The objective of this study is to consider the flow of temperature dependent viscosity and thermal conductivity of free convective heat and mass transfer of viscoelastic fluid over a stretching surface with nth order of chemical reaction and thermophoresis. The effect of the temperature dependent dynamic viscosity and thermal conductivity together with modified thermal and solutal Grashof numbers are properly accounted for in order to enhance the transport phenomenon. Similarity transformations are used to convert and parameterize the non-linear partial differential equation to a system of coupled non-linear ordinary differential equation. The approximate analytical solutions of the corresponding BVP are obtained through Optimal Homotopy Analysis Method (OHAM). The effect of some pertinent parameters is tested on velocity, temperature, concentration profiles. It is observed from the computation that, the thickness of the velocity and thermal boundary layer increases with an increase in temperature dependent variable viscosity and thermal conductivity parameters when modified thermal and solutal Grashof numbers and are less than zero. It is also observed that the concentration layer becomes thinner with increasing thermophoresis parameter when the chemical reaction parameter is greater than zero for both cases of first and second order of chemical reaction i.e. when n=1,2.

Abstract:
The dynamics of steady, two-dimensional magnetohydrodynamics (MHD) free convective
flow of micropolar fluid along a vertical porous surface embedded in a thermally
stratified medium is investigated. The ratio of pressure drop caused by liquid-solid interactions to that of pressure drop caused by viscous resistance are equal;
hence, the non-Darcy effect is properly accounted for in the momentum equation.
The temperature at the wall and at the free stream which best accounts for thermal
stratification are adopted. Similarity transformations are used to convert the nonlinear
partial differential equation to a system of coupled non-linear ordinary differential
equation and also to parameterize the governing equations. The approximate
analytical solution of the corresponding BVP are obtained using Homotopy Analysis
Method (HAM). The effects of stratification parameter, thermal radiation and other
pertinent parameters on velocity, angular velocity and temperature profiles are
shown graphically. It is observed that increase in the stratification parameter leads to
decrease in both velocity and temperature distribution and also makes the microrotation
distribution to increase near the plate and decrease away from the plate. The
influence of both thermal stratification and exponential space dependent internal
heat source on velocity, micro-rotation and temperature profiles are presented. The
comparison of the solutions obtained using analytical techniques (HAM) and
MATLAB package (bvp4c) is shown and a good agreement is observed.

Abstract:
Investments are one of the main factors of agricultural development and agricultural development can be considered a function of investment. Agriculture, as one of the most important economic areas in Serbia is need of investments which can intensify agricultural production. The whole progress of humanity caused by the continuous separation of part of the surplus and its investment in the steady development of productive forces of every society, regardless of the mode of production and productive relations that govern it.

Abstract:
In this paper a macroscopic model of tumor cord growth is developed, relying on the mathematical theory of deformable porous media. Tumor is modeled as a saturated mixture of proliferating cells, extracellular fluid and extracellular matrix, that occupies a spatial region close to a blood vessel whence cells get the nutrient needed for their vital functions. Growth of tumor cells takes place within a healthy host tissue, which is in turn modeled as a saturated mixture of non-proliferating cells. Interactions between these two regions are accounted for as an essential mechanism for the growth of the tumor mass. By weakening the role of the extracellular matrix, which is regarded as a rigid non-remodeling scaffold, a system of two partial differential equations is derived, describing the evolution of the cell volume ratio coupled to the dynamics of the nutrient, whose higher and lower concentration levels determine proliferation or death of tumor cells, respectively. Numerical simulations of a reference two-dimensional problem are shown and commented, and a qualitative mathematical analysis of some of its key issues is proposed.

Abstract:
This paper concerns multiphase models of tumor growth in interaction with a surrounding tissue, taking into account also the interplay with diffusible nutrients feeding the cells. Models specialize in nonlinear systems of possibly degenerate parabolic equations, which include phenomenological terms related to specific cell functions. The paper discusses general modeling guidelines for such terms, as well as for initial and boundary conditions, aiming at both biological consistency and mathematical robustness of the resulting problems. Particularly, it addresses some qualitative properties such as a priori nonnegativity, boundedness, and uniqueness of the solutions. Existence of the solutions is studied in the one-dimensional time-independent case.

Abstract:
In this paper we systematically apply the mathematical structures by time-evolving measures developed in a previous work to the macroscopic modeling of pedestrian flows. We propose a discrete-time Eulerian model, in which the space occupancy by pedestrians is described via a sequence of Radon positive measures generated by a push-forward recursive relation. We assume that two fundamental aspects of pedestrian behavior rule the dynamics of the system: On the one hand, the will to reach specific targets, which determines the main direction of motion of the walkers; on the other hand, the tendency to avoid crowding, which introduces interactions among the individuals. The resulting model is able to reproduce several experimental evidences of pedestrian flows pointed out in the specialized literature, being at the same time much easier to handle, from both the analytical and the numerical point of view, than other models relying on nonlinear hyperbolic conservation laws. This makes it suitable to address two-dimensional applications of practical interest, chiefly the motion of pedestrians in complex domains scattered with obstacles.

Abstract:
This paper deals with the early results of a new model of pedestrian flow, conceived within a measure-theoretical framework. The modeling approach consists in a discrete-time Eulerian macroscopic representation of the system via a family of measures which, pushed forward by some motion mappings, provide an estimate of the space occupancy by pedestrians at successive time steps. From the modeling point of view, this setting is particularly suitable to treat nonlocal interactions among pedestrians, obstacles, and wall boundary conditions. In addition, analysis and numerical approximation of the resulting mathematical structures, which is the main target of this work, follow more easily and straightforwardly than in case of standard hyperbolic conservation laws, also used in the specialized literature by some Authors to address analogous problems.

Abstract:
In this paper we investigate the ability of some recently introduced discrete kinetic models of vehicular traffic to catch, in their large time behavior, typical features of theoretical fundamental diagrams. Specifically, we address the so-called "spatially homogeneous problem" and, in the representative case of an exploratory model, we study the qualitative properties of its solutions for a generic number of discrete microstates. This includes, in particular, asymptotic trends and equilibria, whence fundamental diagrams originate.

Abstract:
This paper deals with a Boltzmann-type kinetic model describing the interplay between vehicle dynamics and safety aspects in vehicular traffic. Sticking to the idea that the macroscopic characteristics of traffic flow, including the distribution of the driving risk along a road, are ultimately generated by one-to-one interactions among drivers, the model links the personal (i.e., individual) risk to the changes of speeds of single vehicles and implements a probabilistic description of such microscopic interactions in a Boltzmann-type collisional operator. By means of suitable statistical moments of the kinetic distribution function, it is finally possible to recover macroscopic relationships between the average risk and the road congestion, which show an interesting and reasonable correlation with the well-known free and congested phases of the flow of vehicles.

Abstract:
In this paper a comparison between first order microscopic and macroscopic differential models of crowd dynamics is established for an increasing number $N$ of pedestrians. The novelty is the fact of considering massive agents, namely particles whose individual mass does not become infinitesimal when $N$ grows. This implies that the total mass of the system is not constant but grows with $N$. The main result is that the two types of models approach one another in the limit $N\to\infty$, provided the strength and/or the domain of pedestrian interactions are properly modulated by $N$ at either scale. This fact is consistent with the idea that pedestrians may adapt their interpersonal attitudes according to the overall level of congestion.