First integrals and exact solutions of the SIRI and tuberculosis models

The first integrals and exact solutions of mathematical models of epidemiology: a susceptible‐infected‐recovered‐infected (SIRI) model and a tuberculosis model with demographic growth are analyzed. These models are represented by systems of first‐order nonlinear ordinary differential equations, and this system is replaced by one which contains a second‐order ordinary differential equation. The partial Lagrangian approach is then utilized to derive the first integrals of these models. Several cases arise. Then, we utilize the derived first integrals to construct exact solutions for the models under investigation and determine new solutions. The dynamic properties of these models are studied too. Copyright © 2016 John Wiley & Sons, Ltd.     

Using a Functional Model to Develop a Mathematical Formula

The unifying theme of models was incorporated into a required Science Capstone course for pre‐service elementary teachers based on national standards in science and mathematics. A model of a teeter‐totter was selectedfor use as an example of a functional model for gathering data as well as a visual model of a mathematical equation for developing the mathematical relationship for a Class 1 lever, M1D1=M1D1. In this study, 20 student groups (n=72) collected data using the model in an inquiry‐based activity. All groups developed the qualitative relationship, 13 groups developed a correct mathematical formula, 6 groups developed one‐half of the relationship (X = mass × distance), and 1 group attempted to develop a procedural relationship. The pre‐service elementary teachers used a variety of model types in the activity including visual/pictorial, functional/physical and mathematical‐both graphs and formulas. The use of the teeter‐totter model as a visual and functional model of a mathematical formula was a factor in developing the mathematical relationship.     

Numerical stability study and error estimation for two implicit schemes in a moving boundary problem

Two algorithms are described [Ferris D. H. (fixed time‐step method) and Gupta and Kumar (variable time‐step method)] that solve a mathematical model for the study of the one‐dimensional moving boundary problem with implicit boundary conditions. Landau's transformation is used, in order to work with a fixed number of nodes at each time‐step. The p.d.e. is discretized using an implicit finite difference scheme. The mathematical model describes the oxygen diffusion in absorbing tissues. An important application is the estimation of time‐variant radiation treatments of cancerous tumors. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 42–61, 2000     

A compact local one‐dimensional scheme for solving a 3D N‐carrier system with Neumann boundary conditions

We consider a mathematical model for thermal analysis in a 3D N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for micro heat transfer. To solve numerically the complex system, we first reduce 3D equations in the model to a succession of 1D equations by using the local one‐dimensional (LOD) method. The obtained 1D equations are then solved using a fourth‐order compact finite difference scheme for the interior points and a second‐order combined compact finite difference scheme for the points next to the boundary, so that the Neumann boundary condition can be applied directly without discretizing. By using matrix analysis, the compact LOD scheme is shown to be unconditionally stable. The accuracy of the solution is tested using two numerical examples. Results show that the solutions obtained by the compact LOD finite difference scheme are more accurate than those obtained by a Crank‐Nicholson LOD scheme, and the convergence rate with respect to spatial variables is about 2.6. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On the existence of time-optimal control of mechanical manipulators

This paper is concerned with the problem of the existence and structure of time-optimal control for models derived from Lagrange equations of motion of mechanical systems involving links. The condition which ensures the existence of time-optimal control is demonstrated. The study conducted in this paper involves a highly nonlinear mathematical model of a two-degree-of-freedom mechanical system. However, the procedure and the results presented in this paper can be extended to mechanical systems with any finite number of degrees of freedom.The authors wish to thank Professor D. G. Hull and the reviewers for their most valuable comments and suggestions.     

Generalization of the Maxwell equation and relation to the elastic equation

In this paper, we propose a hyperbolic system of first‐order pseudo‐differential equations as generalization of the Maxwell equation. We state basic properties of this system corresponding to the ones of the (usual) Maxwell equation and explain that several known generalized Maxwell equations presented by some researchers can be integrated into the system. Namely, their equations can be regarded as our equation in special cases. Their generalized equations admit not only transversal but also longitudinal waves and are examined from the physical viewpoint. Using the present system, from the mathematical viewpoint, we interpret the meaning for presence of the longitudinal wave (with the transversal one) in their generalized equations. This presence means existence of more than one non‐zero characteristic root for the system (ie, non‐zero eigenvalue of the symbol). We prove also that our system becomes a first‐order expression of (generalized) elastic equations. Furthermore, it is shown that introducing the elastic equations implies expressing the generalized Maxwell equations by the potentials.     

The second‐order upwind finite difference fractional steps method for moving boundary value problem of oil‐water percolation

The research of the three‐dimensional (3D) compressible miscible (oil and water) displacement problem with moving boundary values is of great value to the history of oil‐gas transport and accumulation in basin evolution, as well as to the rational evaluation in prospecting and exploiting oil‐gas resources, and numerical simulation of seawater intrusion. The mathematical model can be described as a 3D‐coupled system of nonlinear partial differential equations with moving boundary values. For a generic case of 3D‐bounded region, a kind of second‐order upwind finite difference fractional steps schemes applicable to parallel arithmetic is put forward. Some techniques, such as the change of variables, calculus of variations, and the theory of a priori estimates, are adopted. Optimal order estimates in l² norm are derived for the errors in approximate solutions. The research is important both theoretically and practically for model analysis in the field, for model numerical method and for software development. Thus, the well‐known problem has been solved.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1103–1129, 2014     

Mathematical modelling of compressible viscous fluid flow in a male rotor‐housing gap of screw machines

This contribution is devoted to the mathematical modelling of a compressible viscous fluid flow through a 2‐D model of the male rotor‐housing gap in screw machines. Numerical solution of the nonlinear conservative system of the compressible Navier‐Stokes equations is obtained by means of the cell‐centred finite volume formulation of the explicit two‐step TVD MacCormack scheme proposed by Causon on a structured quadrilateral grid using the own developed numerical code. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

3‐D viscous Cahn–Hilliard equation with memory

We deal with the memory relaxation of the viscous Cahn–Hilliard equation in 3‐D, covering the well‐known hyperbolic version of the model. We study the long‐term dynamic of the system in dependence of the scaling parameter of the memory kernel ε and of the viscosity coefficient δ. In particular we construct a family of exponential attractors, which is robust as both ε and δ go to zero, provided that ε is linearly controlled by δ. Copyright © 2008 John Wiley & Sons, Ltd.     

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid‐saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time‐fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time‐fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.     

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.     

PDE Models for Chemotactic Movements: Parabolic,Hyperbolic and Kinetic

Modeling the movement of cells (bacteria, amoeba) is a long standing subject and partial differential equations have been used several times. The most classical and successful system was proposed by Patlak and Keller & Segel and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on bacteria like Escherichia coli or amoeba like Dictyostelium discoïdeum exhibiting pointwise concentrations.For human endothelial cells, several experiments show the formation of networks that can be interpreted as the initiation of angiogenesis. To recover such patterns a hydrodynamical model seems better adapted.The two systems can be unified by a kinetic approach that was proposed for Escherichia coli, based on more precise experiments showing a movement by jump and tumble. This nonlinear kinetic model is interesting by itself and the existence theory is not complete. It is also interesting from a scaling point of view; in a diffusion limit one recovers the Keller-Segel model and in a hydrodynamical limit one recovers the model proposed for human endothelial cells.We also mention the mathematical interest of analyzing another degenerate parabolic system (exhibiting different properties) proposed to describe the angiogenesis phenomena i.e. the formation of capillary blood vessels.     

A numerical scheme for the one‐dimensional pressureless gases system

In this work, we investigate the numerical approximation of the one‐dimensional pressureless gases system. After briefly recalling the mathematical framework of the duality solutions introduced by Bouchut and James (Comm. Partial Differential Equations 24 (1999), 2173–2189), we point out that the upwind scheme for density and momentum does not satisfy the one‐sided Lipschitz (OSL) condition on the expansion rate required for the duality solutions. Then we build a diffusive scheme which allows the OSL condition to be recovered by following the strategy described by Boudin (SIAM J Math Anal 32 (2000), 172–193) for the continuous model. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

On the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations

We show the existence of an inertial manifold (ie, a globally invariant, exponentially attracting, finite‐dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations.     

ANALYSES FOR A MATHEMATICAL MODEL OF THE PATTERN FORMATION ON SHELLS OF MOLLUSCS

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Existence results for functional first‐order coupled systems and applications

This paper is concerned with the existence of solutions of the first‐order fully coupled system with coupled functional boundary conditions. These functional boundary conditions generalize the usual boundary assumptions and may be applied to most of the classical cases. The arguments used are based on the Arzela‐Ascoli theorem and Schauder's fixed‐point theorem. An application to a mathematical model of the thyroid‐pituitary interaction and their homeostatic mechanism is included.     

Mathematical and Numerical Study of the 3D Atmospheric Boundary Layer Flows Including Pollution Dispersion

The paper presents a mathematical and numerical investigation of the atmospheric boundary layer (ABL) flow over 3D complex terrain part of which is represented by the real topography of the Krkonose mountains located in the Czech Republic. The flow is supposed to be turbulent, non‐stratified, viscous, incompressible and stationary. Two mathematical models have been formulated. The first model is based upon the RANS equations in the conservative form and the second one uses the Boussinesq approximation of RANS equations and takes the non‐conservative form. Also pollution dispersion over the complex 3D terrain has been considered in both models. The problem closure is achieved by an algebraic turbulence model and given boundary conditions.     

On coupled transversal and axial motions of two beams with a joint

In this paper we develop and analyze a mathematical model for combined axial and transverse motions of two Euler-Bernoulli beams coupled through a joint composed of two rigid bodies. The motivation for this problem comes from the need to accurately model damping and joints for the next generation of inflatable/rigidizable space structures. We assume Kelvin-Voigt damping in the two beams whose motions are coupled through a joint which includes an internal moment. The resulting equations of motion consist of four, second-order in time, partial differential equations, four second-order ordinary differential equations, and certain compatibility boundary conditions. The system is re-cast as an abstract second-order differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove the system is well posed, and that with positive damping parameters the resulting semigroup is analytic and exponentially stable. The spectrum of the infinitesimal generator is characterized.     

Analysing quality with generalized kinetic methods

A mathematical structure is developed with the aim of analysing the time evolution of the quality of a composite system such as a medical service inside an hospital. The approach belongs to the so-called Generalized Kinetic Theory, and consists of a set of balanced statistical equations on the probability distribution functions of the system populations over a state variable that represents the perceived quality. Internal and external actions are taken into account by means of direct interactions and ensemble terms. The mathematical framework is developed for a general setting. As a particular case, a model is suggested with reference to the quality of a specific medical service.     

A global attractor for a fluid–plate interaction model accounting only for longitudinal deformations of the plate

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier–Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in‐plane motions on a flexible flat part of the boundary. The main novelty of the model is the assumption that the transversal displacements of the plate are negligible relative to in‐plane displacements. These kinds of models arise in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. Under some conditions this attractor is an exponentially attracting single point. We also show that the corresponding linearized system generates an exponentially stable C₀‐semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results mean that dissipation of the energy in the fluid because of viscosity is sufficient to stabilize the system. Copyright © 2011 John Wiley & Sons, Ltd.