Abstracts
Boundary value problem for a kind of third order quasi-linear
singularly perturbed differential equations
Lihua Chen
Fujian Normal University
Abstract
This paper studied the singularly perturbed boundary value problem
for a kind of third order quasi-linear differential equations by the
method of boundary layer functions. Under the appropriate conditions, the
existence and uniqueness of the solutions are proved And the uniformly
validity of its asymptotic solutions are given out.
Finite element analysis for singularly perturbed advection–diffusion
Robin boundary values problem
Songlin Chen
Anhui University of Technology
Abstract
We consider a singularly perturbed advection–diffusion two-point
Robin boundary value problem whose solution has a single boundary layer.
Based on piecewise linear polynomial approximation, finite element method
is applied on the problem. Estimation of the error between solution and
the finite element approximation is given in energy norm on shishkin-type
mesh.
A perturbation method which achieves accurate point-to-point ray
traces and its context
George Daglish
University of Surrey London, United Kingdom
Abstract
The work to be described here is a small part of a project aimed at
providing, in the first instance, the necessary facets which together
would form a Simulation System in which arrival times for P-wave species
can to be compared with the onsets of Love and Rayleigh waves.
Work on this system is largely motivated by the quest for means by which
“oceanic” Earthquakes can be identified in time to give warning of
possible Tsunamis.
It may well be possible to rapidly collect and process Surface-wave
Information, for Epicentral Location, with a fair degree of automation,
using automated “Picks” of energy onsets within Seismographic
recordings, together with combinatorial scans under the equation system:
 c t   t  
x  a i  R 2 cos  0 i
 ; i  0, n  1; n  6.
R


(0.1)
This system uses great circle arcs to locate Epicentral Parameters as the
vector:
x
y
 
z
 
 t 
 c0 
In this, x   x, y, z  is the Cartesian position of the Epicentre in a
Space-Frame, rotating with the Reference Sphere, whose radius is R. The
origin of this Space-Frame coincides with the centre of the Reference
Sphere. The set
a i 
represents the coordinates of the Seismographic
Stations in the same frame of reference, while the set
ti  represents the
set of onset timings taken relative to the Lead Station such that t0  Zero.
 t and c0 are the Time-to-Origin and an estimate of wave front velocity,
respectively.
To the above overall end, work was undertaken to determine if, and within
what limits, such Epicentral Locations could be found – in time to provide
adequate warnings of possible Tsunamis.
Within this line of investigation, there arose the need to provide a system
for the simulation of rays propagating through layered media, in which
either the refraction was constant with depth (within each layer), or,
in which the refractive power was varying linearly with depth (also
“intra layer”).
In the case of such rays it was necessary to create a scenario for two
cases:
1. Rays emanating from points at given “take-off” angles and thus
defining their own points of arrival.
2. Rays emanating from given points but whose “take-off” angles are
to be modified such that specified arrival points are reached.
In both these cases the rays are to travel through a stratified or layered
structure from a given depth to the surface.
In the simulation under consideration here, the creation of an accurate
Point-to-Point Ray is provided by a Perturbation Method, moreover a
Lagrangian one.
There are several previously described Perturbation Methods for achieving
such results, other than the Lagrangian, which include:
1. The Hamiltonian Formulation Approach.
2. “Hybrid” Perturbation.
3. An Approach using Fermat’s Principle.
However, only the Lagrangian approach that was used in the Simulation,
mentioned above, will be described in detail here.
In this technique the perturbation factor " " seeks to indirectly modify
an initially given “take-off” angle until the successive modifications
cause the ray to strike a prescribed target point p   x, z  . While this
happens " " will have been asymptotically approaching Zero.
[A lead reference for this kind of investigation is: Thurber, C.H. and Rabinowitz,
N., “Advances in Seismic Event Location”, Kluwer Academic Publishers, 2000].
Singular perturbation of three-pointed value problem of higher-order
nonlinear differential equation
Haiyun Ding
East China Normal University
Abstract
A class of singular perturbation of three-pointed value problem
of high-order nonlinear differential equation is examined in this
paper. Using the method of boundary functions, the asymptotic
solution of this problem is shown and proved to be uniformly
effective. The existence and uniqueness of the solution for the
system has also been proved. Numerical result is presented, which
supports the theoretical result.
A novel mathematical modeling of multiple scales for a class of two
dimensional singular perturbed problems
Liangliang Du
Tongji University
Abstract
A novel mathematical modeling of multiple scales (NMMMS) is presented
for a class of singular perturbed problems with both boundary or
transition layers in two dimensions. The original problems are converted
into a series of problems with different scales, and under these different
scales, each of the problems is regular. The rational spectral collocation
method (RSCM) [1] is applied to deal with the problems without
singularities. NMMMS can still work successfully even when the parameter
is extremely small (“  ”= 10-25 or even smaller). A brief error estimate
for the model problem is given in section 2. Numerical examples are
implemented to show the method is of high efficiency and accuracy.
Solvability of a second-order differential equation with integral
boundary condition at resonance on an unbounded domain
Zengji Du
Xuzhou Normal University
Abstract
This paper deals with the solvability of the second-order integral
boundary value problem at resonance on a half-line
x(t )  f (t , x(t ), x(t )), t  (0, ),
1
x(0)   x(t )dt , lim x(t )  0,
0
t 
and
x(t )  f (t , x(t ), x(t ))  e(t ), t  (0, ),
1
x(0)   x(t )dt , lim x(t )  0,
0
t 
where f : [0, )  R  R is a S-Carathéodory function with respect to
2
L1[0, ), e :[0, )  R is continuous. In this paper, both of the boundary
value conditions are responsible for resonance. By using the coincidence
degree theory, we establish a new general existence result.
Slow-fast Bogdanov-Takens bifurcations
Freddy Dumortier
Hasselt University, Belgium
Abstract
The talk deals with perturbations from planar vector fields having
a line of zeros and representing a singular limit of Bogdanov-Takens (BT)
bifurcations.
We introduce, among other precise definitions, the
notion of slow-fast BT-bifurcation and we provide a complete study of the
bifurcation diagram and the related phase portraits. Based on geometric
singular perturbation theory, including blow-up, we get results that are
valid on a uniform neighbourhood both in parameter space and in the phase
plane. The talk is based on joint
work with Peter De Maesschalck.
High order finite volume methods for singular perturbation problems
Congnan He
Guangxi University for Nationalities
Abstract
In this paper we establish a high order finite volume method for the
fourth order singular perturbation problems. In conjunction with the
optimal meshes, the numerical solutions resulting from the method have
optimal convergence order. Numerical experiments are presented to verify
our theoretical estimates.
Stochastic finite element analysis in the impact boundary-layer
Lei Hou
Shanghai University
Abstract
Many materials perform the non-Newtonian property in the micron-scale
rheometry test.
In this paper a stochastic boundary-layer analysis on
the non-Newtonian equation is discussed for the mathematical and virtual
test methods in the auto-crash safety analysis. The stochastic and
asymptotic
analysis
are
given
for
the
Galerkin
solution
under
sufficiently smoothed conditions; the predicted 3-dimensional FEA
simulation and corrected stable Runge-Kutta method are also estimated in
such an iterative scheme.
The internal problems for singularly perturbed integro-differential
equation and difference-differential equation
Chuan Li
East China Normal University
Abstract
In this paper, under some conditions, a kind of singularly perturbed
second order integro-differential equation is considered. For some
specificity of the reduced solution, lead its solution has a corner layer
in the domain [a,b], an asymptotic expansion of the solution is developed
using boundary layer method, justification of the existence of the
solution and error estimates are given, and then an example is showed.
And further discussion is the Tikhonov system with an integrodifferential equation. For some particularity of this problem, we could
use some conclusion of the former problem to this system. For example,
when prove the existence of the solution by differential inequality, the
method of constructing the supper and lower solutions of the former
problem could be used in this.
At the last of this paper, in view of another internal layer, has
talked about the Tikhonov system with a difference-differential equation,
this time the internal layer is like the Contrast Steplike Structure. We
have constructed the asymptotic solutions of the left and right problems,
and proved the existence of the solution of the initial problem in
“connection” method.
On a nonlocal problem modeling Ohmic heating with variable
thermal conductivity
Fei Liang
Southeast University
Abstract
In this paper, we consider the nonlocal parabolic problem of the form
ut    (u 3u ) 
 exp(u 4 )
(  exp(u )dx)
4
2
, x  
2
, t  0,

with a homogeneous Dirichlet boundary condition, where  is a positive
parameter. For  to be an annulus, we prove that for each 0    
2
2
there corresponds a unique steady-state solution and u ( x, t ) is a global
in time-bounded solution, which tends to the unique steady-state solution
as t   uniformly in x . Whereas for   
stationary solution and if   2 
2
2
2 there is no
2 then u ( x, t ) blows up in finite
time for all x .
A numerical investigation of blow-up in reaction-diffusion problems
with traveling heat sources
Kewei Liang
Zhejiang University
Abstract
This paper studies the numerical solution of a reaction-diffusion
differential equation with traveling heat sources. According to the fact
that the locations of heat sources have been known, we present a novel
moving mesh algorithm for solving the problem. Several examples are
provided to demonstrate the efficiency of the new moving mesh method,
especially in two dimensional case. Moreover, numerical results
illustrate the speed of the movement of the heat source is critical for
blow-up.
On the homotopy multiple-variable method and its applications in
the interactions between nonlinear gravity waves
Shijun Liao
Shanghai Jiao Tong University
Abstract
A multiple-variable method is proposed to investigate the
interactions of fully-developed periodic traveling primary
waves.
This multiple-variable technique does not depend upon any small physical
parameters, but it logically contains the famous multiple-scale
perturbation methods.
By means of this technique, we show that the
amplitudes of all wave components are constant even if the wave resonance
condition is exactly satisfied.
Besides, it is revealed, maybe for the
first time, that there exist multiple solutions for the resonant waves.
Furthermore, a generalized resonance condition for n ( 2  n   )
arbitrary periodic traveling waves is given, which logically contains
Phillips' resonance condition and opens a way to investigate the
interaction of more than four traveling waves.
The singular perturbation of boundary value problem for the
third-order nonlinear vector integro-differential equation and its
application
Surong Lin
Fujian Radio and TV University
Abstract
In this paper, the singular perturbation of boundary value problem
to a class of third-order nonlinear vector integro-differential equation
is studied. Using the method of differential inequalities, under certain
conditions, the existence of perturbed solution is proved, the uniformly
valid asymptotic expansion for arbitrary order and the estimation of
remainder term are given. Finally, the results are applied to study
singularly perturbed boundary value problem to a nonlinear vector
fourth-order differential equation. The existence of solution and its
asymptotic estimation can be obtained conveniently.
Spherically symmetric standing waves for a liquid/vapor phase
transition model
Haitao Fan, Georgetown University
Xiao-Biao Lin, North Carolina State University, USA
Abstract
We study fluid flow involving liquid/vapor phase transition in a cone
shaped section, simulating the flow in fuel injection nozzles. Assuming
that the flow is spherically symmetric, and the fluid has high specific
heat, we look for standing wave solutions inside the nozzle. The model
is a system of viscous conservation laws coupled with a reaction-diffusion
equation. We look for two types of standing waves-Explosion and
Evaporation waves. If the diffusion coefficient, viscosity and typical
reaction time are small, the system is singularly perturbed. Transition
from
liquid
mixture to vapor occurs in an internal layer inside the
nozzle. First, matched formal asymptotic solutions are obtained. Internal
layer solutions are obtained by the shooting method. Then we look for a
real solution near the approximation.
Convergence of linear multistep methods for index-2 differential-
algebraic equations with a variable delay
Hongliang Liu
Xiangtan University
Abstract
Linear multistep methods (LMMs) are applied to index-2 nonlinear
differential-algebraic
corresponding
equations
convergence
with
results
are
a
variable
obtained
delay.
and
The
successfully
confirmed by some numerical examples. These results will contribute to
solving nonstandard delay singular perturbation problems.
Nonmonotone interior layer behavior of solutions of some quadratic
singular perturbation problems with high-order turning points
Shude Liu Jingsun Yao Huaijun Chen
Anhui Normal University
Abstract
We consider quadratic singular perturbation problems of the form
x  f (t ) x 2  g (t , x), a  t  b ，
（1）
x(a,  )  A, x(b,  )  B ，
（2）
where   0 is a small parameter, a, b (a  0  b) and A, B are constants.
Assume
[ H1 ]
there
exist
functions
uL (t )  C 2 [a, 0]
and
uR (t )  C 2 [0, b]
satisfying, respectively, the reduced problems
f (t )u  2  g (t , u)  0, u(a)  A
（3）
f (t )u  2  g (t , u)  0, u(b)  B ，
（4）
and
so that u L (0)  u R (0) and u L (0  )  u R (0  ) ;
[ H 2 ] f (t )  C n [a, b] (n  3) satisfying f (0)  f (0)    f ( n 1) (0)  0 and
f ( n ) (0)  0 ，i.e., t  0 is a high-order turning point;
[ H 3 ] g~(v) : g (0, u (0))  g (0, u (0)  v) is a smooth function satisfying

v
0
ｗhere v  v( ) ( 
g~ ( z )dz  0
t

and

v (0)
0
g~ ( z )dz  0 ,
) solves the problem
d 2v ~
 g (v )  0 ,
d 2
dv
(0)  0, v()  0 ,
d
u(t )  u L (t ) in [a ,0] and u(t )  u R (t ) in [0, b] .
Under hypotheses [ H1 ] — [ H 3 ] ， an O (  ) approximation of problem
(1),(2) is constructed using the method of composite expansions. It is
then shown, using the fixed point theorem, that for  sufficiently small,
problem (1),(2) has a solution x(t ,  ) with
x(t ,  )  u (t )  v(
t
)  O(  )

as   0 , uniformly on [a, b] . More precisely, x(t ,  )  u (t ) for x in
[a,0)  (0, b]
and x(0,  )  u (0)  v(0) as   0 , where v(0)  0 . It is to
say x(t ,  ) exhibits spike layer behavior at t  0 .
Now we replace [ H1 ] with
[ H1 ]
there
exist
functions
and
u L (t )
u R (t )
of
C2
on
[a, b] satisfying the reduced problems (3) and（4）respectively， so that
u L (0)  u R (0) (without loss of generality let u L (0)  u R (0) ).
Replacing g~ (v ) in [ H 3 ] with
g~ (v) : g (0, u (0))  g (0, u (0)  v )
L
L
L
L
and rewrite corresponding hypothesis as [ H 3 ] .
In a similar way, we can obtain an O (  ) approximation of problem
(1),(2) under hypotheses [ H1 ] , [ H 2 ] and [ H 3 ] :

u L (t )  v L (
x0 (t ,  )  
u R (t )  v R (

t

t

) , a  t  0,
) , 0  t  b,
where v L ( ) , vR ( ) may be given implicitly by
 
dz
vL
vL ( 0 )
2F ( z)
and
 
vR
vR ( 0 )
dz
2G ( z )
respectively, vL (0)  u L (0)  u R (0) , vR (0)  vL (0)  u L (0)  u R (0), and
F ( z)  
uL (0) z
G( z)  
uR (0) z
uL (0)
uR (0)
[ g (0, w)  g (0, u L (0))]dw,
[ g (0, w)  g (0, u R (0))]dw .
Moreover, we can show that for  sufficiently small, problem (1),(2) has
a solution x(t ,  ) with
x(t ,  )  x0 (t ,  )  O(  )
as   0 ， uniformly on [a, b] . More precisely, x(t ,  )  u L (t ) for x
in [a ,0) , x(t ,  )  u R (t ) for
x
in (0, b]
and
x(0,  )  u L (0)  vL (0)
  0 ,where vL (0)  u L (0)  u R (0)  0 . It is to say
x(t ,  )
as
exhibits
nonmonotone transition layer behavior at t  0 .
Recent applications of stationary-phase method to water wave
problems
Dongqiang Lu
Shanghai University
Abstract
The
generation
inviscid/viscous
and
interaction
gravity
waves,
of
surface
and
capillary-gravity
interfacial
waves
and
flexural-gravity waves due to an submerged body are investigated
analytically.
Based on the assumption for the incompressible fluid the
small-amplitude waves, a linear system is established.
The submerged
body is mathematically represented by a fundamental singularity.
The
integral solutions for the free-surface and interfacial waves are
obtained by means of the joint Laplace-Fourier transform.
Then the
corresponding asymptotic representations are derived for far-field waves
by Lighthill's two-stage method.
The first stage involves the Cauchy
residue while the second the Stokes and Scorer methods of stationary phase.
From the analytical solutions obtained, the principle physical features
of the wave generation and interaction are revealed.
The effects of the
characteristics of the disturbance (location and speed) and the ocean
(stratified layer density and depth ratios), the viscosity of the fluid,
the capillarity and elasticity of fluid surface are discussed in detail.
A class of singular perturbations for second order linear turning
point boundary value problems with high order on infinite interval
Haibo Lu
East China Normal University
Abstract
A class of singular perturbations for second order linear turning
point boundary value problems with high order on infinite interval, which
encountered in Non-Newtonian boundary layer calculation of collision
problem, is considered. Using the matching of asymptotic expansions, the
formal asymptotic solution is constructed. By using the theory of
differential inequality the uniform validity of the asymptotic expansion
for the solution is proved.
Convergence properties of formal power series over singular
perturbations of a formal curve
Daowei Ma
Wichita State University, USA
Abstract
Consider a formal power series
perturbation family
hs ( X ) ,
f ( X , Y ) and a certain singular
where each hs ( X ) is an element of [[ X ]] .
We prove that if f ( X , hs ( X )) is convergent for each s in a compact set
E in the complex plane of positive capacity, then f is convergent. We
also obtain some results about the Hartogs property for real analytic
functions.
This is a joint work with B. Fridman.
Singular perturbations and vanishing passage through a turning
point
Peter De Maesschalck
Hasselt University, Belgium
Abstract
We study the cyclicity of certain slow-fast cycles.
limit periodic sets that are composed of a fast orbit,
with a curve of singular points.
are
glued
together
One part of the singular
curve is
normally attracting, another part is normally repelling,
point is in between.
These
A typical tool to study the
and a contact
cyclicity is the
analysis of the asymptotic behaviour of
orbits near the limit periodic set.
by an integral along the curve of
term diverges however when
appear. Depending on
attracting side
the
divergence integral along
The leading order term is given
singularities.
This leading order
additional singularities in the slow dynamics
the location of the additional singularity
(on the
or repelling side), the obtained limit cycles are
hyperbolically stable or hyperbolically unstable.
In this talk,
we
consider the case of a singularity in the slow dynamics passing from one
side to
the other, through the contact point.
Joint work with F.
Dumortier.
Second order parameter-uniform convergence for a finite difference
method for a singularly perturbed linear parabolic system
John J H Miller
Trinity College, Dublin, Ireland
Abstract
A singularly perturbed linear system of second order ordinary
differential equations of parabolic reaction-diffusion type with given
initial and boundary conditions is considered. The leading term of each
equation is multiplied by a small positive parameter. These singular
perturbation parameters are assumed to be distinct. The components of the
solution exhibit overlapping layers. Shishkin piecewise-uniform meshes
are introduced, which are used in conjunction with a classical finite
difference discretisation, to construct a numerical method for solving
this problem. It is proved that the numerical approximations obtained with
this method are first order convergent in time and essentially second
order convergent in the space variable uniformly with respect to all of
the parameters.
Optimal control problem in contrast structures
Mingkang Ni
East China Normal University
Abstract
Optimal control problem in contrast structures is an important topic
having both deeply meanings in theory and wide applications. The authors
once studied the existence of contrast structures and the construction
of uniformly valid asymptotic solutions for a series of variational
problems with small parameters. But there exists some difficulties in
these problems. Firstly, when one constructs formal asymptotic solutions,
it needs not only asymptotic expansions of differential equations and
initial boundary values but also small parameter expansions of the
performance index functional. Secondly, there are not so many tools that
can be used for demonstrating the uniformly valid of the formal asymptotic
solution.
This paper primarily discusses a class of nonlinear optimal control
problems with small parameters. Under some given conditions, the
existence of step-step solution is proved, and the location of transfer
points is determined by using necessary conditions of the existence of
the optimal solution, and also the uniformly valid asymptotic expression
of step-step solution is obtained by the direct expansion method.
Positivity and general scheme of asymptotic method of differential
inequalities for contrast structures in reaction -diffusion-advection
problems
Nefedov N.N.
Moscow State University, Russia
Abstract
For some cases of initial boundary value problem for the equation
 2 u 
u
 f (u , u , x,  ), x  D  R N , t  0 ,
t
(1)
which plays important role in many applications and is called reactiondiffusion-advection equation we state the conditions which imply the
existence of contrast structures - solutions with internal layers. Among
others we discuss the following problems:
1. Lyapunov stability of stationary solutions.
2. The analysis of local and global domain of stability of contrast
structure.
3. The problem of stabilization of the solution of initial boundary
value problem.
Our investigations are based on asymptotic method of differential
inequalities and general scheme of this method will be presented.
Using two-timing (or multiscale) methods to solve singularly
perturbed problems
Robert O’Malley
University of Washington, USA
Abstract
Although matched expansions is more popular and better understood,
this paper seeks to show through solving specific examples involving
ordinary differential equations that multiscale methods are often
preferable.
On benchmarking the numerical libraries for singular-perturbation
problems
Yuhe Ren
NAG Ltd., United Kingdom
Abstract
D02NEF is a general purpose solver for integrating the initial value
problem for a stiff system of implicit ordinary differential equations
with coupled algebraic equations. D02NEF uses the DASSL implementation
of the Backward Differentiation Formulae (BDF).The NAG SMP Library has
been specially developed and tuned to provide the utmost performance on
SMP platforms. The results for benchmarking D02NEF (DASSL) based on the
NAG SMP Library are shown.
Birth of canard cycles
Robert Roussarie
Universite de Bourgogne (Dijon), France
Abstract
This talk is based on joint works by Freddy Dumortier and myself. We
consider a slow-fast unfolding of Liénard type X  ,  :
x  y  F ( x,  ), y   G ( x,  )
where ( x, y )
(0, 0), 
0 
p
and 
0

. Functions F , G are smooth or
even real analytic for some results.
We
assume
that
:
F ( x,  )  x 2  O( x3 ), G( x,  )  b( )  x  O( x3 ) with
b(0 )  0 and db(0 )  0 . Then, the turning point (0, 0) for X 0 ,0 is a
limiting situation of (generalized) Hopf bifurcations, that we call
slow-fast Hopf point.
We investigate the number of limit cycles that can appear near such
a slow-fast Hopf point and this, under very general conditions. One of
the results states that for analytic unfoldings depending on a finite
number of parameters, there is a finite upper bound for this number of
bifurcating limit cycles. In the smooth case, it could be expected that
a generic slow-fast Hopf unfolding with p parameters (1 ,
,  p ) produces
at most p limit cycles. It is precisely what we intended to prove but
rather surprisingly, this result can only be obtained modulo a conjecture
about a remarkable system of generalized Abelian integrals (and up to now
this conjecture is not proved!).
The treatment is based on blow-up, good normal forms and appropriate
Chebyshev systems of functions. The most difficult problem to deal with
concerns the uniform treatment of the evolution that a limit cycle
undergoes when it grows from a small limit cycle near the turning point
to a large limit cycle near a canard cycle of detectable size. This
explains the title.
In this talk I want to comment some important steps of the proof. In
a first step, we transform by blowing-up the unfolding X  ,  to a simpler
family of 3-dimensional vector fields defined near a critical locus E .
In a second and crucial step, we obtain a good presentation of a difference
map near a singular polycycle   E , defined for the flow of this blown-up
vector field family. It is along  that occurs after blow-up, the
transition between small and large limit cycles. This step uses some new
precise smooth normal form at singular points of the blown-up vector field
family. Next we use the presentation of the difference map to study of
cyclicity  . The main difficulty is that the singular polycycle  is
not contained into the regular part of the critical locus E and that, as
a consequence, the blown-up vector field family does not reduced to a
family of 2-dimensional vector fields, in any neighborhood of  .
The traveling wave solutions of the modified Kdv equation under
higher order perturbation
Desheng Shan
Shandong University of Technology
Abstract
In this paper, the existence and number of periodic traveling wave
solutions
of
higher
order
perturbed
modified
Korteweg-de
Vries
equation(MKdV for short) are studied. By employing the geometric singular
perturbation
theory,
we
reduce
the
three
dimensional
singular
perturbation system, which derived from the higher order perturbed MKdV,
into a general planar perturbed system. Then applying the bifurcation
theory of planar dynamical systems, we obtain that the given system can
have three periodic traveling wave solutions, have two periodic traveling
wave solutions and one solitary wave, or have one periodic traveling wave
solutions and two solitary wave solutions in some given conditions.
Exact solution approach of model problems in singular perturbations
Nico M. Temme
Centrum Wiskunde & Informatica, the Netherlands
Abstract
We consider several model problems from a class of ellipt ic
perturbation equations in two and three dimensions. The domains, the
differential operators, the boundary conditions, and so on, are rather
simple, and are chosen in a way that the solutions can be obtained in the
form of integrals or Fourier series. By using several techniques from
asymptotic analysis (saddle point methods, for instance) we try to
construct asymptotic approximations with respect to the small parameter
that multiplies the differential operator of highest order. In particular
we consider approximations that hold uniformly in the so-called boundary
layers. We also pay attention to how to obtain a few terms in the asymptotic
expansion by using direct methods based on singular perturbation methods.
This lecture is based on joint work with José L. López and Ester Pérez
Sinusía (Pamplona, Spain).
Singularly perturbed problems with the multiple roots of the
degenerate equations
Butuzov V.F.
Moscow State University, Russia
Abstract
This talk is devoted to some new results in the singularly perturbed
problems where Tichonov’s type stability assumptions are violated,
particularly to some important for applications cases when the degenerate
equation has multiple or intersecting roots are considered. The results
for the case of multiple roots of are presented for the boundary value
problem
d 2u
du

  A( x)  f (u, x,  )  0, 0  x  1,
2
dx
dx
du
du
(0,  ) 
(1,  )  0.
dx
dx
2
(1)
(2)
Some new results for the problems when the roots of the degenerate equation
intersect (this case is also referred to as the case of stability change)
are presented for the boundary value problem for the equation
 2(
 2u u
 )  f (u, x, t ,  ), ( x, t )  D  (1  x  1)  R,
x 2 t
(3)
where   0 is small parameter, R  (  t  ) , function f (u , x, t ,  )
is T -periodic in t .
The interior layer problem for a second order nonlinear singularly
perturbed differential-difference equation
Aifeng Wang
East China Normal University
Abstract
The interior layer problem for a kind of second order nonlinear
singularly perturbed differential-difference equation is studied. Using
the methods of boundary function and fractional steps, we construct the
formula asymptotic expansion of the problem and point out that the
boundary layer at t  0 has a great influence upon the interior layer at
t   . At the same time, Based on differential inequality techniques, the
existence of the smooth interior layer solution and the uniform validity
of the asymptotic expansion are proved. The result of this paper is new
and it complements the previously known ones.
Exponential dichotomy in random dynamical systems
Guangwa Wang
Xuzhou Normal University
Abstract
In the theory of singular perturbation, the technique of diagonalization is an important method for reducing a system. This technique is
based on the theory of exponential dichotomy. In fact, besides this,
exponential dichotomy plays important roles in many other fields.
Noticing the significance and importance of exponential dichotomy, in
this talk, we shall study the theory of exponential dichotomy for some
general random dynamical systems. More precisely, we will establish the
Sacker-Sell Spectral Decomposition Theorem in some different frameworks
of
random
dynamical
systems:
finite
dimensional
case,
infinite
dimensional case with compactness and infinite dimensional case with some
kind of weak compactness, respectively. Moreover, in the last part, some
further problems will be presented.
The talk is based on a joint work with Professor Yongluo Cao from
Suzhou University of China.
Second order quasilinear singularly perturbed differential difference
equations with boundary problem
Na Wang
East China Normal University
Abstract
In recently years, solutions with internal layers are very hot in
singularly perturbed theory. Many good consequences about the formal
asymptotic solution and the solution existence have been got yet. And the
existence of the solution in those problems was proved mainly based
on“differential
inequality"
method.
However,
in
practical,
the
variation of most problems is often not isolated in time, the solution's
stability of the system may be change because of the delayed affection.
Therefore, considering the structure and the expression of the uniformly
valid asymptotic solution is particularly important.
In this paper, a kind of second-order quasilinear singularly
perturbed difference-differential equations is considered. Under some
hypothesis conditions, the original problems are patitioned into the two
pure
boundary
layer
singularly
perturbed
problems.
Combinating
the“fractional steps method”with the“boundary layer function method”
in difference-differential equation, formal asymptotic solution is
constructed. By means of sewing orbit smooth, we get the uniformly valid
solution in the whole interval, and consider the error estimation between
the true solution and the asymptotic solution, which proves the existence
of solutions of the original problem. As to deal with the high dimension
of the singularly perturbed problems with the internal layers, this method
is also valid. Finally, a specific example was given to demonstrate
the feasibility of the method.
Existence of positive solutions for nonlinear m-point boundary value
problems on time scales
Limeng Wu
Yanshan University
Abstract
In this paper, by means of fixed point theorems in a cone, we study
the existence of at least two and three positive solutions of m-point
boundary value problem for second order dynamic equations on time scales.
As an application, we also give some examples to demonstrate our results.
The general solution of the modified Korteweg-de-Vries equation
Hanmei Yu & Hongxia Ge
Ningbo University
Abstract
Traffic congestion is related to various density waves, which might
be described by the nonlinear wave equations, such as the Burgers,
Korteweg-de-Vries (KdV) and modified Korteweg-de-Vries (mKdV) Equations.
In this paper，the mKdV equations of three different versions of lattice
hydrodynamic models, which describe the kink-antikink soliton waves are
derived by nonlinear analysis. What's more, the general solution is given,
which is applied to solve a new model - the lattice hydrodynamic with
bidirectional pedestrian flow. The result shows that our general solution
is consistent with that given by previous work.
A novel numerical method for a class of problems with transition
layer and Burgers' equation
Shuoyu Zheng
Tongji University
Abstract
A numerical method for solving a class of quasi-linear singular
two-point boundary value problem with transition layer is presented in
this paper. To the problems appear like  uxx  a  u  f ( x)  ux  b(u, x)  0 , we
develop a multiple scales method. Firstly, this method solves the location
of the transition layer, then it approximates the singular problem with
reduced problems in non-layer domain and pluses a layer corrected problem
which nearly has an effect in the layer domain. Both problems are
transformed to first order problems which can be solved out easily. To
the problems  uxx  b(u, x)  0 , we establish a similar method which
approximate the problem with reduce problems and a two-point boundary
value problem. Unsteady problems are also considered in our paper. We
extend our method to solve Burgers’ Equation problems by catching the
transition layer with the formula of shock wave velocity and approximating
it with a similar process.
Canard cycles for predator-prey competition models
Huaiping Zhu
York University, Canada
Abstract
In this talk, I will present a continuation study of predator-prey
models. There have been extensive stability and bifurcation studies of
classical predator-prey models, yet the study of canard cycles of the
model is rather limited. By using the techniques introduced by Dumortier
and Roussarie about center manifolds and singular perturbation, we study
the bifurcations of canard cycles in a general singular perturbed
predator-prey model, and apply the results to obtain canard cycles in the
model with Holling types of functional response.
This is a joint work with Chengzhi Li.
Some results on optimal control via extension principle
Jinghao Zhu
Tongji University
Abstract
The extension principle is a general untraditional approach to
various control problems. In the case of optimization problems it was
proposed by Vadim Krotov(1962, 1996) and V.I. Gurman(1985,1998) as a
generalization and modern development of Lagrange multi[pliers’ rule
into global sufficient optimization conditions. It served a basis for a
series of new untraditional optimization and optimal control methods. On
the other hand, the canonical duality theory proposed by David
Y.Gao(2000,2004) is a potentially powerful methodology, which can be used
to solve a wide class of global optimization problems. The classical
Lagrangian duality as well as the modern Fenchel- Moreau-Rockafellar type
duality theory cannot be used in isolation for solving nonconvex problems
due to the intrinsic duality gap. Canonical duality theory was developed
from nonconvex analysis and mechanics during the last decade. The
canonical dual transformation can be used to formulate perfect dual
problems without a duality gap.
In this talk, we present some new
researches on global optimization and optimal control via extension
principle and canonical dual method. On the one hand, we introduce a
backward differential flow finding solution to a global optimization and
present analytic solutions for some optimal control problems (J.Zhu(2009,
2010). On the other hand, we use Gurman(2005) improvement process to deal
with some nonconvex global optimizations.