Traveling wave solutions in the mathematical model of blood coagulation

Under normal conditions, blood coagulation provides an effective protective mechanism preventing bleeding in case of vessel damage. Details of its functioning are of particular importance since any blood coagulation disorders lead to severe physiological aggravations. Multiple experimental and computational studies demonstrate the thrombin concentration distribution to determine the spatio-temporal dynamics of clot formation. Propagating from the injury site with constant speed, thrombin concentration profile can be modeled with a traveling wave solution of the system of partial differential equations describing main reactions of the coagulation cascade. In the current study, we derive conditions on the existence and stability of such solutions and provide an analytic approach of their wave speed estimation.     

A moving two-reaction zones model: global existence of solutions

We address some aspects concerning the analysis of a moving-boundary system modeling concrete carbonation. The model is based on the fact that carbonation might be considered as a reaction which is localized on two distinct a priori unknown internal zones which progress into concrete. We report on the existence of local and global weak solutions. The main feature of the problem is that the nonlinear coupling of the system occurs due to the moving boundary and nonlinearity of the involved productions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity

We prove the global‐in‐time existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity, with initial data ψ₀,A₀ ∈ L³,u₀ ∈ L^{3 ∕ 2},u₀ ≥ 0 in three dimension and in two dimension. Copyright © 2014 John Wiley & Sons, Ltd.     

A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients

This paper is devoted to the analysis of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on R⁺. We show global existence and Lipschitz continuity with respect to the model ingredients. In distinction to previous studies, where the L¹ norm was used, we apply the flat metric, similar to the Wasserstein W₁ distance. We argue that analysis using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Lipschitz continuous dependence with respect to the model coefficients and initial data and the uniqueness of the weak solutions are shown under the assumption on the Lipschitz continuity of the kinetic functions. The proof of this result is based on the duality formula and the Gronwall-type argument.     

Finite propagation speed for solutions of the wave equation on metric graphs

We provide a class of self-adjoint Laplace operators ?Δ on metric graphs with the property that the solutions of the associated wave equation satisfy the finite propagation speed property. The proof uses energy methods, which are adaptations of corresponding methods for smooth manifolds.     

Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach

In the present study, we propose a novel multiphysics model that merges two time-dependent problems – the Fluid-Structure Interaction (FSI) and the ultrasonic wave propagation in a fluid-structure domain with a one directional coupling from the FSI problem to the ultrasonic wave propagation problem. This model is referred to as the “eXtended fluid-structure interaction (eXFSI)” problem. This model comprises isothermal, incompressible Navier–Stokes equations with nonlinear elastodynamics using the Saint-Venant Kirchhoff solid model. The ultrasonic wave propagation problem comprises monolithically coupled acoustic and elastic wave equations. To ensure that the fluid and structure domains are conforming, we use the ALE technique. The solution principle for the coupled problem is to first solve the FSI problem and then to solve the wave propagation problem. Accordingly, the boundary conditions for the wave propagation problem are automatically adopted from the FSI problem at each time step. The overall problem is highly nonlinear, which is tackled via a Newton-like method. The model is verified using several alternative domain configurations. To ensure the credibility of the modeling approach, the numerical solution is contrasted against experimental data.     

Three dimensional axisymmetric problems in piezoelectric media: Revisited by a real fundamental solutions based new method

The purpose of the present work is to establish a set of real fundamental solutions for the differential governing equations of three dimensional axisymmetric problems in piezoelectric media. Firstly, conventional complex fundamental solutions are derived by analysis on the eigenvalue problem, and then, Euler’s formula is used to transform them into equivalent real fundamental solutions. As an example of application, the fracture problem of an axisymmetric penny-shaped crack in a piezoelectric layer is resolved by the real fundamental solutions based new method. Theoretical derivation and numerical computation are validated in the special case of a penny-shaped crack in an infinite piezoelectric body. Effects of geometrical parameters and electric-loading coefficient on energy release rates are surveyed and their agreement with the results of existing papers is also indicated. The advantage of such a real fundamental solutions based new method is that it can effectively help to avoid the difficult complex analysis in mixed boundary value problems.     

Some extended analyses concerning the physics and kinematics of wave propagation in moving systems

For studying the details of the physical processes concerning space-time relations of signal exchanges in moving inertial-systems, it is purposeful to make at first a short exposition of the kinematics of electromagnetic wave propagation from a moving source to a stationary or moving field point in the free unbounded space. An introductory analysis on the kinematics of wave propagation from a source to an observer in a moving inertial system follows thereafter, primarily for elucidating the natural causality of the source and the observer remaining centered at the common midpoint of the radiating waves, without undergoing any relative drift. Special emphasis is then put to study the space-time relations of wave propagation from a source to an observer in the moving inertial system. The use of the emitting and receiving signal-cones in the moving system, with their common plane-cut, provides all the details with convincing insights concerning the causality and effect, both in the absolute space and in the Lorentz-space. The essential findings are elucidated with application to numerical examples and corresponding illustrations.      

Some new solutions of the problems of wave propagation in a two-layer fluid

We present new analytic solutions of the problem of wave propagation in a continuously stratified fluid in the Boussinesq approximation. We study the propagation of internal waves in an ideal fluid in systems of homogeneous-layer/continuously stratified layer and homogeneous-layer/continuously stratified half-space type. We obtain the dispersion equations and study several limiting cases. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, Vol. 27, 1997, pp. 132–137.     

On the existence of periodic weak solutions of Navier-Stokes equations in regions with periodically moving boundaries

We prove the existence of periodic weak solutions to the Navier-Stokes equations in regions with moving boundaries using the elliptic regularization.     

A note on the wave propagation in water of variable depth

In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg-de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called “the method of integrating factor” is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed.     

Modeling of three dimensional thermocapillary flows with evaporation at the interface based on the solutions of a special type of the convection equations

Theoretical and numerical study of the convection processes, which are accompanied by evaporation/condensation, in the framework of new non-standard problem is largely motivated by new physical experiments. One of the principal questions is to understand the character and to evaluate the degree of influence of particular factors or their combined action on the structure of the joint flows of liquid and gas-vapor mixture. The flow topology is determined by four main mechanisms: natural and thermocapillary convection, tangential stresses and mass transfer due to evaporation at the interface. The mathematical modeling of the fluid flows in an infinite channel with a rectangular cross section is carried out on the basis of the solution of a special type of the convection equations. The effects of thermodiffusion and diffusive thermal conductivity in the gas phase and evaporation at the thermocapillary interface are taken into consideration. Numerical investigations are performed for the liquid – gas (ethanol – nitrogen) system under normal and low gravity. The fluid flows are characterized as translational and progressively rotational motions and can be realized in various forms.     

A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation

The classical solution of the Dirichlet problem with a continuous boundary function for a linear elliptic equation with Hölder continuous coefficients and right-hand side satisfies the interior Schauder estimates describing the possible increase of the solution smoothness characteristics as the boundary is approached, namely, of the solution derivatives and their difference ratios in the corresponding Hölder norm. We prove similar assertions for the generalized solution with some other smoothness characteristics. In contrast to the interior Schauder estimates for classical solutions, our established estimates for the differential characteristics imply the continuity of the generalized solution in a sense natural for the problem (in the sense of (n-1)-dimensional continuity) up to the boundary of the domain in question. We state the global properties in terms of the boundedness of the integrals of the square of the difference between the solution values at different points with respect to especially normalized measures in a certain class.     

A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor

In this paper we consider the strongly damped wave equation with time-dependent terms

u_tt−Δu−γ(t)Δu_t+β_ε(t)u_t=f(u),     

Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η_t^(I) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.     

A generalized dynamic model of nanoscale surface acoustic wave sensors and its applications in Love wave propagation and shear-horizontal vibration

A generalized dynamic model to depict the wave propagation properties in surface acoustic wave nano-devices is established based on the Hamilton's principle and variational approach. The surface effect, equivalent to additional thin films, is included with the aid of the surface elasticity, surface piezoelectricity and surface permittivity. It is demonstrated that this generalized dynamic model can be reduced into some classical cases, suitable for macro-scale and nano-scale, if some specific assumptions are utilized. In numerical simulations, Love wave propagation in a typical surface acoustic wave device composed of a piezoelectric ceramic transducer film and an aluminum substrate, as well as the shear-horizontal vibration of a piezoelectric plate, is investigated consequently to qualitatively and quantitatively analyze the surface effect. Correspondingly, a critical thickness that distinguishes surface effect from macro-mechanical behaviors is proposed, below which the size-dependent properties must be considered. Not limited as Love waves, the theoretical model will provide us a useful mathematical tool to analyze surface effect in nano-devices, which can be easily extended to other type of waves, such as Bleustein-Gulyaev waves and general Rayleigh waves.