ALBERT

Description: This study develops a spectral theory of the interior transmission problem (ITP) for heterogeneous and anisotropic elastic solids. The subject is central to the so-called qualitative methods for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the ITP for elastic bodies is neither self-adjoint nor elliptic. To help deal with such impediments, earlier studies have established the well-posedness of an elastodynamic ITP under notably restrictive assumptions on the contrast in elastic and mass density parameters between the scatterer and the background solid. Due to lack of self-adjointness of the problem, these analyses were further successful in substantiating the discreteness of the relevant eigenvalue spectrum but not its existence. The aim of this work is to provide a systematic treatment of the ITP for elastic bodies that transcends the limitations of earlier analyses. Considering a broad range of material-contrast configurations, this paper investigates the questions of the solvability of the ITP, the discreteness of its eigenvalues and, for the first time, of the existence of such eigenvalue spectrum. Necessitated by the breadth of material configurations studied, the relevant claims are established via a suite of variational formulations, each customized to meet the needs of a particular subclass of eigenvalue problems.

Description: Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

Description: In this paper, the rigorous linking of exact stochastic models to mean-field approximations is studied. Using a continuous-time Markov chain, we start from the exact formulation of a simple epidemic model on a certain class of networks, including completely connected and regular random graphs, and rigorously derive the well-known mean-field approximation that is usually justified based on biological hypotheses. We propose a unifying framework that incorporates and discusses the details of two existing proofs and we put forward a new ordinary differential equation (ODE)-based proof. The more well-known proof is based on a first-order partial differential equation approximation, while the other, more technical one, uses Martingale and Semigroup theory. We present the main steps of both proofs to investigate their applicability in different modelling contexts and to make these ideas more accessible to a broader group of applied researchers. The main result of the paper is a new ODE-based proof that may serve as a building block to prove similar convergence results for more complex networks. The new proof is based on deriving a countable system of ODEs for the moments of a distribution of interest and proving a perturbation theorem for this infinite system.

Description: In this paper, we extend certain key results from the classical theory of isotropic elasticity to the generalized theory of elasticity for decagonal quasicrystaline composites. These results include: (i) the dependence of the solution on the number of elastic constants, (ii) Green's functions for bimaterials consisting of two bonded half-planes, (iii) Green's functions for a circular elastic inclusion, (iv) the oscillatory singular stress field in the vicinity of an interface crack tip and (v) the inverse problem corresponding to the design of harmonic shapes.

Description: The note develops an approximate approach to mixed boundary value problems in linear elasticity starting from an explicit asymptotic model for the Rayleigh surface wave. It is demonstrated that the original vector mixed problem may be reduced to a scalar problem for the Laplace equation. As an illustration, the steady-state motion of a rigid stamp is analysed. Comparison of asymptotic and exact results is presented.

Description: In this paper, we investigate the advantages of the 2D Navier–Stokes–Voight (NSV) turbulence model for use in algorithms and explore its limits in the context of image inpainting. We begin by giving a brief review of the work of Bertalmio et al. in 2001 when an elegant analogy between the image intensity function for the image inpainting problem and the stream function in 2D incompressible fluid was established. An approximate solution to the inpainting problem was then obtained by numerically approximating the steady-state solution of the 2D Navier–Stokes vorticity transport equation, and simultaneously solving the Poisson problem between the vorticity and stream function, in the region to be inpainted. This elegant approach allows one to produce an approximate solution to the image inpainting problem by using techniques from computational fluid dynamics. Recently, the 3D NSV model of viscoelastic fluid was suggested by Cao et al. as an inviscid regularization to the 3D Navier–Stokes equations (NSEs). We give some background on the NSV model, describe why it is a good candidate sub-grid scale turbulence model and then we propose this model as an alternative partial differential equation for image inpainting. We describe an implementation of the inpainting procedure using the NSV model and then present numerical results comparing the images obtained when using the NSE versus the NSV model. Our results show that the NSV model allows for a larger time step to converge to the steady-state solution, yielding a more efficient numerical process when automating the inpainting process.We compare the quality of the resulting images using a subjective measure (human evaluation) and an objected measure (by calculating the peak signal-to-noise ratio). We also present some new theoretical results based on energy methods comparing the sufficient conditions for numerical stability for the two model equations. These theoretical and numerical studies shed some light on what can be expected from this category of approach when automating the inpainting problem.

Description: On the basis of the general non-linear theory of a hyperelastic material with initial stress, initially without consideration of the origin of the initial stress, we determine explicit expressions for the stress-dependent tensor of incremental elastic moduli. In considering three special cases of initial stress within the general framework, namely hydrostatic stress, uniaxial stress and planar shear stress, we then elucidate in general form the dependence of various elastic moduli on the initial stress. In each case, the effect of initial stress on the wave speed of homogeneous plane waves is studied and it is shown how various special theories from the earlier literature fit within the general framework. We then consider the situation in which the initial stress is a pre-stress associated with a finite deformation and, in particular, we discuss the specialization to the second-order theory of elasticity and highlight connections between several classical approaches to the topic, again with special reference to the influence of higher-order terms on the speed of homogeneous plane waves. Some discrepancies arising in the earlier literature are noted.

Description: In this paper, we introduce a macroscopic model for road traffic accidents along highway sections. We discuss the motivation and the derivation of such a model, and we present its mathematical properties. The results are presented by means of examples where a section of a crowded one-way highway contains in the middle a cluster of drivers whose dynamics are prone to road traffic accidents. We discuss the coupling conditions and present some existence results of weak solutions to the associated Riemann Problems. Furthermore, we illustrate some features of the proposed model through some numerical simulations.

Description: Transverse vibrations of Euler–Bernoulli beams coupled with van der Waals elastic forces are considered for multi-walled carbon nanotubes. Eigenfunctions are obtained for general boundary conditions of interest in fields such as nano-electronics and atomic force microscopy. It is employed a matrix basis generated by a Green matrix function of initial value. Uniform modes are characterized in closed form. Double-walled carbon nanotubes are discussed in detail for a cantilever beam.

Description: In this paper, we prove that the Cauchy problem for a non-linear wave equation (1) has a unique global generalized solutions in and a unique global classical solution in . We also prove that the Cauchy problem for the equation (1) admits a unique global generalized solution in C 2 ([0, ); W m , p ( R )   L ( R ))( m  ≥ 0 is an integer,1 ≤  p  ≤ ) and a unique global classical solution in .