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Discrete & Continuous Dynamical Systems - B

2015 , Volume 20 , Issue 5

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Special section on differential equations: Theory, application, and numerical approximation
Yanzhao Cao, Anping Liu and Zhimin Zhang
2015, 20(5): i-ii doi: 10.3934/dcdsb.2015.20.5i +[Abstract](179) +[PDF](36.7KB)
Abstract:
It is our honor to bring you this special section dedicated to recent advances in computational and applied mathematics in science and engineering. These articles comprise a collection of diverse theoretical results as well as applications, primarily centered on the following areas.

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A first order semi-discrete algorithm for backward doubly stochastic differential equations
Feng Bao, Yanzhao Cao and Weidong Zhao
2015, 20(5): 1297-1313 doi: 10.3934/dcdsb.2015.20.1297 +[Abstract](255) +[PDF](439.7KB)
Abstract:
Numerical solutions of backward doubly stochastic differential equations (BDSDES) and the related stochastic partial differential equations (Zakai equations) are considered. First order algorithms are constructed using a generalized Itô-Taylor formula for two-sided stochastic differentials. The convergence order is proved through rigorous error analysis. Numerical experiments are carried out to verify the theoretical results and to demonstrate the efficiency of the proposed numerical algorithms.
Stability and convergence of time-stepping methods for a nonlocal model for diffusion
Qingguang Guan and Max Gunzburger
2015, 20(5): 1315-1335 doi: 10.3934/dcdsb.2015.20.1315 +[Abstract](216) +[PDF](628.0KB)
Abstract:
A time-dependent nonlocal model for diffusion is considered. A feature of the model is that instead of boundary conditions, constraints over regions having finite measures are imposed. The explicit forward-Euler, implicit backward-Euler, and Crank-Nicolson methods are considered for discretizing the time derivative and piecewise-linear finite element methods are used for spatial discretization. The unconditional stability of the backward-Euler and Crank-Nicolson schemes and the conditional stability of the forward-Euler scheme are proved as are optimal error estimates for all three schemes. Comparisons with the analogous results for classical local diffusion problems, e.g., the heat equation, are provided as are the results of numerical experiments that illustrate the theoretical results.
Error analysis for numerical formulation of particle filter
Xiaoying Han, Jinglai Li and Dongbin Xiu
2015, 20(5): 1337-1354 doi: 10.3934/dcdsb.2015.20.1337 +[Abstract](295) +[PDF](475.6KB)
Abstract:
As an approximation of the optimal stochastic filter, particle filter is a widely used tool for numerical prediction of complex systems when observation data are available. In this paper, we conduct an error analysis from a numerical analysis perspective. That is, we investigate the numerical error, which is defined as the difference between the numerical implementation of particle filter and its continuous counterpart, and demonstrate that the error consists of discretization errors for solving the dynamic equations numerically and sampling errors for generating the random particles. We then establish convergence of the numerical particle filter to the continuous optimal filter and provide bounds for the convergence rate. Remarkably, our analysis suggests that more frequent data assimilation may lead to larger numerical errors of the particle filter. Numerical examples are provided to verify the theoretical findings.
Euler-Maclaurin expansions and approximations of hypersingular integrals
Chaolang Hu, Xiaoming He and Tao Lü
2015, 20(5): 1355-1375 doi: 10.3934/dcdsb.2015.20.1355 +[Abstract](226) +[PDF](455.2KB)
Abstract:
This article presents the Euler-Maclaurin expansions of the hypersingular integrals $\int_{a}^{b}\frac{g(x)}{|x-t|^{m+1}}dx$ and $\int_{a}^{b}% \frac{g(x)}{(x-t)^{m+1}}dx$ with arbitrary singular point $t$ and arbitrary non-negative integer $m$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.
A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential
Hengguang Li and Jeffrey S. Ovall
2015, 20(5): 1377-1391 doi: 10.3934/dcdsb.2015.20.1377 +[Abstract](300) +[PDF](1217.8KB)
Abstract:
We develop an a posteriori error estimate of hierarchical type for Dirichlet eigenvalue problems of the form $(-\Delta+(c/r)^2)\psi=\lambda \psi$ on bounded domains $\Omega$, where $r$ is the distance to the origin, which is assumed to be in $\overline\Omega$. This error estimate is proven to be asymptotically identical to the eigenvalue approximation error on a family of geometrically-graded meshes. Numerical experiments demonstrate this asymptotic exactness in practice.
The dynamics of an HBV epidemic model on complex heterogeneous networks
Meihong Qiao, Anping Liu and Qing Tang
2015, 20(5): 1393-1404 doi: 10.3934/dcdsb.2015.20.1393 +[Abstract](223) +[PDF](303.3KB)
Abstract:
In this paper, an HBV epidemic model on complex heterogeneous networks is proposed. Theoretical analysis of the HBV spreading dynamics is presented via mean-field approximation. Stabilities of the disease-free equilibrium and the endemic equilibrium are studied. The theoretical results reveal that disease propagation is impacted by the heterogeneous connectivity patterns and the underlying network structures.
Polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method for elliptic problems
Lunji Song and Zhimin Zhang
2015, 20(5): 1405-1426 doi: 10.3934/dcdsb.2015.20.1405 +[Abstract](216) +[PDF](389.9KB)
Abstract:
A polynomial preserving recovery technique is applied to an over-penalized symmetric interior penalty method. The discontinuous Galerkin solution values are used to recover the gradient and to further construct an a posteriori error estimator in the energy norm. In addition, for uniform triangular meshes and mildly structured meshes satisfying the $\epsilon$-$\sigma$ condition, the method for the linear element is superconvergent under the regular pattern and under the chevron pattern, while it is superconvergent for the quadratic element under the regular pattern.
Fast finite volume methods for space-fractional diffusion equations
Hong Wang, Aijie Cheng and Kaixin Wang
2015, 20(5): 1427-1441 doi: 10.3934/dcdsb.2015.20.1427 +[Abstract](175) +[PDF](395.1KB)
Abstract:
Fractional diffusion equations model phenomena exhibiting anomalous diffusion that is characterized by a heavy tail or an inverse power law decay, which cannot be modeled accurately by second-order diffusion equations that is well known to model Brownian motions that are characterized by an exponential decay. However, fractional differential equations introduce new mathematical and numerical difficulties that have not been encountered in the context of traditional second-order differential equations. For instance, because of the nonlocal property of fractional differential operators, the corresponding numerical methods have full coefficient matrices which require storage of $O(N^2)$ and computational cost of $O(N^3)$ where $N$ is the number of grid points.
    We develop a fast locally conservative finite volume method for a time-dependent variable-coefficient conservative space-fractional diffusion equation. This method requires only a computational cost of $O(N \log N)$ at each iteration and a storage of $O(N)$. Numerical experiments are presented to investigate the performance of the method and to show the strong potential of these methods.
On the Benilov-Vynnycky blow-up problem
Marina Chugunova, Chiu-Yen Kao and Sarun Seepun
2015, 20(5): 1443-1460 doi: 10.3934/dcdsb.2015.20.1443 +[Abstract](184) +[PDF](2863.3KB)
Abstract:
We study an initial-boundary value problem for a fourth-order parabolic partial differential equation with an unknown velocity. The equation originated from the linearization of a two-dimensional Couette flow model, that was recently proposed by Benilov and Vynnycky. In the case of a $180^{\circ}$-- contact angle between liquid and a moving plate Benilov and Vynnycky conjectured that the speed of the contact line blows up to infinity in finite time. In this paper we present numerical simulations and qualitative analysis of the model. We show that depending on the initial data and parameter values different long time behaviors of velocity can be observed. The speed of the contact line may blow up to infinity or converge to a constant.
A free boundary problem for aggregation by short range sensing and differentiated diffusion
Jan Haškovec and Dietmar Oelz
2015, 20(5): 1461-1480 doi: 10.3934/dcdsb.2015.20.1461 +[Abstract](197) +[PDF](705.9KB)
Abstract:
On the $d$-dimensional torus we consider the drift-diffusion equation, where the diffusion coefficient may take one of two possible values depending on whether the locally sensed density is below or above a given threshold. This can be interpreted as an aggregation model for particles like insect populations or freely diffusing proteins which slow down their dynamics within dense aggregates. This leads to a free boundary model where the free boundary separates densely packed aggregates from areas with a loose particle concentration.
    The paper has a rigorous part and a formal part. In the rigorous part we prove existence of solutions to the distributional formulation of the model. In the second, formal, part we derive the strong formulation of the model including the free boundary conditions and characterize stationary solutions giving necessary conditions for the emergence of stationary plateaus. We conclude that stationary aggregation plateaus in this situation are either spherical, complements of sphericals or stripes, which has implications for biological applications.
    Finally, numerical simulations in one and two dimensions are used to give evidence for the long time convergence to stationary states which feature aggregations.
The improved results on the stochastic Kolmogorov system with time-varying delay
Yangzi Hu and Fuke Wu
2015, 20(5): 1481-1497 doi: 10.3934/dcdsb.2015.20.1481 +[Abstract](209) +[PDF](643.8KB)
Abstract:
This paper discusses the stochastic Kolmogorov system with time-varying delay. Under two classes of sufficient conditions, this paper solves the non-explosion, the moment boundedness and the polynomial pathwise growth simultaneously. This is an important improvement for the existing results, since the moment boundedness and the polynomial pathwise growth do not imply each in general. Moreover, these two classes of conditions only depends on the parameters of the system and are easier to be used. Finally, a two-dimensional Komogorov model is examined to illustrate the efficiency of our result.
Chemotaxis can prevent thresholds on population density
Johannes Lankeit
2015, 20(5): 1499-1527 doi: 10.3934/dcdsb.2015.20.1499 +[Abstract](232) +[PDF](526.1KB)
Abstract:
We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to \begin{align*} u_t = -\nabla \cdot (u \nabla v)+\kappa u-\mu u^2\\ 0 = \Delta v - v + u      \\ \partial_v v|\partial \Omega = \partial_v u|\partial \Omega = 0 ,           u(0,\cdot) = u_0, \end{align*} in balls in $\mathbb{R}^n$. They exist globally in time for $\mu\ge 1$ and, for a certain class of initial data, undergo finite-time blow-up if $\mu<1$.
    We then use this blow-up result to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system - thereby extending recent results of Winkler [26] to the higher dimensional (radially symmetric) case.
Convective nonlocal Cahn-Hilliard equations with reaction terms
Francesco Della Porta and Maurizio Grasselli
2015, 20(5): 1529-1553 doi: 10.3934/dcdsb.2015.20.1529 +[Abstract](159) +[PDF](533.1KB)
Abstract:
We introduce and analyze the nonlocal variants of two Cahn-Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals. The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors.
Remarks on pattern formation in a model for hair follicle spacing
Peter Rashkov
2015, 20(5): 1555-1572 doi: 10.3934/dcdsb.2015.20.1555 +[Abstract](206) +[PDF](704.0KB)
Abstract:
A modified version of the Gierer-Meinhardt reaction-diffusion system (without source terms) is used in a model for hair follicle spacing in mice, proposed by Sick, Reinker, Timmer and Schlake [22]. Global existence of solutions of this model system is shown by computing uniform bounds. Analysis of conditions for emergence of spatially heterogeneous solutions is performed using a limiting form of the original reaction-diffusion system. The conditions for pattern formation given in [22] are improved by including those subregions in the parameter space where far-from-equilibrium heterogeneous solutions occur.
Global behaviour of a delayed viral kinetic model with general incidence rate
Hong Yang and Junjie Wei
2015, 20(5): 1573-1582 doi: 10.3934/dcdsb.2015.20.1573 +[Abstract](169) +[PDF](332.8KB)
Abstract:
This paper aims to show the global behaviour of a viral kinetic model with two time delays and general incidence rate. For the basic reproduction number $R_{0}<1$, the disease-free equilibrium is shown to be globally asymptotically stable by constructing Lyapunov functional and using LaSalle invariance principle. For the basic reproduction number $R_{0}>1$, the interior equilibrium of model exists and is also globally asymptotically stable. Our work show more general conclusion than other known papers on delayed viral models.

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