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:
1)
This system uses great circle arcs to locate Epicentral Parameters as the vector:
In this, is the Cartesian position of the Epicentre in a Space-Frame, rotating with the Reference Sphere, whose radius is The origin of this Space-Frame coincides with the centre of the Reference Sphere. The set represents the coordinates of the Seismographic Stations in the same frame of reference, while the set represents the set of onset timings taken relative to the Lead Station such that and 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 While this happens will have been asymptotically approaching
[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
and
where : is a S-Carathéodory function with respect to 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 integro-
differential 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
with a homogeneous Dirichlet boundary condition, where is a positive parameter. For to be an annulus, we prove that for each there corresponds a unique steady-state solution and is a global in time-bounded solution, which tends to the unique steady-state solution as uniformly in . Whereas for there is no stationary solution and if then blows up in finite time for all .
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 () 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 equations with a variable delay. The corresponding convergence results are obtained and 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
， （1）
， （2）
where is a small parameter, andare constants. Assume
there exist functions and satisfying, respectively, the reduced problems
（3）
and
， （4）
so that and ;
satisfying and ，i.e., is a high-order turning point;
is a smooth function satisfying
and ,
ｗhere solves the problem
,
,
in and in .
Under hypotheses —，an 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 with
as , uniformly on . More precisely, for in and as , where . It is to say exhibits spike layer behavior at .