Subscribe: IMA Journal of Applied Mathematics - current issue
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IMA Journal of Applied Mathematics Current Issue

Published: Sat, 02 Sep 2017 00:00:00 GMT

Last Build Date: Tue, 03 Oct 2017 12:54:18 GMT


Schur complement domain decomposition methods for the solution of multiple scattering problems


We present a Schur complement domain decomposition (DD) algorithm for the solution of frequency domain multiple scattering problems. Just as in the classical DD methods, we (1) enclose the ensemble of scatterers in a domain bounded by an artificial boundary, (2) we subdivide this domain into a collection of non-overlapping subdomains so that the boundaries of the subdomains do not intersect any of the scatterers and (3) we connect the solutions of the subproblems via Robin boundary conditions matching on the common interfaces between subdomains. We use subdomain Robin-to-Robin maps to recast the DD problem as a sparse linear system whose unknown consists of Robin data on the interfaces between subdomains—two unknowns per interface. The Robin-to-Robin maps are computed in terms of well conditioned boundary integral operators, and thus the method of solution proposed in this paper can be viewed as a boundary integral equation (BIE)/BIE coupling via artificial subdomains. Unlike classical DD, we do not reformulate the DD problem in the form a fixed point iteration, but rather we solve the ensuing linear system by Gaussian elimination of the unknowns corresponding to inner interfaces between subdomains via Schur complements. Once all the unknowns corresponding to inner subdomains interfaces have been eliminated, we solve a much smaller linear system involving unknowns on the inner and outer artificial boundary. We present numerical evidence that our Schur complement DD algorithm can produce accurate solutions of very large multiple scattering problems that are out of reach for other existing approaches.

Surface instability of imperfectly bonded multi-layered curved films under van der Waals forces


Using a linear stability analysis and the transfer matrix method, we investigate the surface instability of an imperfectly bonded multi-layered curved film interacting with a curved rigid contactor, another imperfectly bonded multi-layered curved film or an imperfectly bonded multi-layered simply-supported cylindrical shell in each case through the action of attractive van der Waals forces. The imperfect interface is modelled as a linear spring layer with vanishing thickness characterized by normal and tangential imperfect interface parameters. Detailed numerical results are presented to demonstrate the resulting analytical solutions.

$\lambda$ -Symmetries and integrability by quadratures


It is investigated how two (standard or generalized) $\lambda$-symmetries of a given second-order ordinary differential equation can be used to solve the equation by quadratures. The method is based on the construction of two commuting generalized symmetries for this equation by using both $\lambda$-symmetries. The functions used in that construction are related with integrating factors of the reduced and auxiliary equations associated to the $\lambda$-symmetries. These functions can also be used to derive a Jacobi last multiplier and two integrating factors for the given equation.Some examples illustrate the method; one of them is included in the XXVII case of the Painlevé-Gambier classification. An explicit expression of its general solution in terms of two fundamental sets of solutions for two related second-order linear equations is also obtained.

$\mathcal{KL}_*$ -stability for a class of hybrid dynamical systems


This article studies $\mathcal{KL}_*$-stability (the stability expressed by $\mathcal{KL}_*$-class function) for a class of hybrid dynamical systems (HDS). The notions of $\mathcal{KL}_{*}\mathcal{K}_{*}$-property and $\mathcal{KL}_{*}$-stability are proposed for HDS with respect to the hybrid-event-time. The $\mathcal{KL}_{*}$-stability, which is based on $\mathcal{K}$ or $\mathcal{L}$ property of the continuous flow, the discrete jump, and the event in an HDS, extends the $\mathcal{KLL}$-stability and the event-stability reported in the literature for HDS. The relationships between $\mathcal{KL}_{*}\mathcal{K}_{*}$-property and $\mathcal{KL}_{*}$-stability are established via introducing the hybrid dwell-time condition (HDT). The HDT generalizes the average dwell-time condition in the literature. For an HDS with $\mathcal{KL}_{*}\mathcal{K}_{*}$-property consisting of stabilizing $\mathcal{L}$-property and destabilizing $\mathcal{K}$-property, it is shown that there exists a common HDT under which the HDS will achieve $\mathcal{KL}_{*}$-stability. Thus HDT may help to derive some easily tested conditions for HDS to achieve uniform asymptotic stability. Moreover, a criterion of $\mathcal{KL}_{*}$-stability is derived by using the multiple Lyapunov-like functions. Examples are given to illustrate the obtained theoretical results.

Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking


This article is concerned with the invisibility cloaking in electromagnetic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. Our study is based on an interior transmission eigenvalue problem. We propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that there exists an infinite set of incident waves such that the cloaking device is nearly invisible under the corresponding wave interrogation. The set of waves is generated from the Maxwell–Herglotz approximation of the associated interior transmission eigenfunctions. We provide the mathematical design of the cloaking device and sharply quantify the cloaking performance.

Slowly varying, macroscale models emerge from microscale dynamics over multiscale domains


Many physical systems are well described on domains which are relatively large in some directions but relatively thin in other directions. In this scenario, we typically expect the system to have emergent structures that vary slowly over the large dimensions. For practical mathematical modelling of such systems we require efficient and accurate methodologies for reducing the dimension of the original system and extracting the emergent dynamics. Common mathematical approximations for determining the emergent dynamics often rely on self-consistency arguments or limits as the aspect ratio of the ‘large’ and ‘thin’ dimensions becomes unphysically infinite. Here we build on a new approach, previously establish for systems which are large in only one dimension, which analyses the dynamics at each cross-section of the domain with a rigorous multivariate Taylor series. Then centre manifold theory supports the local modelling of the system’s emergent dynamics with coupling to neighbouring cross-sections treated as a non-autonomous forcing. The union over all cross-sections then provides powerful support for the existence and emergence of a centre manifold model global in the large finite domain. Quantitative error estimates are determined from the interactions between the cross-section coupling and both fast and slow dynamics. Two examples provide practical details of our methodology. The approach developed here may be used to quantify the accuracy of known approximations, to extend such approximations to mixed order modelling, and to open previously intractable modelling issues to new tools and insights.

SVIR epidemic model with age structure in susceptibility, vaccination effects and relapse


An SVIR epidemic model with continuous age structure in the susceptibility, vaccination effects and relapse is proposed. The asymptotic smoothness, existence of a global attractor, the stability of equilibria and persistence are addressed. It is shown that if the basic reproductive number $\Re_0<1$, then the disease-free equilibrium is globally asymptotically stable. If $\Re_0>1$, the disease is uniformly persistent, and a Lyapunov functional is used to show that the unique endemic equilibrium is globally asymptotically stable. Combined effects of susceptibility age, vaccination age and relapse age on the basic reproductive number are discussed.

A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application


Compared to the classical first-order Grünwald–Letnikov formula at time $t_{k+1}\; (\text{or}\; t_{k})$, we firstly propose a second-order numerical approximate formula for discretizing the Riemann–Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank–Nicolson scheme for the fractional differential equations with time fractional derivatives. The established formula has the following form RLD0,tαu(t)| t=tk+12=τ−α∑ℓ=0kϖℓ(α)u(tk−ℓτ)+O(τ2),k=0,1,…,α∈(0,1), where the coefficients $\varpi_{\ell}^{(\alpha)}$$(\ell=0,1,\ldots,k)$ can be determined via the following generating function G(z)=(3α+12α−2α+1αz+α+12αz2)α,|z|<1.Next, applying the formula to the time fractional Cable equations with Riemann–Liouville derivative in one and two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.