# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

October 2007 , Volume 8 , Issue 3

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2007, 8(3): i-i doi: 10.3934/dcdsb.2007.8.3i +[Abstract](1286) +[PDF](43.8KB)
Abstract:
This special issue features a selection of papers presented at the 2006 Interna- tional Conference on Applied Mathematics and Interdisciplinary Research-Nankai, June 12-15, 2006, held at the newly inaugurated Chern Institute of Mathematics located on the campus of Nankai University in Tianjin, P. R. China. This con- ference focused on the interface between applied and computational mathematics and a variety of vibrant research areas such as soft matter physics, rheology of complex fluids, fluid dynamics, and new material phases, etc.. It attracted about 60 researchers around the world with expertise in various related disciplines. The meeting provided a unique opportunity for the participants, as well as the faculty and graduate students at Nankai University, to experience an innovatory form of academic interaction: interdisciplinary research.
Interdisciplinary research has emerged as an important and vibrant research for- mat as the boundaries between traditional disciplines are evolving and redefining. This is especially true in science and engineering communities, with which applied mathematics has traditionally maintained a strong tie. Now, it must face the new challenge and actively embrace the emerging interdisciplinary research platform. This conference was held under the backdrop of this pressing need and aimed at bridging applied and computational mathematics with other vibrant disciplines to facilitate cross-disciplinary research activities.

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2007, 8(3): 529-537 doi: 10.3934/dcdsb.2007.8.529 +[Abstract](1527) +[PDF](142.1KB)
Abstract:
We investigate the rational evaluation of foreign currency option price when the underlying price processes are described by exponential Lévy processes. The main contribution of our study is that we give the integro-differential equations for foreign currency option prices.
2007, 8(3): 539-556 doi: 10.3934/dcdsb.2007.8.539 +[Abstract](2247) +[PDF](207.8KB)
Abstract:
This paper is concerned with the dynamics of vesicle membranes in incompressible viscous fluids. Some rigorous theory are presented for the phase field Navier-Stokes model proposed in [7], which is based on an energetic variation approach and incorporates the effect of bending elasticity energy for the vesicle membranes. The existence and uniqueness results of the global weak solutions are established.
2007, 8(3): 557-567 doi: 10.3934/dcdsb.2007.8.557 +[Abstract](1332) +[PDF](174.8KB)
Abstract:
Let X be a Normed space and $S(X) = \{x \in X : \|\|x\|\| = 1\}$ be the unit sphere of X. Following the previous results for the Pythagorean approach in Banach spaces [5], [6], the generalized parameters $E_{\xi, \eta}(X)=$sup${\alpha_{\eta}(\xi x): x in S(X)\}$, $e_{\xi, \eta}(X)=$inf$\{\alpha_{\eta}(\xi x): x \in S(X)\}$, $F_{\xi, \eta}(X)=$sup${\beta_{\eta}(\xi x): x \in S(X)\}$, and $f_{\xi, \eta}(X)=$inf${\beta_{\eta}(\xi x): x \in S(X)\}$, where $\alpha_{\eta}(\xi x) =$sup${||\xi x + \eta y ||^2+ ||\xi x - \eta y ||^{2}: y \in S(X)\}$, $\beta_{\eta}(\xi x) =$inf${\|\|\xi x + \eta y ||^{2}+ ||\xi x - \eta y ||^{2}: y \in S(X)\}$ and $\xi, \eta > 0$ are defined and studied. The values of these parameters of some classical normed spaces are estimated and the relationship of these parameters with other geometric properties are investigated, and some existing results are extended also.
2007, 8(3): 569-587 doi: 10.3934/dcdsb.2007.8.569 +[Abstract](1274) +[PDF](950.4KB)
Abstract:
An optimal control problem for the forced Fisher equation is considered. The control is an artificially introduced genotype and the objective is to match, as well as possible, a specified gene frequency. The existence of a solution of the optimal control problem is proved and an optimality system is derived through the Lagrange multiplier technique. Numerical approximations of the optimality system are defined using finite element methods to effect spatial discretization and a backward Euler method for the time discretiza- tion. Convergence of semi-discrete in time approximations of the state system is proved and a gradient method for solving the nonlinear discrete systems is developed. The results of some preliminary computational experiments are provided.
2007, 8(3): 589-611 doi: 10.3934/dcdsb.2007.8.589 +[Abstract](1282) +[PDF](655.4KB)
Abstract:
In this paper, we present an observer design method for nonlinear systems based on pseudospectral discretizations and a moving horizon strategy. The observer has a low computational burden, a fast convergence rate and the ability to handle measurement noise. In addition to ordinary differential equations, our observer is applicable to nonlinear systems governed by deferential-algebraic equations (DAE), which are considered very difficult to deal with by other designs such as Kalman filters. The performance of the proposed observer is demonstrated by several numerical experiments on a time-varying chaotic nonlinear system with unknown parameters and a nonlinear circuit with a singularity-induced bifurcation.
2007, 8(3): 613-622 doi: 10.3934/dcdsb.2007.8.613 +[Abstract](1287) +[PDF](168.3KB)
Abstract:
The effect of thermal buoyancy on the stability properties of lower branch Tollmein–Schlichting waves are investigated. At moderate values of thermal buoyancy the standard triple deck structure, which describes the evolution of such short wavelength instabilities in a buoyant boundary layer, is unaltered. The leading order eigenrelation is now a function of thermal buoyancy and from it we can derive the new dominant length-and time–scales for the instability in the case when the boundary layer is strongly buoyant. These new scales demonstrate that, in the case of strong wall cooling the lower branch structure is identical to the upper branch structure, thus suggesting that the curve of neutral stability may become closed at some large value of a Reynolds number. In the alternate limit of strong wall heating the evolution of a fixed frequency disturbance is governed by the linearized interactive boundary-layer equations; in this case wave–like disturbances cannot be described by any form of the quasi–parallel approximation theory.
2007, 8(3): 623-648 doi: 10.3934/dcdsb.2007.8.623 +[Abstract](1054) +[PDF](527.7KB)
Abstract:
Growth, shape and texturing dynamics of single 2D spherulites are analyzed using the Landau-de Gennes (LdG) liquid crystal model of isotropic-nematic phase ordering. Direct numerical simulation shows that non-circular nucleation due to anisotropy in the interfacial tension results in non-circular shapes in the early stages of growth. However, interfacial heterogeneities in growth and structure then lead to interfacial defect nucleation and shedding and a reshaping of the interface into a circular shape. The formulated dynamics show that a growing spherulite in the early stage of phase ordering also acquires topological higher charges than expected from the well-known Kibble mechanism. In agreement with experiments under strong quenching, the predicted growth dynamics of a spherulite of characteristic radius R is linear: $R$~$t$. To better understand these computational results, a dynamic shape equation is obtained from the LdG model and is shown to have the same form as the growth and shape equations of crystal growth. The shape equation is used to reveal the mechanisms involved in shape transformations, interfacial defect shedding, and growth dynamics computed by direct numerical simulation of the bulk LdG equations.
2007, 8(3): 649-661 doi: 10.3934/dcdsb.2007.8.649 +[Abstract](1931) +[PDF](601.2KB)
Abstract:
Using phase field methods, we introduce a penalty formulation for restricting the support of solutions of the hydrodynamic Poisson-Nernst-Plank equations to evolving subregions of the domain. The formulation is derived through variational principles from a free energy involving the phase field and electrostatic energy. We validate the model by energetic arguments and several dynamic, finite element simulations of the (linear) Navier-Stokes, Poisson-Nernst-Plank and Allen-Cahn system.
2007, 8(3): 663-676 doi: 10.3934/dcdsb.2007.8.663 +[Abstract](1546) +[PDF](202.2KB)
Abstract:
We study a finite element approximation for the consistent splitting scheme proposed in [11] for the time dependent Navier-Stokes equations. At each time step, we only need to solve a Poisson type equation for each component of the velocity and the pressure. We cast the finite element approximation in an abstract form using appropriately defined discrete differential operators, and derive optimal error estimates for both velocity and pressure under the inf-sup assumption.
2007, 8(3): 677-693 doi: 10.3934/dcdsb.2007.8.677 +[Abstract](1769) +[PDF](226.4KB)
Abstract:
In this paper, we explore three efficient time discretization techniques for the local discontinuous Galerkin (LDG) methods to solve partial differential equations (PDEs) with higher order spatial derivatives. The main difficulty is the stiffness of the LDG spatial discretization operator, which would require a unreasonably small time step for an explicit local time stepping method. We focus our discussion on the semi-implicit spectral deferred correction (SDC) method, and study its stability and accuracy when coupled with the LDG spatial discretization. We also discuss two other time discretization techniques, namely the additive Runge-Kutta (ARK) method and the exponential time differencing (ETD) method, coupled with the LDG spatial discretization. A comparison is made among these three time discretization techniques, to conclude that all three methods are efficient when coupled with the LDG spatial discretization for solving PDEs containing higher order spatial derivatives. In particular, the SDC method has the advantage of easy implementation for arbitrary order of accuracy, and the ARK method has the smallest CPU cost in our implementation.
2007, 8(3): 695-706 doi: 10.3934/dcdsb.2007.8.695 +[Abstract](1198) +[PDF](254.7KB)
Abstract:
In this paper, we develop a mathematical model and a numerical technique to study the coupled turbulent flow and heat transfer process in continuous steel casting under an electromagnetic force. The complete set of field equations are established and solved numerically. The influences of the electromagnetic field on a flow pattern of molten steel and the temperature field as well as steel solidification are presented in the paper.
2007, 8(3): 707-733 doi: 10.3934/dcdsb.2007.8.707 +[Abstract](1316) +[PDF](2460.9KB)
Abstract:
We numerically explore texture (resolved by the second-moment of the orientational distribution) and shear banding of nematic polymers in shear cells, allowing for one-dimensional morphology in the gap between par- allel plates. We solve the coupled Navier-Stokes and Doi-Marrucci-Greco orientation tensor model, considering both confined orientation in the plane of shear and full orientation tensor degrees of freedom, and both primary flow and vorticity (in the full tensor model) components. This formulation makes contact with a large literature on analytical and numerical (cf. the review [41]) as well as experimental (cf. the review [45]) studies of nematic polymer texture and flow feedback. Here we focus on remarkable sensitivity of texture & shear band phenomena to plate anchoring conditions on the orientational distribution. We first explore steady in-plane flow-nematic states at low Peclet (Pe) and Ericksen (Er) numbers, where asymptotic analysis provides exact texture scaling properties [18, 6]. We illustrate that in-plane steady states co-exist with, and are unstable to, out-of-plane steady states, yet the structures and their scaling properties are not dramatically different. Non-Newtonian shear bands arise through orientational stresses. They are explored first for steady states, where we show the strength and gap location of shear bands can be tuned with anchoring conditions. Next, unsteady flow-texture transitions associated with the Ericksen number cascade are explored. We show the critical Er of the steady-to-unsteady transition, and qualitative features of the space-time attractor, are again strongly dependent on wall anchoring conditions. Other simulations highlight unsteady flow-nematic structures over 3 decades of the Ericksen number, comparisons of shear banding and texture features for in-plane and out-of-plane models, and vorticity generation in out-of-plane attractors.

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