Existence of infinitely many solutions for the one-dimensional Perona-Malik model

We establish the existence of infinitely many weak solutions for the the one-dimensional version of the well-known and widely used Perona-Malik anisotropic diffusion equation model in image processing. We establish the existence result under the homogeneous Neumann condition with smooth non-constant initial values. Our method is to convert the problem into a partial differential inclusion problem.     

A mathematical model for induction hardening including mechanical effects

We investigate a mathematical model for induction hardening of steel. It accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We take care of effects like transformation strain and transformation plasticity induced by the phase transitions and allow for physical parameters depending on the respective phase volume fractions.The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L¹ data. We prove existence of a weak solution to the complete system using a truncation argument.     

The effective conductivity of an infinitely interchangeable mixture

We are concerned with the estimation of the effective electrical conductivity of random heterogenous materials. The purpose of this paper is to discuss a property of “statistical symmetry” verified by the symmetric cell materials of Miller. This property will be referred to as infinite interchangeability. The usual way to approach cell materials is through n-point correlation functions. The property of infinite interchangebility permits us to approach cell materials from a completely different point of view. Our main result is a simple algorithm, based on this symmetry property, for computing any coefficient of the perturbation expansion in terms of information from the dilute limit. Specifically, knowledge of the coefficients of the expansion in powers of the volume fraction up to order r allows for computation of the perturbation expansion coefficients up to order (2r + 1). This result, which was previously known for r = 2 in the isotropic case and for r = 1 in the anisotropic case, can also be obtained from the standard correlation function approach, as pointed out by Milton.     

A mathematical model of mechanical contact of bodies across a thin inhomogeneous layer

We obtain the generalized conditions of mechanical contact of bodies that model a thin inhomogeneous isotropic layer situated between them. The technique is based on the application of averaging of the equilibrium equations for the layer and the assumption of a cubic distribution of displacements over its thickness. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 107–109.     

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 mathematical model for unemployment

In this paper we have proposed and analyzed a non-linear mathematical model for unemployment by considering three variables, namely the numbers of unemployed, temporarily employed and regularly employed persons. The model is studied using the stability theory of differential equations. It is found that the model has only one equilibrium, which is non-linearly stable under certain conditions. Numerical simulation of the model has been carried out to confirm the analytical results.     

Existence of infinitely many solutions for sublinear elliptic problems

Let N?2 and be a bounded domain. In the present paper, we show the existence of infinity many solutions of nonlinear Dirichlet boundary value problem

     

A mathematical model of market competition

We consider a mathematical model of decision making by a company attempting to win a market share. We assume that the company releases its products to the market under the competitive conditions that another company is making similar products. Both companies can vary the kinds of their products on the market as well as the prices in accordance with consumer preferences. Each company aims to maximize its profit. A mathematical statement of the decision-making problem for the market players is a bilevel mathematical programming problem that reduces to a competitive facility location problem. As regards the latter, we propose a method for finding an upper bound for the optimal value of the objective function and an algorithm for constructing an approximate solution. The algorithm amounts to local ascent search in a neighborhood of a particular form, which starts with an initial approximate solution obtained simultaneously with an upper bound. We give a computational example of the problem under study which demonstrates the output of the algorithm.     

A mathematical model of clarifier-thickener units

We review a recent analysis of a clarifier-thickener model for flocculated suspensions. The resulting governing PDE is a strongly degenerate convection-diffusion equation with discontinuous flux. A new numerical example illustrates how control actions speed up thickener operation significantly. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical model of retinal neovascularization

A mathematical model based on diffusion equations is presented, which directly relates the production of Vascular Endothelial Growth Factor (VEGF) in the retina to oxygen concentration and consumption, the capillary density growth to VEGF production and concentration, and the oxygen concentration to the growth of capillary density. The effects of local neovascularization on local oxygenation, which in turn affects the vascularization process resulting in biological feedback, are examined.     

A mathematical model of progressive literacy

We consider the two principal determinants of the dynamics of literacy in a developing country such as India: (a) a tendency of each family to improve or retain (but never retrogress) its literacy state with each successive generation, and (b) a differential in fertility, with number of off‐springs rising with decreasing literacy. The interplay of these contradictory trends is investigated in a mathematical model adapted and generalized from earlier work in the literature on modelling of social mobility and graded social systems. The model, besides being practically relevant, is exactly solvable and yields simple criteria and closed‐form solutions for different literacy distributions in terms of fertility and intergenerational transition proportions between male‐female “paired literacy states” of a society.     

A mathematical model of inter-terminal transportation

We present a novel integer programming model for analyzing inter-terminal transportation (ITT) in new and expanding sea ports. ITT is the movement of containers between terminals (sea, rail or otherwise) within a port. ITT represents a significant source of delay for containers being transshipped, which costs ports money and affects a port’s reputation. Our model assists ports in analyzing the impact of new infrastructure, the placement of terminals, and ITT vehicle investments. We provide analysis of ITT at two ports, the port of Hamburg, Germany and the Maasvlakte 1 & 2 area of the port of Rotterdam, The Netherlands, in which we solve a vehicle flow combined with a multi-commodity container flow on a congestion based time–space graph to optimality. We introduce a two-step solution procedure that computes a relaxation of the overall ITT problem in order to find solutions faster. Our graph contains special structures to model the long term loading and unloading of vehicles, and our model is general enough to model a number of important real-world aspects of ITT, such as traffic congestion, penalized late container delivery, multiple ITT transportation modes, and port infrastructure modifications. We show that our model can scale to real-world sizes and provide ports with important information for their long term decision making.     

A mathematical model of silicon-based thermobiosensors

A mathematical model of a silicon glucose thermobiosensor thatdetects changes in temperature produced by a biocatalytic reactionis proposed for calculation of transient temperature and reactantconcentration profiles, time-dependencies of the output signal,and calibration curves. Mathematically the model is reducedto a one-dimensional linear initial-boundary-value problem ofthe heat-conduction equation with a thermal source F(x, t).In order to find F(x, t), a system of the second-order nonlinearpartial differential equations for glucose and oxygen concentrations,describing a combination of diffusion-membrane theory and Michaelis-Mentenenzyme reaction theory, has been solved. The computed dependenciesof transient temperature profile and sensor response to variousconditions such as oxygen buffer concentration, membrane thickness,enzyme loading, and operation mode are analysed for the optimaldesign of the tensor.     

A mathematical model of infectious diseases

In the formulation of models for the spread of communicable diseases which include removal and population dynamics, it is necessary to distinguish between removal through recovery with immunity and removal by death due to disease. This distinction must be made because of the difference in the effect on the population dynamics of the different kinds of removal and because there are significant differences in the behavior of the models. We have formulated a class of models which allow recovery with immunity for a fraction of the infective and permanent removal by death from disease for the remainder. Earlier models of this type have postulated an increased death rate for infective, but such models are restricted to exponentially distributed-infective periods. Because of the differences in behavior between models with recovery and models with permanent removal do not arise when the infective period is exponentially distributed, we have chosen to formulate a different type of model which is sufficiently general to admit qualitative differences.     

LS model on thermal shock problem of generalized magneto-thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity

In the present paper, an estimation is made to investigate the transient phenomena in magneto-thermoelastic model in the context of the theory of generalized thermoelasticity LS model with variable thermal conductivity. FEM is proposed to analyze the problem and obtain the numerical solutions for the displacement, temperature, and radial and hoop stresses. The boundary conditions for the mechanical and Maxwell’s stresses at the internal and outer surfaces is considered. An application of an infinitely long annular cylinder is investigated for the inner surface is traction free and subjected to thermal shock, while the outer surface is traction free and thermally isolated. Finally, the displacement, incremental temperature, the stress components are obtained and then presented graphically.     

A mathematical model for asset pricing

Asset price dynamics is studied by using a system of ordinary differential equations which is derived by utilizing a new excess demand function introduced by Caginalp [4] for a market involving more information on demand and supply for a stock rather than their values at a particular price. Derivation is based on the finiteness of assets (rather than assuming unbounded arbitrage) in addition to investment strategies that are based on not only price momentum (trend) but also valuation considerations. For this new model and the older models which were extracted using the classical excess demand function by Caginalp and Balenovich [2] and [3], time evolutions of asset price are compared through numerical simulations.     

A mathematical model for continuous crystallization

J. Banasiak We discuss a mixed‐suspension, mixed‐product removal crystallizer operated at thermodynamic equilibrium. We derive and discuss the mathematical model based on population and mass balance equations and prove local existence and uniqueness of solutions using the method of characteristics. We also discuss the global existence of solutions for continuous and batch mode. Finally, a numerical simulation of a continuous crystallizer in steady state is presented. Copyright © 2015 John Wiley & Sons, Ltd.