bluuebunny/crossref_metadata_2025 huggingface dataset:
Global solution to a phase field model with irreversible and constrained phase evolution
Authors: Luterotti Fabio , Schimperna Giulio , Stefanelli Ulisse
Published: December 2016

This note deals with a nonlinear system of PDEs describing some irreversible phase change phenomena that account for a bounded limit velocity of the phase transition process. An existence result is established by using time discretization, compactness arguments, and techniques of subdifferential operators.
Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor
Authors: Ly Hung V. , Tran Hien T.
Published: December 2016

Proper orthogonal decomposition (which is also known as the Karhunen-Loève decomposition) is a reduction method that is used to obtain low-dimensional dynamic models of distributed parameter systems. Roughly speaking, proper orthogonal decomposition (POD) is an optimal technique of finding a basis that spans an ensemble of data, collected from an experiment or a numerical simulation of a dynamical system, in the sense that when these basis functions are used in a Galerkin procedure, they will yield a finite-dimensional system with the smallest possible degrees of freedom. Thus, the technique is well suited to treat optimal control and parameter estimation of distributed parameter systems. In this paper, the method is applied to analyze the complex flow phenomenon in a horizontal chemical vapor deposition (CVD) reactor. In particular, we show that POD can be used to efficiently approximate solutions to the compressible viscous flows coupled with the energy and the species equations. In addition, we also examine the feasibility and efficiency of the POD method in the optimal control of the source vapors to obtain the most uniform deposition profile at the maximum growth rate. Finally, issues concerning the implementation of the method and numerical calculations are discussed.
The initial-value problem for the Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers with generalized polytrope laws
Authors: Brown Kevin G. , Choudhury S. Roy
Published: December 2016

The general initial-value problem for the linear Kelvin-Helmholtz instability of arbitrarily compressible magnetized anisotropic velocity shear layers is considered. The time evolution of the physical quantities characterizing the layer is treated using Laplace transform techniques. Singularity analysis of the resulting equations using Fuchs-Frobenius theory yields the large-time asymptotic solutions. Since all the singular points turned out to be real, the instability is found to remain, within the linear theory, of the translationally convective shear type. No onset of rotational or vortex motion, i.e., formation of “coherent structures” occurs because there are no imaginary singularities.
High frequency waves near cusp caustics
Authors: Kalligiannaki E. , Katsaounis Th. , Makrakis G. N.
Published: December 2016

It is well known that the usual harmonic ansatz of geometrical optics fails near caustics. However, uniform expansions exist which are valid near and on the caustics, and reduce asymptotically to the usual geometric field far enough from them. In this paper, we apply the Kravtsov-Ludwig technique for computing high-frequency fields near cusp caustics. We compare these fields, with those predicted by the geometrical optics, for a couple of model problems: first, the cusp generated by the evolution of a parabolic initial front in a homogeneous medium, a problem which arises in the high-frequency treatment of cylindrical aberrations, and second, the cusp formed by refraction of the rays emitted from a point source in a stratified medium with a weak interface. It turns out that inside and near the cusp, the geometrical optics solution is significantly different than the Kravtsov-Ludwig solution, but far enough from the caustic that the two solutions are, in fact, in very good agreement.
A primal-dual active set algorithm for bilaterally control constrained optimal control problems
Authors: Hintermüller Michael
Published: December 2016

A generalized Moreau-Yosida based primal-dual active set algorithm for the solution of a representative class of bilaterally control constrained optimal control problems with boundary control is developed. The use of the generalized Moreau-Yosida approximation allows an efficient identification of the active and inactive sets at each iteration level. The method requires no step-size strategy and exhibits a finite termination property for the discretized problem class. In infinite as well as in finite dimensions a convergence analysis based on an augmented Lagrangian merit function is given. In a series of numerical tests the efficiency of the new algorithm is emphasized.
𝐿-derivative of an approximate solution to Φ’=𝐀(𝐭)Φ: series and product formulae for left corrections
Authors: Najfeld Igor , Lakin William
Published: December 2016

Given an initial approximation

Φ
0

{\Phi _0}

to the fundamental matrix of solutions for

Φ
′

=

A

(
t
)

Φ

\Phi ’ = \textrm {A}\left ( t \right )\Phi

, it is shown that a left correction,

Γ

Φ
0

\Gamma {\Phi _0}

, is locally more accurate than a right correction,

Φ
0

Γ

{\Phi _0}\Gamma

. For each relative error function considered, there is a left correction

Γ
\Gamma

and the associated differential equation. The common feature is the same integrable part whose forcing function is the difference between L-derivatives of the exact and the initial solution.
Numerical approach to the waiting time for the one-dimensional porous medium equation
Authors: Nakaki Tatsuyuki , Tomoeda Kenji
Published: December 2016

We consider the nonlinear degenerate diffusion equation. The most striking manifestation of the nonlinearity and degeneracy is an appearance of interfaces. Under some condition imposed on the initial function, the interfaces do not move on some time interval

[

0
,

t
∗

]

\left [ {0,{t^*}} \right ]

. In this paper, from numerical points of view, we try to determine the value of

t
∗

{t^*}

, which is called the waiting time.
Linear wellposedness of the evolution equation of a mixing strip of two fluids with initially sharp interface in a porous medium
Authors: Duchon Jean , Robert Raoul
Published: December 2016

An illposed sharp interface equation, describing the evolution of two well separated fluids in a porous medium, is replaced by a linearly wellposed Cauchy problem for the evolution of a mixing strip described in terms of the level curves of the volume proportion of one fluid.
Kernel sections for damped non-autonomous wave equations with linear memory and critical exponent
Authors: Zhou Shengfan
Published: December 2016

We prove the existence of kernel sections for the process generated by a non-autonomous wave equation with linear memory when there is nonlinear damping and the nonlinearity has a critically growing exponent; we also obtain a more precise estimate of upper bound of the Hausdorff dimension of the kernel sections. And we point out that in the case of autonomous systems with linear damping, the obtained upper bound of the Hausdorff dimension decreases as the damping grows for suitable large damping.
Exponential stability and global existence in thermoelasticity with radial symmetry
Authors: Rieger Marc Oliver
Published: December 2016

In this paper we consider equations of linear and nonlinear thermoelasticity with various boundary conditions. We assume radial symmetry of the initial data to prove exponential decay and to show the global existence of solutions of the nonlinear problem for small initial data.
Spatial discretization of the dusty gas equations
Authors: Gottlieb David , Temam Roger
Published: December 2016

In this article we consider a semi-discretization of the dusty gas equations corresponding to a spatial discretization by finite differences or pseudo-spectral methods and discuss the stability and well-posedness of the corresponding system of ordinary equations.
On a modified shock front problem for the compressible Navier-Stokes equations
Authors: Blokhin A. M. , Trakhinin Yu. L.
Published: December 2016

We discuss the possibility of considering the shock wave in a compressible viscous heat conducting gas as a strong discontinuity on which surface the generalized Rankine-Hugoniot conditions hold. The corresponding linearized stability problem for a planar shock lacks boundary conditions; i.e., the shock wave in a viscous gas viewed as a (fictitious) strong discontinuity is like undercompressive shock waves in ideal fluids and, therefore, it is unstable against small perturbations. We propose such additional jump conditions so that the stability problem becomes well-posed and its trivial solution is asymptotically stable (by Lyapunov). The choice of additional boundary conditions is motivated by a priori information about steady-state solutions of the Navier-Stokes equations which can be calculated, for example, by the stabilization method. The established asymptotic stability of the trivial solution to the modified linearized shock front problem can allow us to justify, at least on the linearized level, the stabilization method that is often used, for example, for steady-state calculations for viscous blunt body flows.
A transmission problem for thermoelastic plates
Authors: Muñoz Rivera Jaime E. , Portillo Oquendo Higidio
Published: December 2016

In this paper we study a transmission problem for thermoelastic plates. We prove that the problem is well-posed in the sense that there exists only one solution which is as regular as the initial data. Moreover, we prove that the local thermal effect is strong enough to produce uniform rate of decay of the solution. More precisely, there exist positive constants

C
C

and

γ
\gamma

such that the total energy

E

(
t
)

E\left ( t \right )

satisfies
\[

E

(
t
)

≤
C
E

(
0
)

e

−
γ
t

E\left ( t \right ) \le CE\left ( 0 \right ){e^{ - \gamma t}}

\]
.
Mapping onto the plane of a class of concave billiards
Authors: Liboff Richard L.
Published: December 2016

Three results are reported in this work. The first addresses the four ’elemental-polygon’ billiards with sides replaced by circular non-overlapping concave elements. Any orbit of the resulting concave billiard is mapped onto a trajectory in the plane that is shown to diverge from the trajectory of the related polygon billiard. This mapping permits application of Lyapunov exponents relevant to an unbounded system to be applied to the bounded concave elemental polygon-billiards. It is shown that Lyapunov exponents for concave elemental polygon-billiards go to zero as the curvature of the concave billiard segments go to zero. The second topic considers the quantum analogue of this problem. A conjecture is introduced which implies that a characteristic quantum number exists below which the adiabatic theorem applies and above which quantum chaos ensues. This parameter grows large as side curvature of the given billiard grows small. Lastly, a correspondence property between classical and quantum chaos for the concave elemental-polygon billiards is described.
Quasistatic viscoelastic contact with friction and wear diffusion
Authors: Shillor M. , Sofonea M. , Telega J. J.
Published: December 2016

We consider a quasistatic problem of frictional contact between a deformable body and a moving foundation. The material is assumed to have nonlinear viscoelastic behavior. The contact is modeled with normal compliance and the associated law of dry friction. The wear takes place on a part of the contact surface and its rate is described by the Archard differential condition. The main novelty in the model is the diffusion of the wear particles over the potential contact surface. Such phenomena arise in orthopaedic biomechanics where the wear debris diffuse and influence the properties of joint prosthesis and implants. We derive a weak formulation of the model which is given by a coupled system with an evolutionary variational inequality and a nonlinear evolutionary variational equation. We prove that, under a smallness assumption on some of the data, there exists a unique weak solution for the model.
Close-in orbits in the restricted problem of three bodies
Authors: Barrar Richard B.
Published: December 2016

Utilizing a transformation due to Birkhoff [3], we establish for the restricted problem of three bodies the existence of conditionally periodic orbits that move in a small neighborhood of one of the primaries. These orbits result from perturbation of elliptic orbits and are valid for all mass ratios of the two primaries.
Necessary and sufficient conditions for bounded global stability of certain nonlinear systems
Authors: Brayton R. K.
Published: December 2016

Nonlinear systems having the form
\[

x
˙

=
−
A
x
+
B
y

\dot x = - Ax + By

\]

\[

y
˙

=
C
x
−
f

(
y
)

\dot y = Cx - f\left ( y \right )

\]
, where

∂
f

/

∂
y

\partial f/\partial y

is a symmetric matrix, are considered. Such systems include the class of nonlinear reciprocal networks where the nonlinearity is voltage (or current) controlled. Also included, provided

c
T

b
≠
0

{c^T}b \ne 0

, are the equations of nonlinear feedback systems,
\[

x
˙

=
A
x
+
b
f

(

c
T

x

)

\dot x = Ax + bf\left ( {{c^T}x} \right )

\]
, considered by Aizerman [1], A type of stability called bounded global stability is considered which requires that all bounded solutions decay as

t
→
∞

t \to \infty

to the set of equilibrium points. A necessary and sufficient condition on the linear parts of these systems for their bounded global stability is given. It is also shown that this condition insures the existence of at least one stable equilibrium point.
Conjugate approximation functions in finite-element analysis
Authors: Brauchli H. J. , Oden J. T.
Published: December 2016

In this paper a theory of conjugate approximations is developed which provides a fundamental basis for most methods of continuous piecewise approximation. It is shown that for a given finite set of base functions used in an approximation there corresponds another set of conjugate functions which play a significant role in approximate methods of analysis. In the case of finite-element approximations, it is shown that the domain of the conjugate functions includes the entire assembly of elements, and, consequently, the established method of computing stresses locally in elements based on displacement approximations is not strictly valid. Indeed, the domain of such “local” stress fields is the entire connected system of elements. Procedures for computing derivatives and discrete analogues of linear operators which are consistent with the theory of conjugate functions are also discussed. For a given linear operator equation, the significance of the conjugate approximations in connection with the adjoint problem is also discussed.
Numerical determination of displacement fields in elastic disks with stress boundary conditions
Authors: Feldmann Jay W.
Published: December 2016

An earlier method [5] of establishing consistent finite difference equations is extended to obtain equations of higher degrees of precision. Again, the main advantage over the customary finite element method is that this new technique uses a uniform grid so that the boundary of the disk need not consist of grid segments. Numerous displays of pertinent quantities are given to facilitate the application of the method to specific stress boundary value problems.
Asymptotic features of viscous flow along a corner
Authors: Pal Alexander , Rubin Stanley G.
Published: December 2016

The asymptotic behavior of the equations governing the viscous flow along a right-angle corner is considered. It is demonstrated that consistent asymptotic series exist for the inner corner layer region. These expansions satisfy the corner layer equations and associated boundary conditions. They exhibit algebraic decay of all the flow properties into the boundary layer away from the corner, and prescribe algebraic decay of the cross flow velocities into the outer potential flow. Of course the streamwise velocity and vorticity are constrained to decay exponentially into the potential flow. The form of this algebraic behavior is required in order to facilitate numerical solution of the corner layer equations. Of particular significance is the use of symmetry as a means of providing a boundary condition, predicting the appearance of logarithmic terms, and specifying the occurrence of arbitrary constants. These constants can only be determined from the complete corner layer solution.
An asymptotic fundamental solution of the reduced wave equation on a surface
Authors: Steele C. R.
Published: December 2016

The physical problem of steady-state heat conduction in a thin shell is described by the “reduced wave equation” in which the differential operator is the (generally noneuclidean) Laplacian for the surface. A similar equation gives the approximation for steady-state waves in a prestressed curved membrane. A modification of the “geometric optics” asymptotic expansion, involving a Bessel function, is given for the fundamental point source solution. This is proven to be uniformly valid in the large, until a “caustic” is reached. Various features of the solution for a surface, which do not occur for the plane, are discussed.
Mean convergence of approximation to a function by general finite sums
Authors: Ho Hwa Shan
Published: December 2016

The approximation of a function by a general finite sum (linear combination of non-orthogonal functions) is considered here. It is shown that the mean error of such an approximation, defined in the sense of any weighted inner product in the Hilbert space, is positive semi-definitely decreasing as the number of terms in the expansion increases. Conditions under which the mean error is stationary are thoroughly discussed. Some interesting properties of such approximations are revealed by related theorems. The theorems are proven for complex variables, and are valid of course for real variables.
Remarks on the root-clustering of a polynomial in a certain region in the complex plane
Authors: Jury E. I. , Ahn S. M.
Published: December 2016

A general formulation for the root clustering of a polynomial is given. An attempt has been made to answer an open question raised by Kalman.
Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing
Authors: Lardner R. W.
Published: December 2016

The method of averaging and the two-time procedure are outlined for a class of hyperbolic second-order partial differential equations with small nonlinearities. It is shown that they both lead to the same integro-differential equation for the lowest-order approximation to the solution. For the special case of the nonlinear wave equation, this lowest-order solution consists of the superposition of two modulated travelling waves, and separate integro-differential equations are derived for the amplitudes of these two waves. As an example, the wave equation with Van der Pol type of nonlinearity is considered.
An iterative technique for solution of the Thomas-Fermi equation utilizing a nonlinear eigenvalue problem
Authors: Luning C. D. , Perry W. L.
Published: December 2016

Development of an iterative solution technique for a certain nonlinear eigenvalue problem supplies an iterative solution technique for the ion case and isolated neutral atom case boundary-value problems for the Thomas-Fermi equation.
Numerical integration methods for the solution of singular integral equations
Authors: Theocaris P. S. , Ioakimidis N. I.
Published: December 2016

The evaluation of the stress intensity factors at the tips of a crack in a homogeneous isotropic and elastic medium may be achieved with higher accuracy and much less computation if the Lobatto-Chebyshev method of numerical solution of the corresponding system of singular integral equations is used instead of the method of Gauss-Chebyshev commonly applied to such problems. Comparison of results obtained by the two numerical methods when applied to the problem of a cruciform crack in an infinite medium proves the potentialities of the new approach.
Complementary variational principles for a boundary-value problem in twisting of ring sectors
Authors: Anderson N. , Arthurs A. M.
Published: December 2016

Complementary variational principles associated with the boundary-value problem for stresses in close-coiled helical springs are presented. An accurate variational solution is obtained for an illustrative problem.
Linear time-invariant transformations of some nonstationary random processes
Authors: Tjøstheim D. , Thomas J. B.
Published: December 2016

We consider the class of nonstationary processes

Y

(
t
)

Y\left ( t \right )

which can be represented as

Y

(
t
)

=
B
X

(
t
)

Y\left ( t \right ) = BX\left ( t \right )

, where

X

(
t
)

X\left ( t \right )

is wide sense stationary and

B
B

is a bounded self-adjoint operator with a bounded inverse. An equivalent characterization of this class of processes is given and is used to construct examples of nonstationary processes belonging to this class. A functional analytic treatment is given for describing the effects of linear time-invariant transformations.
On the linear capillary gravity wave problem
Authors: Murray J. C.
Published: December 2016

A general initial-boundary value problem is formulated for capillary-gravity waves which includes the derivation of a boundary condition at the liquid-solid intersection. Conditions on the bounding surface geometry which ensure uniqueness are also established.
Variational solutions for two nonlinear boundary-value problems for diffusion with concentration-dependent coefficients
Authors: Reeves R. I.
Published: December 2016

The theory of complementary variational principles is used to obtain variational solutions for two nonlinear boundary-value problems which arise in the theory of diffusion with concentration-dependent coefficients.
The stability of vector renewal equations pertaining to heterogeneous chemical reaction systems
Authors: DeGance A. E. , Johns L. E.
Published: December 2016

For a class of vector renewal equations arising in the theory of spatially nonuniform chemical reaction processes, stability conditions are given in terms of the physicochemical operators D and K. In particular, we provide conditions for the stability of an initially uniform, multicomponent film bounding a planar catalytic surface, and for the asymptotic stability of a reaction system composed of a population of catalyst particles. We obtain such results by identifying a sup norm which is naturally induced from the factors that symmetrize D and/or K. Further, we exhibit conditions under which a special class of vector renewal equations has positive solutions.
Network problems, 𝑀-functions and uniformly monotone networks
Authors: Porsching T. A.
Published: December 2016

Four classes of network problems—conductive, resistive, conductive boundary value, and resistive boundary value—are considered in this paper. In each case solution of the network problem is tantamount to determining a zero of a nonlinear system of equations. Under certain monotonicity assumptions, it is shown that the nonlinear Gauss-Seidel iterative procedure is globably convergent when applied to these systems.
A singular perturbation solution to a problem of extreme temperatures imposed at the surface of a variable-conductivity halfspace: small surface conductivity
Authors: Cooper Leonard Y.
Published: December 2016

The transient temperature field resulting from a constant and uniform temperature

T
s

{T_s}

(or time-dependent heat flux

H
=
h

t

−
1

/

2

H = h{t^{ - 1/2}}

) imposed at the surface of a halfspace initially at uniform temperature

T
0

{T_0}

is considered. A temperature-dependent thermal conductivity variation,

k

(
T
)

=

k
0

exp
⁡

[

λ
(
T
−

T
0

)

/

T
0

]

k\left ( T \right ) = {k_0}\exp \left [ {\lambda (T - {T_0})/{T_0}} \right ]

, and a constant product of density and specific heat,

ρ
C

\rho C

, are assumed to be accurate models for the halfspace for some useful temperature range. The problem is initially formulated in terms of the dimensionless conductivity

ϕ
=
k

(
T
)

/

k
0

\phi = k\left ( T \right )/{k_0}

. Attention is then focused on the singular problem resulting from the limits

ϕ
s

=
ϕ

(

T
s

)

↓
0

{\phi _s} = \phi \left ( {{T_s}} \right ) \downarrow 0

and

ϕ
s

→
∞

{\phi _s} \to \infty

. This work considers the use of matched asymptotic expansions to solve the problem under the first of these limits. In particular, Fraenkel’s interpretation [5] of Van Dyke’s method of inner and outer expansions [6] is carefully applied to the problem under consideration. Besides obtaining a uniformly valid solution to the problem, a particularly interesting explicit result is deduced, namely
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript phi Subscript s Baseline down-arrow 0 Endscripts h equals minus left-parenthesis 1.182754 dot dot dot right-parenthesis left-parenthesis upper T 0 slash lamda right-parenthesis left-bracket rho upper C k 0 slash 2 right-bracket Superscript 1 slash 2 Baseline plus upper O left-parenthesis phi Subscript s Baseline l n phi Subscript s Baseline right-parenthesis">

lim

ϕ
s

↓
0

h
=
−
(
1.182754
⋅
⋅
⋅
)
(

T
0

/

λ
)

[
ρ
C

k
0

/

2

]

1

/

2

+
O
(

ϕ
s

l
n

ϕ
s

)

\lim \limits _{{\phi _s} \downarrow 0} h = - (1.182754 \cdot \cdot \cdot )({T_0}/\lambda ){[\rho C{k_0}/2]^{1/2}} + O({\phi _s}ln{\phi _s})

\]
Variational-Lagrangian irreversible thermodynamics of nonlinear thermorheology
Authors: Biot M. A.
Published: December 2016

A principle of virtual dissipation generalizing d’Alembert’s principle to nonlinear irreversible thermodynamics provides a unifying foundation which leads to an extremely general variational-Lagrangian analysis of dissipative phenomena. Thus a synthesis is achieved between thermodynamics and classical mechanics. The present paper applies this principle to the nonlinear thermomechanics of continua with dissipation and heat conduction. Field equations, constitutive equations and Lagrangian equations with generalized coordinates are derived for nonlinear thermoviscoelastcity, nonlinear thermoelasticity and heat conduction, plasticity, and compressible heat conducting fluids with Newtonian and non-Newtonian viscosity. The thermodynamics of instability is also analyzed from the same fundamental viewpoint.
On the asymptotic stability of oscillators with unbounded damping
Authors: Artstein Zvi , Infante E. F.
Published: December 2016

Through a technique inspired by the invariance principle of LaSalle, a general growth condition on the damping coefficient

h

(
t
)

h\left ( t \right )

of the equation
\[

x
¨

+
h

(
t
)

x
˙

+
k
x
=
0
,
k
>
0
,
h
(
t
)
≥
ϵ
>
0

\ddot x + h\left ( t \right )\dot x + kx = 0, k > 0,h(t) \ge \epsilon > 0

\]
, is given which is sufficient for the global asymptotic stability of the origin yet permits this coefficient to grow to infinity with time. The methods used do not depend on linearity, and are applied to obtain similar results to the nonlinear analogue of this equation.
Variational irreversible thermodynamics of heat and mass transfer in porous solids: new concepts and methods
Authors: Biot M. A.
Published: December 2016

A recently developed variational principle of virtual dissipation along with a new approach to the thermodynamics of open systems is applied to coupled mass and heat transfer in a porous solid containing a fluid. General differential field equations are derived directly from the variational principle. A general energy flux theorem is formulated. Vapor-liquid phase transition and capillary condensation are discussed. Field equations for nonequilibrium adsorption are also obtained. Lagrangian equations with generalized coordinates are derived directly from the variational principle without use of the field equations. They provide the foundation of finite-element methods as well as of many other techniques particularly suitable in geothermal systems analysis.
Hopf-Friedrichs bifurcation and the hunting of a railway axle
Authors: Huilgol R. R.
Published: December 2016

After deriving the equations of motion which govern the lateral and yaw motions of a railway axle, these are cast in the form of a system of first-order nonlinear differential equations. To this system the Hopf-Friedrichs bifurcation theory is applied to determine when a periodic orbit will bifurcate from the equilibrium position. Sufficient conditions to guarantee the stability of the orbit are investigated.
An electrostatic problem in bi-cyclide coordinates
Authors: Aikawa K. , Hisamoto T. , Suganuma T.
Published: December 2016

This paper deals with an electrostatic problem for the field between two charged conductors

±

u
0

\pm {u_0}

maintained at potential

±
V

\pm V

in bi-cyclide coordinates (

u
,
v
,
ψ

u, v, \psi

). In this coordinate system, Heine functions are used, of which something is known. Heine functions are the solutions of Heine differential equations. Though the problem is to be solved in the same manner as the problem in the case of bispherical coordinates, it has not been clarified because of the complexity of Heine functions. A Heine differential equation is solved to satisfy the boundary condition that the functions and their derivatives are bounded at the ends of interval, and eigenvalues and eigenfunctions are evaluated. A formula giving the capacity between two electrodes is presented and numerically calculated.
On some problems of homogenization
Authors: Burgers J. M.
Published: December 2016

This article concerns a type of partial differential equations in which coefficients occur that are periodic functions of the basic independent variables or coordinates

x
1

{x_1}

,

x
2

{x_2}

(we restrict ourselves to two-dimensional problems). The period length

H
H

, here assumed to be the same in both directions, is supposed to be small in comparison with the scale of the general field, and interest is directed to what happens when

H
→
0

H \to 0

. An example is the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma StartFraction partial-differential Over partial-differential x Subscript j Baseline EndFraction left-parenthesis a Subscript i j Baseline StartFraction partial-differential u Over partial-differential x Subscript i Baseline EndFraction right-parenthesis equals f left-parenthesis x 1 comma x 2 right-parenthesis">

Σ

∂

∂

x
j

(

a

i
j

∂
u

∂

x
i

)

=
f

(

x
1

,

x
2

)

\Sigma \frac {\partial }{{\partial {x_j}}}\left ( {{a_{ij}}\frac {{\partial u}}{{\partial {x_i}}}} \right ) = f\left ( {{x_1},{x_2}} \right )

\]
with the boundary condition

u
=
0

u = 0

on a given closed curve

C
C

in the

X
1

{X_1}

,

x
2

{x_2}

-plane. Size and shape of

C
C

are independent of

H
H

, while the coefficients

a

11

{a_{11}}

,

a

12

{a_{12}}

,

a

21

{a_{21}}

,

a

22

{a_{22}}

are periodic functions of auxiliary variables

ξ
i

=

x
1

/

H

{\xi _i} = {x_1}/H

with period 1 in

ξ
1

{\xi _1}

,

ξ
2

{\xi _2}

. The problem of interest is whether the unknown function

u
u

, which must react upon all the fluctuations of the

a

i
j

{a_{ij}}

, can be related to a function

U

(

x
1

,

x
2

)

U\left ( {{x_1}, {x_2}} \right )

which is determined by a partial differential equation with constant coefficients, and how these coefficients can be obtained from the

a

i
j

{a_{ij}}

. This is called the problem of "homogenization".
Some functional differential equations
Authors: Chambers Ll. G.
Published: December 2016

Sufficient conditions are given for the solution of the functional differential equations with associated boundary conditions
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d y slash d x equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts a Subscript n Baseline y left-parenthesis mu Superscript n Baseline x right-parenthesis comma y left-parenthesis 0 right-parenthesis equals 1 comma d y slash d x equals integral Subscript 0 Superscript normal infinity Baseline a left-parenthesis u right-parenthesis y left-parenthesis mu Superscript u Baseline x right-parenthesis d u comma y left-parenthesis 0 right-parenthesis equals 1">

d
y

/

d
x
=

∑

n
=
0

∞

a
n

y

(

μ
n

x

)

,

y

(
0
)

=
1
,

d
y

/

d
x
=

∫
0
∞

a

(
u
)

y

(

μ
u

x

)

d
u

,

y

(
0
)

=
1

dy/dx = \sum \limits _{n = 0}^\infty {{a_n}y\left ( {{\mu ^n}x} \right ),\qquad y\left ( 0 \right ) = 1, \\ dy/dx = \int _0^\infty {a\left ( u \right )y\left ( {{\mu ^u}x} \right )du},\qquad y\left ( 0 \right ) = 1}

\]
A discussion is also given of some possible solutions to the differential equations which do not satisfy the boundary conditions.
The shape of the strongest column and some related extremal eigenvalue problems
Authors: Barnes Earl R.
Published: December 2016

We determine the shape of the strongest column in the class of columns of length

l
l

, volume

V
V

, and having similar cross-sectional areas

A
(
x
)

A(x)

satisfying

a
≤
A

(
x
)

≤
b

a \le A\left ( x \right ) \le b

where

a
a

and

b
b

are prescribed positive bounds. In the special case where there are no constraints on the areas of cross-sections the problem has been solved by Keller [1] and by Takjbakhsh and Keller [2]. These authors observed that the problem is equivalent to an extremal eigenvalue problem and developed a variational technique for solving such problems. We treat a slightly more general class of extremal eigenvalue problems and give sufficient conditions for a given function to be a solution. Our work on the strongest constrained column demonstrates a procedure for finding functions satisfying these conditions.
Stability and uniqueness for a turbulence model of Burgers
Authors: Horgan C. O. , Olmstead W. E.
Published: December 2016

In his early work on mathematical models of turbulence, J. M. Burgers proposed a nonlinear system, coupling an ordinary and a partial differential equation, to simulate flow in a channel. The now well-known Burgers equation arose in his work from a simplification of this system. The original system has some interesting features not shared by the Burgers equation. This investigation establishes results on the stability of the “laminar” stationary solution and uniqueness of the nonstationary solution of the system.
Global asymptotic stability of certain models for protein synthesis and repression
Authors: Banks H. T. , Mahaffy J. M.
Published: December 2016

We consider a class of mathematical models involving nonlinear differential equations with hereditary terms. Included as special cases are a number of models that have been proposed as qualitative models for protein synthesis in eukaryotic cells. We establish global stability for these models and discuss the implications of our results.
Asymptotic properties of best 𝐿₂[0,1] approximation by splines with variable knots
Authors: Barrow D. L. , Smith P. W.
Published: December 2016

Let

S
N
k

S_N^k

be the set of

k
k

th-order splines on

[

0
,
1

]

\left [ {0, 1} \right ]

having at most

N
−
1

N - 1

interior knots, counting multiplicities. We prove the following sharp asymptotic behavior of the error for the best

L
2

[

0
,
1

]

{L_2}\left [ {0, 1} \right ]

approximation of a sufficiently smooth function

f
f

by the set

S
N
k

S_N^k

:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript upper N right-arrow normal infinity Endscripts upper N Superscript k Baseline d i s t left-parenthesis f comma upper S Subscript upper N Superscript k Baseline right-parenthesis equals left-parenthesis StartAbsoluteValue upper B Subscript 2 k Baseline EndAbsoluteValue slash left-parenthesis 2 upper K right-parenthesis factorial right-parenthesis Superscript 1 slash 2 Baseline left-parenthesis integral Subscript 0 Superscript 1 Baseline StartAbsoluteValue f Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis tau right-parenthesis EndAbsoluteValue Superscript sigma Baseline d tau right-parenthesis Superscript 1 slash sigma">

lim

N
→
∞

N
k

d
i
s
t

(

f
,

S
N
k

)

=

(

|

B

2
k

|

/

(

2
K

)

!

)

1

/

2

(

∫
0
1

|

f

(
k
)

(
τ
)

|

σ

d
τ

)

1

/

σ

\lim \limits _{N \to \infty } {N^k}dist\left ( {f, S_N^k} \right ) = {\left ( {\left | {{B_{2k}}} \right |/\left ( {2K} \right )!} \right )^{1/2}}{\left ( {{{\int _0^1 {\left | {{f^{\left ( k \right )}}\left ( \tau \right )} \right |} }^\sigma }d\tau } \right )^{1/\sigma }}

\]
, where

σ
=

(

k
+
1

/

2

)

−
1

\sigma = {\left ( {k + 1/2} \right )^{ - 1}}

and

B

2
k

{B_{2k}}

is the

2
k

2k

th Bernoulli number. Similar results have previously been obtained for piecewise polynomial (i.e., with no continuity constraints) approximation, but with different constant before the integral term. The approach we use is first to study the asymptotic behavior of dist(

f
f

,

S
N
k

(
t
)

S_N^k\left ( t \right )

), where

S
N
k

(
t
)

S_N^k\left ( t \right )

is the linear space of

k
k

th-order splines having simple knots determined from the fixed function

t
t

by the rule

t
i

=
t

(

i

/

N

)

,
i
=
0
,
.
.
.
,
N

{t_i} \\ = t\left ( {i/N} \right ),i = 0,...,N

.
A global particular solution to the initial-value problems of stellar dynamics
Authors: Kurth Rudolf
Published: December 2016

An explicit global particular solution of the initial-value problem of stellar dynamics is presented.
The non-uniform motion of a thin smooth rigid inclusion through an elastic solid
Authors: Brock L. M.
Published: December 2016

A two-dimensional example of non-uniform motion by a smooth, relatively thin, symmetric rigid inclusion through an elastic solid is analyzed. Integral solutions to the related boundary-initial value problem are obtained for the case of edge speeds bounded by the dilatational wave speed at large times. The wavefront patterns, including the possibility of Mach waves, are noted. Some attention is focused on the case of the straight wedge, and logarithmic singularities are found at its edge for subsonic speeds.
On optimal strain paths in linear viscoelasticity
Authors: Gurtin Morton E. , MacCamy Richard C. , Murphy Lea F.
Published: December 2016

For a viscoelastic material the work

W

(
e
)

W\left ( e \right )

needed to produce a given strain

e
0

{e_0}

in a given time

T
T

depends on the strain path

e

(
t
)

,
0
≤
t
≤
T

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

, connecting the unstrained state with

e
0

{e_0}

. We here ask the question: Of all strain paths of this type, is there one which is optimal,

1

^{1}

that is, one which renders

W
W

a minimum? In answer to this question we show that: (i) There is no smooth optimal strain path. (ii) There exists a unique optimal path in

L
2

(

0
,
T

)

{L_2}\left ( {0,T} \right )

; this path is smooth on the open interval

(

0
,
T

)

\left ( {0,T} \right )

, but suffers jump discontinuities

2

^{2}

at the end points 0 and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis i period e period comma e left-parenthesis 0 Superscript plus Baseline right-parenthesis not-equals 0 comma e left-parenthesis upper T Superscript minus Baseline right-parenthesis not-equals e 0 right-parenthesis">

T

(

i
.
e
.
,
e

(

0
+

)

≠
0
,

e

(

T
−

)

≠

e
0

)

T\left ( {i.e.,e\left ( {{0^ + }} \right ) \ne 0, \\ e\left ( {{T^ - }} \right ) \ne {e_0}} \right )

. (iii) For a Maxwell material the optimal path is linear on

(

0
,
T

)

\left ( {0,T} \right )

.
Nonmonotonic solutions of the Falkner-Skan boundary layer equation
Authors: Troy William C.
Published: December 2016

We investigate the equation

f
′

+
f
f
+
β
(
1
−

f

′
2

)
=
0

f’ + ff + \beta (1 - f’^2) = 0

together with boundary conditions

f

(
0
)

=

f
′

(
0
)

=
0

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

and

f
′

(
∞
)

=
1

f’\left ( \infty \right ) = 1

. Here

β
\beta

is negative. Previous results are summarized which describe solutions which satisfy

|

f
′

|

>
1

\left | f’ \right | > 1

for all

η
≥
0

\eta \ge 0

. It is shown that there is a sequence

{

β
j

}

j
∈

N

\left \{ \beta _j \right \}_{j \in \mathbb {N}}

of decreasing, negative values of

β
\beta

, and a corresponding sequence

{

f
j

}

j
∈

N

\left \{ f_j \right \}_{j \in \mathbb {N}}

of solutions such that for each

j
∈

N

j \in \mathbb {N}

the equation

f
j
′

−
1
=
0

f_j’ - 1 = 0

has exactly

j
j

positive solutions and for some

μ
j

>
0
,

f
j
′

=
1
+
o

(
exp
⁡

(
−

μ
j

η
)

)

\mu _j > 0, f_j’ = 1 + o\left ( \exp \left ( - \mu _j\eta \right ) \right )

as

η
→
∞

\eta \to \infty

.
Homogenization and random evolutions: applications to the mechanics of composite materials
Authors: Bécus Georges A.
Published: December 2016

The technique of homogenization is used to derive the effective properties of laminated composites. A new probabilistic justification for homogenization using the concept of random evolutions is provided and indicates that the effective properties of deterministic periodic composite and those of a randomly perturbed periodic composite are the same.
Elastic realizability of force and displacement systems in structures
Authors: Spillers W. R. , Lefcochilos E.
Published: December 2016

This paper discusses the constraints imposed upon equilibrium and compatibility solutions in structures through the use of constitutive equations. It is shown, for example, that “realizable” force systems are bounded by statically determinate force systems for the case of trusses. The analysis used depends heavily upon the concept of “basic solutions” (statically determinate substructures) for linear systems which appears most commonly in the theory of linear programming.
Further study on one-dimensional shock waves in nonlinear elastic media
Authors: Ting T. C. T.
Published: December 2016

One may define the growth of a shock wave by the growth in the amplitude of discontinuity in the velocity (denoted by

[
v
]

\left [ v \right ]

) across the shock wave as the shock wave propagates. One may also define the growth of a shock wave by the growth in the amplitude of discontinuity in the stress

[
σ
]

\left [ \sigma \right ]

, strain

[
ϵ
]

\left [ \epsilon \right ]

, or entropy

[
η
]

\left [ \eta \right ]

, It is shown that one definition predicts the growth of the shock wave while others may predict its decay. In this paper we derive the transport equations for one-dimensional shock waves in nonlinear elastic media in which the shock wave can be defined as the amplitude of either

[
ϵ
]

2

{\left [ \epsilon \right ]^2}

,

(
b
)

\left ( b \right )

,

[
v
]

\left [ v \right ]

or

[
η
]

\left [ \eta \right ]

. Moreover, the dependent quantity can be any one of, or a linear combination of, the seven quantities behind the shock wave. It is shown that when the region ahead of the shock wave is under a homogeneous deformation, the amplitudes of

[
v
]

\left [ v \right ]

,

[
σ
]

\left [ \sigma \right ]

and

[
ϵ
]

\left [ \epsilon \right ]

grow or decay simultaneously if

(
a
)

\left ( a \right )

[
ϵ
]

2

{\left [ \epsilon \right ]^2}

is a strictly increasing function of

[
η
]

\left [ \eta \right ]

, or

(
b
)

\left ( b \right )

the purely mechanical theory of shock waves is employed in which the effect of the entropy is ignored. Regardless of whether the effect of the entropy is ignored or not, there is no assurance that the amplitudes of

[
v
]

\left [ v \right ]

,

[
σ
]

\left [ \sigma \right ]

and

[
ϵ
]

\left [ \epsilon \right ]

grow or decay simultaneously if the region ahead of the shock wave is not under a homogeneous deformation.
The swallowtail and butterfly cuspoids and their application in the initial post-buckling of single-mode structural systems
Authors: Hui David , Hansen Jorn S.
Published: December 2016

This paper investigates the swallowtail and butterfly catastrophes from the point of view which is applicable in the theory of elastic stability. Thus, the results are concerned with the various forms of these instabilities as well as the determination of the critical load surfaces which are of engineering significance. It is demonstrated that the results are applicable to an axially loaded beam resting on a nonlinear elastic foundation.
Wave propagation in an elastic half-space due to surface pressure over a non-uniformly changing circular zone
Authors: Brock L. M.
Published: December 2016

Two problems of pressure distributions applied to an elastic half-space over a circular pressure zone whose center is fixed but whose radius changes non-uniformly with time are considered. In one case, the pressure depends only on time; in the other case, the pressure varies with the radial distance from the pressure zone center. Complete transform solutions are obtained and several wave propagation aspects are briefly studied, with emphasis on the Rayleigh pole contributions and the associated propagating singularities. The effects of some specific zone time-histories on the Rayleigh pole disturbances at the half-space surface are considered. Some characteristics of a given time-history appear to be manifested in the corresponding disturbance.
Beam bending problems on a Pasternak foundation using reciprocal variational inequalities
Authors: Kikuchi Noboru
Published: December 2016

The present study is concerned with a class of two-body contact problems in linear elasticity. The model problem is a bending problem of the beam resting unilaterally upon a Pasternak foundation. A variational formulation is given by the mini-max principle of the functional, and proof of the existence of saddle points is given by the compatibility condition for applied forces and moments on the beam. It has been found that the compatibility condition for equilibrium of the beam and foundation can be achieved by arguments of coerciveness of the functional on the admissible set. An approximation and example of the problem by finite element methods and a numerical method for its solution are also introduced.
Synchronous solutions of power systems: existence, uniqueness and stability
Authors: Skar Sherwin J.
Published: December 2016

This paper contains a study of the asymptotic stability and uniqueness of equilibrium solutions of multi-dimensional Hamiltonian-like systems. The results are applied to the swing equations, the classical model for power systems. By developing some results in matrix theory, it is shown that asymptotically stable equilibrium solutions may exist even though most rotor angle pairs are more than

90
∘

{90^ \circ }

, some even

180
∘

{180^ \circ }

, out of phase. In contrast to the numerical criteria usually used, an analytic criterion for the existence of asymptotically stable equilibrium solutions of the swing equations is given.
Large deformations of a heavy cantilever
Authors: Wang Chang Yi
Published: December 2016

A cantilever of uniform cross-section and density is held at an angle

α
\alpha

at one end. The shape of the cantilever depends heavily on

α
\alpha

and a nondimensional parameter

K
K

which represents the relative importance of density and length so that of flexural rigidity. Perturbations on the elastica equations for small and large

K
K

show good agreement with exact numerical integration. It is found that whenever

K
K

reaches a critical value, bifurcations of the solutions occur. This nonuniqueness can be observed by the flipping phenomena as

α
\alpha

is increased.
On a second-order boundary value problem arising in combustion theory
Authors: Holmes Philip
Published: December 2016

We obtain existence and uniqueness results for the boundary-value problem
\[

y
=

x
2

−

y
2

,

y

(
x
)

∼
∓
x

a
s

x
→
±
∞

y = {x^2} - {y^2}, \qquad y\left ( x \right ) \sim \mp x \qquad as \qquad x \to \pm \infty

\]
. Our main result shows that there are precisely two solutions

y
+

(
x
)

>
−

|
x
|

{y_+} \left ( x \right ) > - \left | x \right |

and

y
−

(
x
)

>
−

|
x
|

{y_-}\left ( x \right ) > - \left | x \right |

. Only the latter is of physical interest in the problem in combustion theory from which the equation arises.
On steady stratified flows in porous media
Authors: Yih Chia Shun
Published: December 2016

Steady flows of an incompressible and nondiffusive fluid stratified in viscosity and in density that take place in porous media are considered, and conditions necessary for similarity between any two such flows are given with proof. Results on the vorticity and circulation in such flows are also given.