as diverse as number theory, gas dynamics, heat conduction, elasticity, etc. Probably the most important reason for this is that the complete and explicit solution of this equation became known in 1950. That solution, however, applies only to the homogeneous part of Eq. (1). In an attempt to tackle the nonhomogeneous case, we relate Eq. (1) to a Riccati equation, through a similarity transformation. Via this route, it is shown that solutions to the nonhomogeneous equation can be obtained.
Mean field variation in random media
Authors: Beran Mark J. , McCoy John J.
Published: December 2016

We consider here the basic equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential Over partial-differential x Subscript i Baseline EndFraction left-bracket epsilon left-parenthesis upper X right-parenthesis StartFraction partial-differential phi Over partial-differential x Subscript i Baseline EndFraction left-parenthesis upper X right-parenthesis right-bracket equals rho left-parenthesis upper X right-parenthesis">

∂

∂

x
i

[

ϵ

(

X

)

∂
ϕ

∂

x
i

(

X

)

]

=
ρ

(

X

)

\frac {\partial }{{\partial {x_i}}}\left [ { \epsilon \left ( \textrm {X} \right )\frac {{\partial \phi }}{{\partial {x_i}}}\left ( \textrm {X} \right )} \right ] = \rho \left ( \textrm {X} \right )

\]
, where

∈

(

X

)

\in \left ( \textrm {X} \right )

is a random function of position and

ρ

(

X

)

\rho \left ( \textrm {X} \right )

is a prescribed source term. A formal equation is derived that governs

{

ϕ

(

X

)

}

\left \{ {\phi \left ( \textrm {X} \right )} \right \}

, where the braces indicate an ensemble average. The equation, which depends on the boundary conditions of the stochastic problem, is presented in terms of an infinite sequence of correlation functions associated with

∈

(

X

)

\in \left ( \textrm {X} \right )

. The equation is investigated first for the case of an infinite dielectric where isotropy may be assumed. An impulse response function is obtained and an explicit form of this response function is presented for the limit of small perturbations. Further, it is shown that the equation governing

{

ϕ

(
x
)

}

\left \{ {\phi \left ( x \right )} \right \}

is greatly simplified for the case in which all characteristic lengths associated with

{

ϕ

(
x
)

}

\left \{ {\phi \left ( x \right )} \right \}

are large compared to all correlation lengths

l
i

{l_i}

associated with the

∈

(

X

)

\in \left ( \textrm {X} \right )

field. The question of boundary conditions is next considered and as an example a spherical boundary (radius

R
R

) is studied. It is demonstrated, in this case, that if

R
≫

l
i

R \gg {l_i}

the effects of the boundary conditions on the equation governing

{

ϕ

(
x
)

}

\left \{ {\phi \left ( x \right )} \right \}

are negligible except at points within a thin layer near the boundary. The relationship between the ensemble average and the local volume average is also discussed.
Potential and Rayleigh-scattering theory for a spherical cap
Authors: Miles John W.
Published: December 2016

Harmonic functions are constructed for spherical-harmonic prescriptions of either a potential or its normal derivative on a spherical cap. The dipole-moment tensor and the Rayleigh-scattering properties of a spherical bowl, including the limiting case of a Helmholtz resonator, are determined. The results are uniformly valid with respect to the polar angle of the cap and resolve certain discrepancies in the existing literature.
A note on angular representations for scattered waves
Authors: Censor Dan
Published: December 2016

Differential-operator representations for the scattered field were given by Twersky [1], [3], [4], and Burke, Censor and Twersky [2]. Essentially, from the data provided on a circle (in two dimensions) or a sphere (in three dimensions), the field may be reconstructed for arbitrary distances.
On the structure of dissipative waves in two and three dimensions
Authors: Halabisky Lorne , Sirovich Lawrence
Published: December 2016

The problem of two- and three-dimensional small disturbances in a dissipative gas is considered. Explicit forms in terms of known functions are obtained for the time-asymptotic theory in each case. Although the far field solutions may be nonlinear in one dimension, they are always self-consistently linear for two and three dimensions.
A table of solutions of the one-dimensional Burgers equation
Authors: Benton Edward R. , Platzman George W.
Published: December 2016

The literature relating to the one-dimensional Burgers equation is surveyed. About thirty-five distinct solutions of this equation are classified in tabular form. The physically interesting cases are illustrated by means of isochronal graphs.
Nonlinear effect of initial stress on crack propagation between similar and dissimilar orthotropic media
Authors: Biot M. A.
Published: December 2016

The theory of crack propagation in orthotopic media is developed by applying the theory of incremental deformations in the vicinity of a state of initial stress. This is carried out in the context of a new approach to analytical methods and a physical analysis which takes into account plastic deformation under prestress. The state of initial stress is triaxial along the directions of elastic symmetry, and the crack is parallel to these directions. An additional shear component for the initial stress is also taken into account and general conditions are derived for crack propagation, including the case of fluid injection into the crack. The analysis is first carried out for an homogeneous medium. The nonlinear influence of the initial stress appears in two ways: first, through a fundamental purely elastic effect related to the occurrence of surface instability, and second, through the influence of the initial stress on plastic behavior. The particular cases of an isotropic elastic medium with finite initial strain and an orthotropic incompressible medium are discussed. The analysis is extended to a crack between dissimilar orthotropic media with initial stress. The method of analysis leads to a number of simplifications and brings out new properties of the solutions for this type of problem. For incompressible media without initial stress, the typical oscillatory behavior disappears. Uniqueness of the solutions is also derived.
Solution of quadratic matrix equations by least-squares method
Authors: Lee Charles C. , Niu H. P.
Published: December 2016

A least-square method is described for obtaining the solutions

A
A

and

B
B

for the matrix equations

Y
2

−
Y
D
+
S
=
0

{Y^2} - YD + S = 0

and

Y
2

−
D
Y
+
S
=
0

{Y^2} - DY + S = 0

respectively. No limitation is set on

A
A

,

B
B

,

D
D

, and

S
S

except that they be square matrices. An illustrated example, including computing procedures, is discussed. The mathematical solutions presented should prove useful for solving the eigenvalue problem

(

λ
2

I
+
λ
D
+
S

)

X
=
0

\left ( {{\lambda ^2}I + \lambda D + S} \right )X = 0

, especially when the dimension of the matrices is large.
On the computation of the Cauchy index
Authors: Anderson Brian D. O.
Published: December 2016

The Cauchy index of a real rational function can be computed by evaluating the signature of a certain Hankel matrix. Alternative procedures for its computation are presented here, one of which offers greater computational simplicity.
On the numerical solution of singular integral equations
Authors: Erdogan F. , Gupta G. D.
Published: December 2016

In this paper a pair of Gauss-Chebyshev integration formulas for singular integrals are developed. Using these formulas a simple numerical method for solving a system of singular integral equations is described. To demonstrate the effectiveness of the method, a numerical example is given. In order to have a basis of comparison, the example problem is solved also by using an alternate method.
Dual extremum principles relating to cooling fins
Authors: Bhargava S. , Duffin R. J.
Published: December 2016

Under consideration is a differential equation

(

p

u
′

)

′

=
q
u

{\left ( {pu’} \right )’} = qu

of the Sturm—Liouville type where the function

q

(
x
)

>
0

q\left ( x \right ) > 0

is given. The problem is to find a function

p

(
x
)

>
0

p\left ( x \right ) > 0

in

0
≤
x
>
b

0 \le x > b

, a constant

b
b

and a solution

u

(
x
)

u\left ( x \right )

of the corresponding differential equation such that the energy functional

∫
0
b

[

p

(

u
′

)

2

+
q

u
2

]

d
x

\int _0^b {\left [ {p{{\left ( {u’} \right )}^2} + q{u^2}} \right ]} dx

is maximized when

p

(
x
)

p\left ( x \right )

is subject to the constraint

∫
0
b

p
ρ

d
x
≤

K
ρ

\int _0^b {{p^\rho }dx \le {K^\rho }}

and

u
u

is subject to the boundary conditions

u
=
1

u = 1

at

x
=
0

x = 0

and

p

(

d
u

/

d
x

)

=
0

p\left ( {du/dx} \right ) = 0

at

x
=
b

x = b

. Here

K
>
0

K > 0

and

ρ
≥
1

\rho \ge 1

are constants. A key relation

|

d
u

/

d
x

|

=
λ

p

(

ρ
−
1

)

/

2

\left | {du/dx} \right | = \lambda {p^{\left ( {\rho - 1} \right )/2}}

, where

λ
\lambda

is a positive constant, is found. This criterion leads to explicit solution of the problem. A further consequence of this criterion together with a pair of dual extremum principles is a “duality inequality” giving sharp upper and lower estimates of the maximum value of the energy functional.
A steering problem
Authors: Synge J. L.
Published: December 2016

If two circular gear-wheels of different sizes engage, this may be regarded as a mechanism which generates a linear relationship between the angles turned through, the ratio of the angles being constant. In this paper it is shown that, if the circular gear wheels are replaced by suitably shaped oval wheels (or cams) which engage without slipping, it is possible to generate an arbitrary functional relationship between the angles turned through. It is further shown how this mechanism might be used in the steering of a four-wheeled vehicle with theoretically perfect satisfaction of the condition that, at any instant, the vehicle as a whole has a unique centre of rotation, so that no side-slip of the wheels occurs. The angle through which the plane of the inner front wheel is turned might be as great as a right angle, this limit being much greater than that attainable with the usual Ackermann linkage.
Divergence instability of multiple-parameter circulatory systems
Authors: Huseyin K. , Leipholz H. H. E.
Published: December 2016

The free vibrations and stability of a discrete linear non-dissipative system subjected to the combined effect of several independent circulatory and conservative loads are studied. Attention is restricted to systems which exhibit only divergence-type instability and are incapable of flutter. Two theorems concerning the basic properties of the characteristic surfaces (loading-frequency relationship) and the stability boundary are established. One of the several other theorems insures that the stability boundary of the corresponding conservative system will always provide a lower bound for the actual stability boundary of the non-conservative system. This particular result is valid for all types of divergence-instability problems, including those which can exhibit flutter after divergence. The theorems established are quite general in the sense that they are valid for all systems in the class considered, without further analytical conditions. Practical implications of the general results are briefly discussed.
Stability criteria for generalized Mathieu equations
Authors: Infeld E.
Published: December 2016

A class of differential equations that generalizes Mathieu’s is treated. Stability criteria for solutions are obtained. The method can be applied to a wider class of Hill equations.
Laminar boundary layer at a discontinuity in wall curvature
Authors: Messiter A. F. , Hu J. J.
Published: December 2016

The discontinuity in pressure gradient predicted for two-dimensional inviscid subsonic or supersonic flow at a jump discontinuity in wall curvature is smoothed by means of local solutions which take into account the interaction of a laminar boundary layer with the external flow.
Lagrangian formulation of bubble dynamics
Authors: Hsieh D. Y.
Published: December 2016

The dynamical problem of two ideal fluids separated by an interface is formulated in terms of general coordinates in Lagrangian variables. The same problem is shown to be equivalent to a Hamiltonian variational principle which takes into account explicitly the surface energy of the interface between the two fluids. The formulation is applied to the motion of slightly nonspherical bubbles. It is shown that, although we start from an entirely different set of differential equations, the results of the linear stability analysis in Eulerian formulation are recovered.
Convergence in iterative design
Authors: Spillers W. R. , Al-Banna S.
Published: December 2016

Earlier results for the monotone convergence of iterative design for a simple model of the truss problem are generalized for the case in which the objective function is a homogeneous, convex function.
Coupled pairs of dual integral equations with trigonometric kernels
Authors: Khadem R. , Keer L. M.
Published: December 2016

The solution is given to a system of two pairs of dual integral equations with constant coefficients involving trigonometric kernels. The method is analogous to that applied to Bessel function kernels and involves reduction to a single Wiener—Hopf equation for which a solution is available. The example of an indenter moving with friction present is worked out by this method and also by means of equivalent reduction of the system of equations to a singular integral equation.
On quadrature formulas for singular integral equations of the first and the second kind
Authors: Krenk Steen
Published: December 2016

In this paper it is shown that by proper choice of the collocation points singular integral equations of the first and the second kind can be integrated by use of the usual Gauss-Jacobi quadrature formula. Detailed formulas are given for various values of the index.
Modified complementary variational principles for a boundary-value problem in twisting of ring sectors
Authors: Arthurs A. M. , Duggan R. C.
Published: December 2016

New dual extremum principles and error bounds for the problem of stress in close-coiled helical springs are presented. An accurate variational solution is obtained in an illustrative calculation.
Comparison theorems for gravity waves in basins of variable depth
Authors: Yih Chia Shun
Published: December 2016

Surface waves of a homogeneous liquid and internal waves of a stratified liquid in basins of variable depth are considered. Inequalities involving the frequencies of oscillation are obtained when a container with one size or geometry is compared with another with a different size or geometry, when waves with one wavelength are compared with waves with another wavelength, or when one stratification is compared with another. Since exact solutions for gravity waves in basins of variable depth are so rare, one hopes the comparison theorems presented herein will be useful.
Uniform asymptotic approximations to the solutions of the Dunn-Lin equations
Authors: Ng B. S.
Published: December 2016

In this paper, we develop a uniform asymptotic theory for the Dunn-Lin equations which govern the stability of compressible boundary layers at moderate Mach numbers. “First approximations” to the solutions are derived when the Prandtl number is equal to unity, in which case the structure of the approximations is especially simple. The rapidly varying parts of the approximations can be expressed in terms of certain generalized Airy functions and the slowly varying parts can be expressed in terms of quantities all but one of which are well-known from the older heuristic theories. The approximations obtained in this way are uniformly valid in a full neighborhood of the turning point. Because of the simplicity of the present theory, it is expected that the techniques developed in this paper can also be applied to other more general stability equations for compressible boundary layers, such as the Lees-Lin equations.
Functional differential equations of Peano-Baker type
Authors: Allen R. C. , Kyner W. T.
Published: December 2016

The application of the moment method to a class of Fredholm integral equations leads to the study of both an infinite system of ordinary differential equations and a functional differential equation. Sufficient conditions are given that the solution to a truncated system of differential equations approximates the solutions to the original problem. A numerical algorithm for constructing approximate solutions is discussed.
Mixed finite-element approximations of linear boundary-value problems
Authors: Reddy J. N. , Oden J. T.
Published: December 2016

A theory of mixed finite-element/Galerkin approximations of a class of linear boundary-value problems of the type

T
∗
T
u
+
k
u
+
f
=
0

T*Tu + ku + f = 0

is presented, in which appropriate notions of consistency, stability, and convergence are derived. Some error estimates are given and the results of a number of numerical experiments are discussed.
Wave phenomena in inhomogeneous media
Authors: Filippi Paul
Published: December 2016

The difference between the fundamental solutions of differential equations governing the same physical phenomenon in two different physical media is investigated. A stationary expression of this difference is established, leading to a Ritz-Galerkin procedure. The Ritz-Galerkin system is solved analytically, providing a representing series for the difference of the two fundamental solutions as a functional of either one or the other of these functions. The convergence of the series is a consequence of its construction itself. Physical examples are considered which show that the convergence rate can be partially controlled.
A structural optimization solution to a branch-and-bound problem
Authors: Lev O. E.
Published: December 2016

A simple algorithm, developed for a least-weight structural optimization problem, is used to force the selection of the same

n
n

components of the vectors

X
X

and

Y
Y

, containing

b
b

elements

(
b
>
n
)

(b > n)

so that the objective function

L
~

max

x
i
,
y
i

{

|
X
|

,

|
Y
|

}

\tilde L {\max _{xi,yi}}\left \{ {\left | X \right |,\left | Y \right |} \right \}

is minimized subject to

n
n

equality constraints on each vector,

A
X
=

b
1

AX = {b_1}

,

A
Y
=

b
2

AY = {b_2}

. The method has an obvious advantage over integer programming or branch-and-bound techniques that would, in this case, seek the best selection of

n
n

out of

b
b

elements which satisfy the constraints.
Singularity at the apex of pyramidal notches with three equal angles
Authors: Keer L. M. , Parihar K. S.
Published: December 2016

A Green’s function approach is employed to study the potential theory problem of determining the strength of singularity at the apex of a pyramid with three equal angles. The problem is reduced to finding the eigenvalue of a singular integral equation. Numerical results are obtained and compared with available literature.
Variational thermodynamics of viscous compressible heat-conducting fluids
Authors: Biot M. A.
Published: December 2016

A new variational principle of virtual dissipation generalizing d’Alembert’s principle to nonlinear irreversible thermodynamics is applied to compressible heat-conducting fluids with Newtonian and non-Newtonian viscosity. The principle is applied in the context of Eulerian formalism where the flow is described with reference to a fixed coordinate system. New concepts of entropy displacement and mass displacement are used as well as a new definition of the chemical potential which avoids the usual ambiguities of the classical thermodynamic approach. The variational principle is used to derive a novel form of field differential equations for the coupled fluid dynamics and convective heat transfer.
De Giorgi’s counterexample in elasticity
Authors: Guidugli Paolo Podio
Published: December 2016

The framework of linear elastostatics is used to interpret De Giorgi’s example concerning the non-regularity of extremals of a certain Dirichlet integral with measurable and bounded coefficients.
Periodic solutions of the sunflower equation: 𝑥̈+(𝑎/𝑟)𝑥+(𝑏/𝑟)sin𝑥(𝑡-𝑟)=0
Authors: Somolinos Alfredo S.
Published: December 2016

In 1967 Israelsson and Johnsson proposed a model for the geotropic circumnutations of Helianthus annus. The existence of a geotropic reaction time is reflected in the delay

r
r

of the equation. Numerical computations suggested the existence of periodic solutions. In this paper, we prove the existence of periodic solutions for a range of the values of the parameters

a
,
b
,
r

a,b,r

. We use Razumikhin-type functions to prove the boundedness of all solutions. We then prove the existence of periodic solutions of small amplitude using the Hopf bifurcation theorem. Finally, we use a fixed-point theorem on a cone to prove the existence of periodic solutions of large amplitude.
Two-timing procedure for higher-order modulations of near-linear dispersive wave trains with an application to plasma waves
Authors: Gatignol P.
Published: December 2016

An asymptotic procedure is proposed to describe the slow modulations of dispersive wave trains when the dispersive effects and the nonlinear distorsion are of the same order of magnitude. For a model equation, the system of modulation equations is derived up to the second order. At this order of approximation, it is seen that the dispersion relation includes partial derivatives not only of the amplitude but also of the wave vector components. Under some assumptions, a partial differential equation is obtained for the complex amplitude. This equation reveals common features with the modified Korteweg-de Vries equation and with a nonlinear Schrödinger equation. At the first order, it reduces to the cubic Schrödinger equation which had been directly obtained by several authors. Finally, the theory is applied to a plasma wave example.
The bifurcation of periodic solutions in the Hodgkin-Huxley equations
Authors: Troy William C.
Published: December 2016

We consider the current clamped version of the Hodgkin-Huxley nerve conduction equations. Under appropriate assumptions on the functions and parameters we show that there are two critical values of

I
I

, the current parameter, at which a Hopf bifurcation of periodic orbits occurs.
Sobolev-type lower bounds on ∥∇𝜓∥² for arbitrary regions in two-dimensional Euclidean space
Authors: Rosen Gerald
Published: December 2016

This note reports the derivation of lower bounds of the Sobolev type on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar nabla psi double-vertical-bar squared identical-to integral Subscript upper R Baseline left-parenthesis partial-differential psi slash partial-differential x 1 right-parenthesis squared plus left-parenthesis partial-differential psi slash partial-differential x 2 right-parenthesis squared right-parenthesis d x 1 d x 2">

‖

∇
ψ

‖

2

≡
∫

R

(
∂
ψ

/

∂

x
1

)
2

+

(
∂
ψ

/

∂

x
2

)
2

)
d

x
1

d

x
2

{\left \| {\nabla \psi } \right \|^2} \equiv \smallint {}_R{(\partial \psi /\partial {x_1})^2} + {(\partial \psi /\partial {x_2})^2})d{x_1}d{x_2}

for generic real scalar

ψ
=
ψ
(

x
1

,

x
2

)

\psi = \psi ({x_1},{x_2})

of function class

C
0

{C^0}

piecewise

C
2

{C^2}

which vanish over the boundary of the (bounded or unbounded) region

R
R

in Euclidean 2-space.
The instability of superposed flow
Authors: Liang S. S. , Seidel B. S.
Published: December 2016

The interfacial instability of a stratified two-phase flow is studied. Two independent cases are investigated: first, the instability of an idealized atmosphere (density decreasing exponentially with height) over a liquid; second, the instability of an ideal gas over a liquid. The unbounded and incompressible cases may be obtained by specializing the above results.
Dual extremum principles for a nonlinear diffusion problem
Authors: Anderson N. , Arthurs A. M.
Published: December 2016

Maximum and minimum principles for a nonlinear boundary value problem in diffusion with concentration-dependent coefficient

D

(
c
)

D\left ( c \right )

are derived in a unified manner from the theory of dual extremum principles. The results are illustrated by a calculation in the case

D

(
c
)

=
exp
⁡
c

D\left ( c \right ) = \exp c

.
Buckling of cylindrical shells with small curvature
Authors: Mallet-Paret John
Published: December 2016

We consider the bifurcation buckling of a rectangular plate with an imperfection of magnitude

α
\alpha

under an applied lateral force of magnitude

λ
\lambda

. The analysis allows the parameters

(

λ
,
α

)

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

to vary independently in a neighborhood of some

(

λ
0

,
0

)

\left ( {{\lambda _0}, 0} \right )

, and describes all buckled states of small magnitude. If the plate is represented by the domain

(

0
,

2

)

×

(

0
,
1

)

\left ( {0, \sqrt 2 } \right ) \times \left ( {0, 1} \right )

in

R
2

{R^2}

, then the lateral force is applied to the edges

x
=
0

x = 0

,

2

\sqrt 2

, and the imperfection is a small vertical displacement of the form

z
=

(

α

/

2

)

y
2

(

σ
x
+
τ

)

z = \left ( {\alpha /2} \right ){y^2}\left ( {\sigma x + \tau } \right )

, where

σ
\sigma

and

τ
\tau

are fixed. Roughly, then, the plate has a small curvature in the

y
y

-direction, of magnitude

α

(

σ
x
+
τ

)

\alpha \left ( {\sigma x + \tau } \right )

.
Dynamic buckling of a damped imperfect column on a nonlinear foundation
Authors: Lockhart Deborah F.
Published: December 2016

The dynamic buckling of a finite damped imperfect column which rests on a nonlinear foundation and which is subjected to axial step-loading is studied using a perturbation method. The imperfection is considered to be small. A simple asymptotic expression for the dynamic buckling load is obtained in terms of the Fourier component of the imperfection in the shape of the classical buckling mode. This leads to simple relations, independent of the imperfection, between the dynamic and static buckling loads for the damped and undamped column.