bluuebunny/crossref_metadata_2025 huggingface dataset:
On a nonhomogeneous system of pressureless flow
Authors: Ding Yi , Huang Feimin
Published: December 2016

In this paper, a nonhomogeneous system of pressureless flow
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript t Baseline plus left-parenthesis rho u right-parenthesis Subscript x Baseline equals 0 comma left-parenthesis rho u right-parenthesis Subscript t Baseline plus left-parenthesis rho u squared right-parenthesis Subscript x Baseline equals rho x">

ρ
t

+

(
ρ
u
)

x

=
0
,

(
ρ
u
)

t

+

(
ρ

u
2

)

x

=
ρ
x

{\rho _t} + {\left ( \rho u \right )_x} = 0, \qquad {\left ( \rho u \right )_t} + {\left ( \rho {u^2} \right )_x} = \rho x

\]
is investigated. It is found that there exists a generalized variational principle from which the weak solution is explicitly constructed by using the initial data; i.e.,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho left-parenthesis x comma t right-parenthesis equals minus StartFraction partial-differential Over partial-differential x squared EndFraction min Underscript y Endscripts upper F left-parenthesis y semicolon x comma t right-parenthesis comma rho left-parenthesis x comma t right-parenthesis u left-parenthesis x comma t right-parenthesis equals StartFraction partial-differential squared Over partial-differential x partial-differential t EndFraction min Underscript y Endscripts upper F left-parenthesis y semicolon x comma t right-parenthesis">

ρ

(
x
,
t
)

=
−

∂

∂

x
2

min
y

F

(
y
;
x
,
t
)

,

ρ

(
x
,
t
)

u

(
x
,
t
)

=

∂
2

∂
x
∂
t

min
y

F

(
y
;
x
,
t
)

\rho \left ( x, t \right ) = - \frac {\partial }{{\partial {x^2}}}\min \limits _y F\left ( y; x, t \right ), \qquad \rho \left ( x, t \right )u\left ( x, t \right ) = \frac {{{\partial ^2}}}{{\partial x\partial t}}\min \limits _y F\left ( y; x, t \right )

\]
hold in the sense of distributions, where

F

(

y
;
x
,
t

)

F\left ( {y; x, t} \right )

is a functional depending on the initial data. The weak solution is unique under an Oleinik-type entropy condition when the initial data is of measurable function. It is further shown that the solution

u

(
x
,
t
)

u\left ( x, t \right )

converges to

x
x

as

t
t

tends to infinity. The proofs are based on the generalized variational principle and careful studies on the generalized characteristics introduced by Dafermos [5].
Nonlinear reptation in molecular based hysteresis models for polymers
Authors: Banks H. T. , Medhin Negash G. , Pinter Gabriella A.
Published: December 2016

We extend the linear “stick-slip” models of Doi-Edwards and Johnson-Stacer to nonlinear tube reptation models. We then show that such models, when combined with probabilistic formulations allowing distributions of relaxation times, provide a good description of dynamic experiments with highly filled rubber in tensile deformations. A connection to other applications including dielectric polarization and reptation in other viscoelastic materials (e.g., living tissue) is noted.
On linear ordinary differential equations with exponential coefficients
Authors: Hegemier G. A.
Published: December 2016

A system of linear ordinary differential equations with exponential coefficients is considered; several theorems are presented concerning general solutions and their asymptotic forms for large values of the independent variable; elementary examples are given and applications cited to illustrate, and demonstrate the utility of, the solution forms obtained. The relevance of the results to the solution of problems of structural stability and wave propagation is briefly discussed.
Characterization of single-blow temperature responses by first moments.
Authors: Kohlmayr Gerhard F.
Published: December 2016

Transient temperature response functions, given as analytic solutions of the single-blow problem in transient heat transfer analysis, are characterized by the first moment of the difference between downstream and upstream fluid temperatures. Both the single-blow problem and the corresponding inverse problem are formulated in terms of Volterra integral equations. Monotonicity and boundedness properties of the response functions are derived. It is shown that the first moment of the temperature difference is well suited for an indirect solution of the curve matching problem.
Parametric synthesis of statistical communication nets
Authors: Frank H. , Hakimi S. L.
Published: December 2016

The traffic within the branches of a communication network is often a random variable with an approximate multivariate normal distribution. The analysis of such systems has been discussed in a previous paper. In this paper, we consider the optimum synthesis problem. Nonlinear and linear programming is used to guarantee that a given flow rate probability between a specified pair of terminals is achieved. In particular, a Uniformly Most Powerful Statistical Test provides the basis for an optimum synthesis procedure that appears to be very efficient. The procedure is formulated as a concave program with quadratic and linear constraints.
Upper and lower bounds of eigenvalues: Mode-clamping theorems
Authors: Sparks M.
Published: December 2016

Upper and lower bounds for the eigenvalues of three types of matrices

M
M

are established. If

M
M

is written as the sum of a diagonal matrix

D
D

plus a matrix

A
A

, the real parts of the eigenvalues of

M
M

must lie between the real parts of the neighboring diagonal elements of

D
D

, no matter how large the elements of

A
A

or how closely spaced the diagonal elements of

D
D

.
Recursive formulas for the evaluation of certain complex integrals.
Authors: Ȧström K. J.
Published: December 2016

This paper presents recursive formulas which admit simultaneous tests of stability and evaluation of quadratic loss functions for linear discrete-time dynamical systems. The method admits a significant reduction of the number of computations in comparison with previously known methods.
Surface wave incidence on a plane structure having a multi-mode discontinuity in impedance
Authors: Morgan Richard C. , Karp Samuel N.
Published: December 2016

The phenomenological theory of multi-mode surface wave propagation is applied to a plane structure having a multi-mode discontinuity in impedance. The resulting boundary-value problem is reduced to the solution of a Wiener–Hopf equation whose factorization is given in terms of the factorization that occurred in the one-mode case. Despite the complexity of the solution, the magnitudes of the surface wave excitation coefficients are elementary functions, as is the cylindrical power flow.
Comparison theorems for second-order nonlinear differential equations
Authors: Eliason Stanley B.
Published: December 2016

Comparison theorems for a nonlinear eigenvalue problem as well as a Lyapunov type of inequality are derived. They are used to establish upper and lower bounds for various integral functionals associated with real solutions of the nonlinear boundary value problem

y
+
p

(
x
)

y

2
n
+
1

=
0
,
y

(
a
)

=

y
′

(
b
)

=
0

y + p\left ( x \right ){y^{2n + 1}} = 0, y\left ( a \right ) = y’\left ( b \right ) = 0

, where

a
>
b

a > b

are real,

n
n

is a positive integer and

p
p

is positive and continuous on

[

a
,
b

]

\left [ {a,b} \right ]

. Some of the results are analogues of a distance between zeros problem for the linear case of

n
=
0

n = 0

.
Application of the Sonin-Pólya oscillation theorem
Authors: Laitone E. V. , Lin Wen Fan
Published: December 2016

The necessary conditions for the Sonin-Pólya theorem, which predicts the variation of the successive relative maxima of the oscillating solutions of a second-order ordinary differential equation, are evaluated for the confluent hypergeometric functions to prove that the criteria for their decreasing sequence, as given in [1], [2] and [3], are incorrect. The Sonin-Pólya theorem is then applied to the differential equation for the linearized shallow-water edge wave that is produced by a beach of constant slope. The final application is to Allen’s differential equation for the angle of attack oscillations of a re-entry ballistic missile, and new stability criteria are obtained for both ascending and descending missiles that are coasting through the atmosphere.
Extremum properties of the generalized Rayleigh quotient associated with flutter instability
Authors: Huseyin K. , Plaut R. H.
Published: December 2016

The extremum properties of the generalized Rayleigh quotient related to flutter instability are investigated. It is shown that, in addition to the well-known stationary property, under certain circumstances the quotient exhibits maximum-minimum properties which are in contrast to those of the classical Rayleigh quotient. One consequence is that an approximate method of stability analysis using these results leads to a lower bound as opposed to an upper bound in the classical case. The results are applied to multiple-parameter systems and a physical interpretation is given for the generalized Rayleigh quotient, leading to the proof of a convexity theorem.
Multiple Fourier analysis in rectifier problems. II
Authors: Sternberg Robert L. , Sheets Michael R. , Sternberg Helen M. , Shigematsu Abraham , Sternberg Alice L.
Published: December 2016

The nonlinear problem of the multiple Fourier analysis of the output from a cut-off power-law rectifier responding to a two-frequency input, reviewed in general in Part I of this study [1], is further scrutinized here for the special case of a zero-power-law device; i.e., a bang-bang device or a total limiter. Solutions for the modulation product amplitudes or multiple Fourier coefficients as in Part I appear as Bennett functions, and line graphs of the first fifteen basic functions for the problem are given. The new functions

A

m
n

(
0
)

(

h
,
k

)

{A_{mn}}^{\left ( 0 \right )}\left ( {h,k} \right )

studied, being based on a discontinuous device, then, together with the functions

A

m
n

(
1
)

(

h
,
k

)

{A_{mn}}^{\left ( 1 \right )}\left ( {h,k} \right )

studied in Part I, provide approximate solutions to the two-frequency modulation product problem for an arbitrary piecewise continuous nonlinear modulator, and the solution for this general problem is outlined. Finally, numerical tables of the zeroth-kind functions

A

m
n

(
0
)

(

h
,
k

)

{A_{mn}}^{(0)}\left ( {h,k} \right )

graphed have been prepared and are available separately in the United States and Great Britain. As before, the entire theory is based on the original multiple Fourier methods introduced by Bennett in 1933 and 1947.
A result on the singularities of matrix functions
Authors: Reid William T.
Published: December 2016

For

F

(
t
)

=
F

(

t
1

,
.
.
.
,

t
p

)

F\left ( t \right ) = F\left ( {{t_1},...,{t_p}} \right )

an

n
×
n

n \times n

complex-valued matrix function which is continuous on an open neighborhood of

t
0

=

(

t
α

0

)

(

α
=
1
,
.
.
.
,
p

)

{t^0} = \left ( {{t_\alpha }^0} \right )\left ( {\alpha = 1,...,p} \right )

and singular at

t
0

{t^0}

there is presented a necessary and sufficient condition for

F

(
t
)

F\left ( t \right )

to be non-singular on a deleted neighborhood of

t
0

{t^0}

. If, in addition,

F
(
t
)

F(t)

is differentiate at

t
0

{t^0}

then a corollary to this criterion yields a differential condition that is sufficient for such isolation of a point of singularity. Applications of corollary are given, including in particular for

n
=
1

n = 1

the correction of a result stated by George and Gunderson [1].
On the stability of solutions of a second-order differential equation
Authors: Infeld E.
Published: December 2016

This paper deals with the Hill differential equation
\[

d
2

y

/

d

x
2

+

r

1
−
2
a
cos
⁡
x
+

a
2

y
=
0

{d^2}y/d{x^2} + \frac {r}{{1 - 2a\cos x + {a^2}}}y = 0

\]
Although this equation looks more difficult than Mathieu’s, it can be dealt with some-what more simply than the latter. Stability criteria are obtained in terms of

r
r

and

a
a

(at least in principle).
The notion of approximate eigenvalues applied to an integral equation of laser theory
Authors: Landau H. J.
Published: December 2016

The integral operator with kernel

(

i
η

/

π

)

1

/

2

exp
⁡

[

−
i
η

(

x
−
y

)

2

]

{\left ( {i\eta /\pi } \right )^{1/2}}\exp \left [ { - i\eta {{\left ( {x - y} \right )}^2}} \right ]

on the interval

|
x
|

\left | x \right |

,

|
y
|

≤
1

\left | y \right | \le 1

serves to model the behavior of a class of lasers. Although the kernel is simple, it is not Hermitian; this presents a major obstacle to a theoretical understanding of the equation—indeed, even the existence of eigenvalues is difficult to prove. We here introduce a definition of approximate eigenvalues and eigenfunctions, and argue that these will model the physical problem equally well. We then show that, for

η
\eta

sufficiently large, each point

λ
\lambda

with

|
λ
|

=
1

\left | \lambda \right | = 1

is an approximate eigenvalue, and that the number of mutually orthogonal approximate eigenfunctions corresponding to

λ
\lambda

grows faster than any constant multiple of

η

\sqrt \eta

This confirms a conjecture of J. A. Cochran and E. W. Hinds, supported by numerical evidence. In physical terms, it shows that for large Fresnel number the laser cannot be expected to settle to a single mode.
Some exact solutions of Burgers-type equations
Authors: Sachdev P. L.
Published: December 2016

A class of Burgers-type equations, reducible through a generalized Hopf—Cole transformation to a linear diffusion equation, are treated by similarity methods. New exact solutions of these equations are obtained and related to the wellknown solutions of the standard Burgers equation. Physical applications of these solutions are indicated.
Complementary variational principles for large deflections of a cantilever beam
Authors: Reeves R. I.
Published: December 2016

Recent results on complementary variational principles and error bounds are applied to two problems concerning the large deflection of a horizontal cantilever. The results are illustrated by obtaining accurate variational solutions in the form of simple polynomials.
On convergence of the finite-element method for a class of elastic-plastic solids
Authors: Havner K. S. , Patel H. P.
Published: December 2016

A proof of convergence of the finite-element method in rate-type, quasistatic boundary value problems is presented. The bodies considered may be discretely heterogeneous and elastically anisotropic, their plastic behavior governed by history-dependent, piecewise-linear yield functions and fully coupled hardening rules. Elastic moduli are required to be positive-definite and plastic moduli nonnegative-definite. Precise and complete arguments are given in the case of bodies whose surfaces are piecewise plane.
An integro-differential equation with application in heat flow
Authors: MacCamy R. C.
Published: December 2016

The problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline left-parenthesis x comma t right-parenthesis equals integral Subscript 0 Superscript t Baseline a left-parenthesis t minus tau right-parenthesis StartFraction partial-differential Over partial-differential x EndFraction sigma left-parenthesis u Subscript x Baseline left-parenthesis x comma tau right-parenthesis right-parenthesis d tau plus f left-parenthesis x comma t right-parenthesis comma 0 greater-than x greater-than 1 comma t greater-than 0 comma u left-parenthesis 0 comma t right-parenthesis identical-to u left-parenthesis 1 comma t right-parenthesis identical-to 0 u left-parenthesis x comma 0 right-parenthesis equals u 0 left-parenthesis x right-parenthesis">

u
t

(

x
,
t

)

=

∫
0
t

a

(

t
−
τ

)

∂

∂
x

σ

(

u
x

(

x
,
τ

)

)

d
τ
+
f

(

x
,
t

)

,

0
>
x
>
1
,

t
>
0
,

u

(

0
,
t

)

≡
u

(

1
,
t

)

≡
0

u

(

x
,
0

)

=

u
0

(
x
)

{u_t}\left ( {x, t} \right ) = \int _0^t {} a\left ( {t - \tau } \right )\frac {\partial }{{\partial x}}\sigma \left ( {{u_x}\left ( {x,\tau } \right )} \right )d\tau + f\left ( {x, t} \right ), \qquad 0 > x > 1, \qquad t > 0, \\ u\left ( {0,t} \right ) \equiv u\left ( {1,t} \right ) \equiv 0 \qquad u\left ( {x, 0} \right ) = {u_0}\left ( x \right )

\]
is considered. Asymptotic stability theorems for the solution are established under appropriate conditions on

a
a

,

σ
\sigma

and

f
f

. The conditions on

a
a

are of frequency domain type and are related to ones used previously in the study of Volterra integral equations,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove u With dot equals minus integral Subscript 0 Superscript t Baseline a left-parenthesis t minus tau right-parenthesis g left-parenthesis u left-parenthesis tau right-parenthesis right-parenthesis d tau plus f left-parenthesis t right-parenthesis">

u
˙

=
−

∫
0
t

a

(

t
−
τ

)

g

(

u

(
τ
)

)

d
τ
+
f

(
t
)

\dot u = - \int _0^t a \left ( {t - \tau } \right )g\left ( {u\left ( \tau \right )} \right )d\tau + f\left ( t \right )

\]
on a Hilbert space. An existence theorem for the problem is established under smallness assumptions on

f
f

and

u
0

{u_0}

This theorem is related to one by Nishida for the damped non-linear wave equation,
\[

u

t
t

+
α

u
t

−

∂

∂
x

σ

(

u
x

)

=
0

{u_{tt}} + \alpha {u_t} - \frac {\partial }{{\partial x}}\sigma \left ( {{u_x}} \right ) = 0

\]
. It is shown that the problem is related to a theory of heat flow in materials with memory.
Zone estimates in the elastic-plastic torsion problem
Authors: Yin Wan Lee
Published: December 2016

Using continuity and jump conditions across the elastic-plastic interface, we show that the curvature of the shearing stress lines in the plastic zone of a simply connected bar under torsion is bounded above by a number proportionate to the twisting angle. For sufficiently large torsion the above geometrical condition defines a region in the cross-section which is a priori elastic. This lower bound for the elastic zone in the sense of set inclusion is supplemented by an upper bound for the zone area.
Transient response of an acoustic half-space to a rotating point load
Authors: Watanabe Kazumi
Published: December 2016

This paper presents a solution of three-dimensional problems for nonuniformly moving loads. By using Cagniard’s technique, the problem is reduced to a change in the order of integration. The technique has no restrictions on the velocity variation and on the trajectory of the load.
A model for one-dimensional, nonlinear viscoelasticity
Authors: MacCamy R. C.
Published: December 2016

The problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t t Baseline equals a left-parenthesis 0 right-parenthesis sigma left-parenthesis u Subscript x Baseline right-parenthesis Subscript x Baseline plus integral Subscript 0 Superscript t Baseline ModifyingAbove a With dot left-parenthesis t minus tau right-parenthesis sigma left-parenthesis u Subscript x Baseline right-parenthesis Subscript x Baseline d tau plus f comma 0 greater-than x greater-than 1 comma t greater-than 0 comma u left-parenthesis 0 comma t right-parenthesis identical-to u left-parenthesis 1 comma t right-parenthesis identical-to 0 comma u left-parenthesis x comma 0 right-parenthesis equals u Subscript o Baseline left-parenthesis x right-parenthesis comma u Subscript t Baseline left-parenthesis x comma 0 right-parenthesis equals u 1 left-parenthesis x right-parenthesis">

u

t
t

=
a

(
0
)

σ

(

u
x

)

x

+

∫
0
t

a
˙

(

t
−
τ

)

σ

(

u
x

)

x

d
τ
+
f
,

0
>
x
>
1
,

t
>
0
,

u

(

0
,
t

)

≡
u

(

1
,
t

)

≡
0
,

u

(

x
,
0

)

=

u
o

(
x
)

,

u
t

(

x
,
0

)

=

u
1

(
x
)

{u_{tt}} = a\left ( 0 \right )\sigma {\left ( {{u_x}} \right )_x} + \int _0^t {\dot a} \left ( {t - \tau } \right )\sigma {\left ( {{u_x}} \right )_x}d\tau + f, \qquad 0 > x > 1, \qquad t > 0, \\ u\left ( {0, t} \right ) \equiv u\left ( {1, t} \right ) \equiv 0, \\ u\left ( {x, 0} \right ) = {u_o}\left ( x \right ), \qquad {u_t}\left ( {x, 0} \right ) = {u_1}\left ( x \right )

\]
is considered. The essential hypotheses are that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a left-parenthesis t right-parenthesis equals a Subscript normal infinity Baseline plus upper A left-parenthesis t right-parenthesis comma a Subscript normal infinity Baseline greater-than 0 comma upper A element-of upper L Superscript 1 Baseline left-parenthesis 0 comma normal infinity right-parenthesis comma left-parenthesis negative 1 right-parenthesis Superscript k Baseline a Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis t right-parenthesis greater-than-or-equal-to 0 comma k equals 0 comma 1 comma 2 comma sigma left-parenthesis 0 right-parenthesis equals 0 comma sigma prime left-parenthesis xi right-parenthesis greater-than-or-equal-to epsilon greater-than 0">

a

(
t
)

=

a
∞

+
A

(
t
)

,

a
∞

>
0
,
A
∈

L
1

(

0
,
∞

)

,

(

−
1

)

k

a

(
k
)

(
t
)

≥
0
,
k
=
0
,
1
,
2
,
σ

(
0
)

=
0
,

σ
′

(
ξ
)

≥
ϵ
>
0

a\left ( t \right ) = {a_\infty } + A\left ( t \right ), {a_\infty } > 0, A \in {L^1}\left ( {0, \infty } \right ), \\ {\left ( { - 1} \right )^k}{a^{\left ( k \right )}}\left ( t \right ) \ge 0, k = 0, 1, 2, \sigma \left ( 0 \right ) = 0, \sigma ’\left ( \xi \right ) \ge \epsilon > 0

\]
. It is shown that the problem has a unique classical solution for all

t
t

if the data are sufficiently small and, if

f
f

is suitably restricted, this solution tends to zero as

t
t

tends to infinity. It is shown that the problem provides a special model for elastic materials which exhibit a memory effect.
Some maximum principles for nonlinear elliptic boundary-value problems
Authors: Schaefer Philip W.
Published: December 2016

The Hopf maximum principles are utilized to obtain maximum principles for functions which are defined on solutions of nonlinear, second-order elliptic equations subject to Dirichlet, Robin, or mixed boundary conditions. The principles derived may be used to deduce bounds on important quantities in physical problems of interest.
On the material time derivative of arbitrary order
Authors: Tao L. N.
Published: December 2016

The material time derivative of arbitrary order of a function of space and time is established. In the case of a function of a single space variable, it is reduced to Faa de Bruno’s formula for derivatives of a function of a function.
Large finite strain membrane problems
Authors: Wu Chien H.
Published: December 2016

Nonlinear membrane problems involving large finite strains are considered. It is found that explicit asymptotic solutions are possible for a rather large class of problems. Two distinct types of asymptotic solutions, roughly depending on whether the strain energy density function is dominated by

I
1

{I_1}

or

I
2

{I_2}

, are found to exist.
Fundamental solutions for ideal fluids in uniform motion
Authors: Dragoş Lazăr
Published: December 2016

In this paper the perturbations produced by mass, momentum and energy sources in the uniform flow of an ideal fluid are determined. The case of instantaneous sources is considered. The matrix solutions thus determined are by definition the fundamental matrices of the systems of equations of fluid mechanics. As an application, the perturbations caused by sources acting permanently for

t
>
0

t > 0

as well as the perturbations produced by sources moving along the direction of the free flow or perpendicular to this direction are determined.
A similarity solution to a nonlinear diffusion equation of the singular type: a uniformly valid solution by perturbations
Authors: Babu D. K. , van Genuchten M. Th.
Published: December 2016

The one-dimensional nonlinear diffusion equation is solved by a perturbation technique. It is assumed that the diffusivity varies as a nonnegative power of the concentration, while the concentration at the supply surface varies as another power of time. The resulting similarity solution that has been derived via a perturbation scheme remains valid for all times and all distances. Explicit series formulae are also derived for the location of the concentration front. Since diffusivity vanishes at zero concentration, the study here pertains to a singular problem.
Nonlinear analysis of a twisted axially loaded elastic rod
Authors: Zachmann David W.
Published: December 2016

A slender, inextensible elastic rod is acted upon by a twisting couple and an axial load. The position of the rod’s centerline is determined by two fourth-order, coupled, nonlinear boundary value problems, each of which contains two eigenparameters. These equilibrium equations admit the trivial solution for all values of the eigenparameters, i.e., for any axial load and any twisting couple. The linearized equilibrium equations have a countable number of eigencurves. Through using the implicit function theorem for Banach spaces it is shown that from each of the eigencurves of the linear problem there bifurcates a two-parameter sheet of nontrivial solutions of the nonlinear equilibrium equations.
On the numerical solution of singular integro-differential equations
Authors: Ioakimidis N. I. , Theocaris P. S.
Published: December 2016

A method of numerical solution of a sufficiently wide class of Cauchy-type singular integrodifferential equations along a straight finite interval is presented. This method consists of approximating the integrals in such an equation by using appropriate numerical integration rules and appropriately-selected collocation points and reducing such an equation to a system of linear algebraic equations. This technique constitutes a direct generalization of the corresponding methods of numerical solution of Cauchy-type singular integral equations and presents some advantages over the classical Multhopp method of numerical solution of Cauchy-type singular integrodifferential equations, to which it reduces in some special cases. An application to a specific equation is also made.
Infinitesimal contact transformations and their applications to abstract growth patterns
Authors: Frolow Igor
Published: December 2016

Several abstract mathematical models of two-dimensional objects or “organisms” are introduced. The objects are abstractions, created by the combination of pattern-theoretic generators. They are deformed by what are shown to be infinitesimal contact transformations. Analytical methods are utilized to determine some of the resulting forms and a computer is programmed to display graphically the developing patterns.
Second-order solution of a nonlinear wave equation
Authors: Lardner R. W.
Published: December 2016

The method of multiple scales is used to investigate the nonlinear wave equation

u

t
t

−

u

x
x

=
ϵ

(

2

u
x

u

x
x

−
2
λ

u
t

)

{u_{tt}} - {u_{xx}} = \epsilon \left ( {2{u_x}{u_{xx}} - 2\lambda {u_t}} \right )

.
Vibrations of a pendulum consisting of a bob suspended from a wire: the method of integral equations
Authors: Ahn Hyun J.
Published: December 2016

The vibrations of a vertical pendulum consisting of a bob suspended from a wire are studied by the method of integral equations and the composite method, respectively. The composite method combines the minimum principles and the method of integral equations. This problem consists of the fourth-order differential equation and the boundary conditions dependent on the eigenvalue parameter. Lower bounds are established for the lowest natural frequencies by both methods. Numerical results are presented. Both theoretical and computational efficiencies are illustrated and the method of integral equations is stressed.
Variational solutions of the Thomas-Fermi equation
Authors: Anderson N. , Arthurs A. M.
Published: December 2016

Variational solutions of the Thomas-Fermi equation are examined in the context of complementary extremum principles. A new one-parameter trial function is found to provide an accurate representation of the solution.
Evolution problems for a class of dissipative materials
Authors: Suquet Pierre-M.
Published: December 2016

This work examines the problems of dynamic and quasi-static evolution for a large class of dissipative materials, including viscoplastic, viscoelastic, and elastic perfectly plastic materials. We show that when the potential of dissipation is regular, the displacement solution is regular; however, in the case of perfect plasticity, where the potential is irregular, the solution can be discontinuous. A suitable framework is used in order to account for these discontinuities. Existence theorems are stated, and the boundary conditions are discussed. The evolution equations encountered in this work are strongly nonlinear but with a monotone time-dependent nonlinearity. A direct method of resolution is proposed, since the known results do not apply in this case.
Wrinkling in finite plane-stress theory
Authors: Wu Chien H. , Canfield Thomas R.
Published: December 2016

A general and complete formulation is given for the wrinkling phenomenon in the context of finite plane-stress theory. The planar portion of the true three-dimensional displacement field, called the pseudo-displacement field, is used as a basis for the necessary kinematic analysis. It is assumed that the principal directions associated with the pseudo-deformation field are the same as those associated with the true stress field. The true stress field is governed by equilibrium and the assumption that one of the principal stresses vanishes, and hence is statically determinate. The difference between the pseudo-strain and the true strain calculated from the true stress is a new tensor, called the wrinkle-strain tensor, and serves as a measure of the wrinkliness of the surface.
On the vibrations of a heterogeneous string
Authors: Synge J. L.
Published: December 2016

It is shown that the vibrations of a heterogeneous string can be represented by an infinite series, each term of which is the result of applying a linear integral operator to a function of position and time furnished by the initial data. The method applies also to plane waves of compression or shear in a heterogeneous elastic solid for which the elastic constants and density are functions of only one coordinate and the waves move in the direction of that coordinate.
Normal compression waves scattering at a flat annular crack in an infinite elastic solid
Authors: Shindo Yasuhide
Published: December 2016

The Hankel transform is used to obtain a complete solution for the dynamic stresses and displacements around a flat annular surface of a crack embedded in an infinite elastic solid, which is excited by normal compression waves. The singular stresses near the crack tips are obtained in closed elementary forms, while the magnitude of these stresses, governed by the dynamic stress-intensity factors, is calculated numerically from a singular integral equation of the first kind. The variations of the dynamic stress-intensity factors with the normalized frequency for the ratio of the inner radius to the outer one and Poisson’s ratio are shown graphically.
Some uniqueness theorems for linear water waves
Authors: Wen Shih Liang , Sastry M. S. K.
Published: December 2016

Uniqueness theorems, under less restrictive conditions at infinity, for water of uniform finite depth, for the cliff problem and for waves on uniform sloping beaches are proved using the Phragmén-Lindelöf principle.
Three-dimensional Oseen flow past a flat plate
Authors: Homentcovschi Dorel
Published: December 2016

On the basis of the Oseen approximation the three-dimensional flow of a viscous incompressible fluid past a flat plate is studied. A system of two integral equations for determining the drag and the lateral force on the plate and an integral equation for the lift are obtained. The paper gives the asymptotic form of the integral equation for the lift, for high Reynolds numbers. In the inviscid limit the integral equation of the lifting surface theory is obtained. Lifting line theory and slender wing theory in weak viscous flow are discussed. Viscosity corrections are given for some particular wings: elliptic wings of high aspect ratio and slender delta wings.
On a mixed problem for the M‘Kendrick-Von Foerster equation
Authors: Yamada Akio , Funakoshi Hiroumi
Published: December 2016

A mixed problem for the M’Kendrick–Von Foerster equation is solved explicitly: existence, uniqueness and regularity theorems are proved as well as several integral formulas.
On static similarity deformations for isotropic materials
Authors: Hill James M.
Published: December 2016

For static deformations of isotropic materials satisfying the principle of material indifference, the governing equations remain invariant under arbitrary orthogonal rotations of the material and spatial coordinate systems. Moreover, the basic equations are also invariant under the same change of length scale for both coordinate systems. The general functional form of the similarity deformation corresponding to these invariances is deduced. Although these invariances involve seven arbitrary constants, it is shown that by an appropriate selection of coordinates only three arbitrary constants are involved in an essential way.
The approach to normality of the solutions of random boundary and eigenvalue problems with weakly correlated coefficients
Authors: Boyce William E. , Xia Ning Mao
Published: December 2016

A general class of linear self-adjoint random boundary value problems with weakly correlated coefficients is considered. The earlier result that the distribution function of the solution approaches the normal as the correlation length

ϵ
\epsilon

tends to zero is generalized somewhat. Correction terms are derived that yield estimates for the distribution function when

ϵ
\epsilon

is small but nonzero. The results are also applied to the eigenvalues and eigenfunctions of a corresponding class of random eigenvalue problems. The discussion is given in terms of second-order equations, but extensions to higher-order problems are readily apparent.
Alternative variational formulations for first order partial differential systems
Authors: Geffen Nima
Published: December 2016

Two simple alternative variational principles are derived for a first order differential system with appropriate initial and boundary conditions. The problem is assumed to be well posed, and may be nonlinear, nonhomogeneous and of any type (i.e. elliptic, hyperbolic or mixed). Primitive variables are used, which allows for non-smooth solutions. Redundancy in the system is considered, and applications to fluidynamics and electrodynamics fields given.
A nonstandard nonlinear boundary value problem for harmonic functions
Authors: Geffen Nima
Published: December 2016

Existence and uniqueness are proved for a nonstandard, nonlinear boundary-value problem for 2-dimensional harmonic functions. The problem models an ideal flow-field, and a few cases of applied interest are considered. Slight generalizations are derived in the appendix.
A perturbation method for solving a quadratic evolution equation
Authors: Hilgers John W. , Spahn Robert J.
Published: December 2016

A quadratic evolution equation of the form
\[

u
˙

=
L
u
+
ϵ
Q
u

\dot u = Lu + \epsilon Qu

\]
is considered where

L
L

and

Q
Q

are particular linear and quadratic integral operators respectively. This equation has been proposed to describe the variation with time of

u
(
x
,
t
)

u(x,t)

, the volume density of an ensemble of particles undergoing concurrent coalescence and fracture.
On an integral formulation for moving boundary problems
Authors: Dewynne Jeffrey N. , Hill James M.
Published: December 2016

The classical moving boundary problems arising in the freezing or chemical reaction of spheres, cylinders and slabs are considered. An integral method is employed to formally effect the integration of the motion of the moving boundary. This formal integration permits upper and lower bounds to be deduced for the motion and in particular simple upper and lower bounds are established for the time to complete freezing or reaction (that is, when the moving boundary reaches the centre of the sphere or cylinder). In addition an improved second upper bound on the motion is achieved by demonstrating that the dimensionless temperature or concentration is bounded above by the standard pseudo steady state approximation. The use of the integral formulation as an iterative scheme and the generalisations for a time dependent surface condition and a non nlinear diffusivity are also briefly considered.
Nonuniform motion of an edge dislocation in an anisotropic solid. I
Authors: Markenscoff Xanthippi , Ni Lu Qun
Published: December 2016

The two-dimensional problem of the nonuniform motion of an edge dislocation in an anisotropic solid (regular hyperbolic case) is solved by means of Laplace transforms with inversion according to the Cagniard-de Hoop technique. The solution is also evaluated asymptotically at the saddle points on the Cagniard-de Hoop contour which lies on a multi-sheet Riemann surface, the singular points of which are examined in detail also in connection to the slowness surface. The stress field is square root singular near the wavefront for a motion of constant velocity starting from rest, and 2/3 singular near the cusp-tips. For general nonuniform motion the stress at the wavefront is obtained as well and an example is given for a motion starting with constant acceleration.
On periodic traveling wave solutions of Boussinesq equation
Authors: Liu Bao Ping , Pao C. V.
Published: December 2016

This paper is concerned with periodic traveling wave solutions of a generalized Boussinesq equation in the form

u

t
t

=
α

u

x
x
x
x

+

(

f
0

(
u
)

)

x
x

{u_{tt}} = \alpha {u_{xxxx}} + {({f_0}(u))_{xx}}

. The basic approach to this problem is to establish an equivalence relation between a corresponding periodic boundary value problem for the traveling wave solution and a Hammerstein type integral equation. This integral representation generates a compact operator in the space of continuous periodic functions of the given period. It is shown by restricting the integral operator on a suitable domain that the Boussinesq equation has the trivial solution as well as a nonconstant periodic traveling wave solution. Special attention is given to the traditional Boussinesq equation where

f
0

(
u
)
=
a

u
2

{f_0}(u) = a{u^2}

. Both of the so called “good” and “bad” Boussinesq equation are treated and the existence of nonconstant traveling wave solution is discussed.
A uniformly convergent series for Sturm-Liouville eigenvalues
Authors: Cope Davis
Published: December 2016

For the regular Sturm-Liouville problem with equation

y
+
(
λ
−
q
(
x
)
)
y
=
0

y + (\lambda - q(x))y = 0

on

0
≤
x
≤
π

0 \le x \le \pi

, there are well-known asymptotic expansions for the eigenvalues and eigenfunctions. We show that these asymptotic expansions can be replaced by convergent series for sufficiently large eigenvalues. Convergence is uniform on the interval

0
≤
x
≤
π

0 \le x \le \pi

and uniform with respect to the eigenvalues, in the sense that a single majorant bounds all series. The basic idea is to replace the asymptotic results, which use an expansion of powers of

n

−
1

o
r

(
n
+
1

/

2

)

−
1

{n^{ - 1}}or{(n + 1/2)^{ - 1}}

for integers

n
n

, by a series in powers of

μ

−
1

{\mu ^{ - 1}}

, where

μ
2

{\mu ^2}

is an eigenvalue for the corresponding constant coefficient Sturm-Liouville problem with equation

y
+
λ
y
=
0

y + \lambda y = 0

.
Asymptotic solution near the apex of an elastic wedge with curved boundaries
Authors: Ting T. C. T.
Published: December 2016

When the boundaries of an elastic wedge are straight lines, the asymptotic solution near the apex

r
=
0

r = 0

of the wedge is simply a series of eigenfunctions of the form

r
λ

f
(
θ
,
λ
)

{r^\lambda }f(\theta ,\lambda )

in which

(
r
,
θ
)

(r,\theta )

is the polar coordinate with origin at the wedge apex and

λ
\lambda

is the eigenvalue. When the wedge boundaries are curved, the eigenvalues remain the same but the curvatures of the boundaries change the form of the eigenfunctions. The eigenfunction associated with a

λ
\lambda

contains not only the term

r
λ

{r^\lambda }

, but also

r

λ
+
1

,

r

λ
+
2

,
.
.
.

{r^{\lambda + 1}},{r^{\lambda + 2}},...

In some cases it also contains the term

r

λ
+
1

(
l
n
r
)

{r^{\lambda + 1}}(ln r)

. Therefore, the second and higher order terms of asymptotic solution are not simply the second and next eigenfunctions. Examples are given for the first few terms of asymptotic solution for wedges with wedge angle

π
\pi

and

2
π

2\pi

. The latter corresponds to a crack with curved free boundaries and we show that there exists a term

r

1

/

2

(
I
n
r
)

{r^{1/2}}(In r)

besides the familiar terms

r

−
1

/

2

{r^{ - 1/2}}

.
Modulational instability in magnetic fluids
Authors: Malik S. K. , Singh M.
Published: December 2016

A weakly nonlinear evolution of two dimensional wave packets on the surface of a magnetic fluid in the presence of a tangential magnetic field is presented taking into account the surface tension. It is shown that the magnetic field has a stabilizing influence on the modulational instability.
Approximate linear stability analysis of a model of adiabatic shear band formation
Authors: Burns Timothy J.
Published: December 2016

The formation of adiabatic shear bands in ductile metals under dynamic loading conditions is generally thought to result from a material instability, which is associated with a peak in the curve of engineering plastic flow stress vs. engineering shear strain. This instability arises from the effect of thermal softening, caused by irreversible adiabatic heating, which counteracts the tendency of the material to harden with increasing plastic strain. An approximate linear stability analysis of a one-dimensional rigid-thermoviscoplastic model, based on data taken from dynamic torsion experiments on thin-walled tubes of mild steel, shows that shear band formation in this situation can be interpreted as a bifurcation from a homogeneous simple shearing deformation which occurs at the peak in the homogeneous stress vs. strain curve. The asymptotic method of multiple scales is used to show that the growth rate of small perturbations on the homogeneous deformation is controlled by the ratio of the slope of the homogeneous stress vs. strain curve to the material viscosity, i.e., the rate of change of the plastic flow stress with respect to the strain-rate. In addition, it is shown that this growth rate is essentially independent of wavelength in any small perturbation. Numerical methods are used to show that this growth rate beyond the bifurcation point may not be sufficiently large for the model to account for the experimental data, and some suggestions are made on how to modify the constitutive equation so that it better fits the experimental data.
A note on Langford’s cylinder functions 𝑐_{𝑛}(𝑧,𝑧₀) and 𝑒_{𝑛}(𝑧,𝑧₀)
Authors: Hill James M. , Dewynne Jeffrey N.
Published: December 2016

New general expressions are given for Langford’s cylinder functions, which occur in solutions of the Cauchy problem for the heat equation in cylindrical co-ordinates. These formulae are deduced by means of generating functions. In addition a new technique is used to obtain Langford’s formal series, new basic formulae connecting Langford’s various cylinder functions are established and their relevance in a formal series solution of a moving boundary problem is noted.
The dynamic 2D analysis of a concentrated force near a semi-infinite crack
Authors: Brock L. M.
Published: December 2016

An exact dynamic 2D solution for a concentrated force near a stationary semi-infinite crack in an unbounded plane can be used in the transient analysis of wave-scattering problems. Direct approaches to obtaining the solution, however, are complicated by the existence of a characteristic length. A less direct approach is used here which circumvents these complications. As an example, the dynamic stress intensity factors are derived and studied for their behavior w.r.t. time and concentrated force-crack edge orientation.
Some results on the optimal spacing of measurements in the identification of structural systems
Authors: Udwadia F. E.
Published: December 2016

This paper deals with finding the optimal measurement locations for a structural system modelled by a single-degree-of-freedom oscillator, so that any one of the parameters to be identified can be estimated with a minimum variance. The measurements are assumed to be taken in a noisy environment, and the paper addresses both linear and nonlinear, nonhysteretic systems. Besides the analytical relations deduced for the optimal measurement locations, it is found that, in general, there may exist measurement locations at which no additional information on the parameter under consideration is generated. For the linear case, the optimal measurement locations are found to be independent of the system response and the actual values of the parameters to be identified. They solely depend on the nature of the excitation used in the identification procedure. Analytical results relating to the optimal measurement locations for minimizing the sum of the variances of the estimates of some of the parameters are also provided.
Invariance group properties and exact solutions of equations describing time-dependent free surface flows under gravity
Authors: Sachdev P. L. , Philip Varughese
Published: December 2016

Using the method of infinitesimal transformations, a 6-parameter family of exact solutions describing nonlinear sheared flows with a free surface are found. These solutions are a hybrid between the earlier self-propagating simple wave solutions of Freeman, and decaying solutions of Sachdev. Simple wave solutions are also derived via the method of infinitesimal transformations. Incomplete beta functions seem to characterize these (nonlinear) sheared flows in the absence of critical levels.
Electrohydrodynamic solitons in Kelvin-Helmholtz flow: the case of a normal field in the absence of surface charges
Authors: Elshehawey Elsayed
Published: December 2016

Nonlinear electrohydrodynamic Kelvin—Helmholtz instability conditions are investigated. A charge-free surface separating two semi-infinite dielectric streaming fluids influenced by a normal electric field is subjected to nonlinear deformations. The method of multiple-scale perturbations is used in order to obtain two nonlinear Schrödinger (NLS) equations describing the behavior of the disturbed system. The stability and instability conditions of the perturbed system are discussed analytically. One of the two NLS equations is used to obtain the electrohydrodynamic (EHD) nonlinear cutoff wave number separating stable and unstable disturbances while the other equation is used to describe analytically the necessary condition for stability and instability for the system. For unstable cases, the solution starting with a solitary wave degenerates into a finite number of EHD solitons.
A third mixed boundary value problem on a sphere
Authors: Johnson Harry L.
Published: December 2016

The paper is concerned with the existence of a classical solution of a mixed third boundary value problem on a sphere. The existence is proved by reducing the problem to a Fredholm integral equation that has a unique solution. Various consequences of the existence theorem are mentioned and some numerical results are given.
A uniqueness theorem for thermoelastic shells with generalized boundary conditions
Authors: Rubin M. B.
Published: December 2016

The uniqueness of the solution of initial, mixed boundary value problems for linear thermoelastic shells is reconsidered within the context of recent developments in the thermomechanical theory of a Cosserat surface [4]. Fairly general boundary conditions are considered which allow mechanical contact with linear elastic media and thermal radiation on the boundary curve of the Cosserat surface and on the major surfaces of the shell.