scholarly journals MODELING THE DYNAMICS OF THE TUMOR DEVELOPMENT PROCESS

2020 ◽  
pp. 9-13
Author(s):  
VLADIMIR VLADIMIROVICH SELVINSKY ◽  
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Tumor Cell ◽  
Ordinary Differential Equations ◽  
Development Process ◽  
Tumor Development ◽  
Living Organism ◽  
Multivariable System ◽  
Lymphocyte Populations

This article discusses the well-known mathematical model of American scientists Rescigno and De Lisi, describing the growth of a tumor of a living organism. The model is a multivariable system of ordinary differential equations with respect to variables characterizing tumor cell and lymphocyte populations. On the basis of the mathematical package of MathCad, a detailed study of various options for the possible development of the tumor is carried out.

Center-Focus Problem for Discontinuous Planar Differential Equations

2003 ◽  
Vol 13 (07) ◽  
pp. 1755-1765 ◽  
Author(s):  
Armengol Gasull ◽  
Joan Torregrosa
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Critical Point ◽  
Limit Cycles ◽  
New Method ◽  
Polar Coordinates ◽  
Mechanical Problem ◽  
Weak Focus ◽  
Elastic Walls

We study the center-focus problem as well as the number of limit cycles which bifurcate from a weak focus for several families of planar discontinuous ordinary differential equations. Our computations of the return map near the critical point are performed with a new method based on a suitable decomposition of certain one-forms associated with the expression of the system in polar coordinates. This decomposition simplifies all the expressions involved in the procedure. Finally, we apply our results to study a mathematical model of a mechanical problem, the movement of a ball between two elastic walls.

Determination of Nonchaotic Behavior for Some Classes of Polynomial Jerk Equations

2020 ◽  
Vol 30 (08) ◽  
pp. 2050117
Author(s):  
Marcelo Messias ◽  
Rafael Paulino Silva
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Simple Form ◽  
Chaotic Dynamics ◽  
Third Order ◽  
Algebraic Criterion

In this work, by using an algebraic criterion presented by us in an earlier paper, we determine the conditions on the parameters in order to guarantee the nonchaotic behavior for some classes of nonlinear third-order ordinary differential equations of the form [Formula: see text] called jerk equations, where [Formula: see text] is a polynomial of degree [Formula: see text]. This kind of equation is often used in literature to study chaotic dynamics, due to its simple form and because it appears as mathematical model in several applied problems. Hence, it is an important matter to determine when it is chaotic and also nonchaotic. The results stated here, which are proved using the mentioned algebraic criterion, corroborate and extend some results already presented in literature, providing simpler proofs for the nonchaotic behavior of certain jerk equations. The algebraic criterion proved by us is quite general and can be used to study nonchaotic behavior of other types of ordinary differential equations.

scholarly journals Rhythm Drivers, Physical Properties, Mechanisms of Formation

2021 ◽  
Vol 248 ◽  
pp. 01007
Author(s):  
Mikhail Mazurov
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Nonlinear System ◽  
Differential Equations ◽  
Physical Properties ◽  
Ordinary Differential Equations ◽  
Numerical Study ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Mechanisms Of Formation

A mathematical model of the pacemaker is presented in the form of a nonlinear system of ordinary differential equations and in the form of a system of partial differential equations for distributed pacemakers. For the numerical study of the properties of the pacemaker, a modified axiomatic Wiener-Rosenbluth method was used using the properties of uniform almost periodic functions. Physical foundations, mechanisms of formation, properties of point and distributed pacemakers are described in detail.

scholarly journals NUMERICAL APPROACHES FOR A MATHEMATICAL MODEL OF AN FCCU REGENERATOR

2008 ◽  
Vol 7 (1) ◽  
pp. 71
Author(s):  
J. C. Penteado ◽  
C. O. R. Negrao ◽  
L. F. S. Rossi
Keyword(s):  
Mathematical Model ◽  
Finite Difference ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Finite Difference Methods ◽  
Implicit Finite Difference ◽  
Conservation Principles ◽  
Continuously Stirred Tank Reactor ◽  
Time Changes ◽  
Numerical Approaches

This work discusses a mathematical model of an FCCU (Fluid Catalytic Cracking Unit) regenerator. The model assumes that the regenerator is divided into two regions: the freeboard and the dense bed. The latter is composed of a bubble phase and an emulsion phase. Both phases are modeled as a CSTR (Continuously Stirred Tank Reactor) in which ordinary differential equations are employed to represent the conservation of mass, energy and species. In the freeboard, the flow is considered to be onedimensional, and the conservation principles are represented by partial differential equations to describe space and time changes. The main aim ofthis work is to compare two numerical approaches for solving the set of partial and ordinary differential equations, namely, the fourth-order Runge-Kutta and implicit finite-difference methods. Although both methods give very similar results, the implicit finite-difference method can be much faster. Steady-state results were corroborated by experimental data, and the dynamic results were compared with those in the literature (Han and Chung, 2001b). Finally, an analysis of the model’s sensitivity to the boundary conditions was conducted.

Full-size mathematical model of the fuel metering unit

2011 ◽  
Vol 8 (1) ◽  
pp. 249-256
Author(s):  
E.Sh. Nasibullaeva ◽  
E.V. Denisova ◽  
I.Sh. Nasibullayev
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Pressure Gradient ◽  
Ordinary Differential Equations ◽  
Constant Pressure ◽  
Initial Conditions ◽  
Control Valve ◽  
Nonlinear Mathematical Model ◽  
Full Size

The paper presents a nonlinear mathematical model for the operation of the fuel metering unit, which takes into account the operation of the control valve, which includes two pistons and three fuel circuits. A technique for determining the initial conditions for a system of ordinary differential equations describing the movements of a servo piston, a piston of a constant pressure gradient valve and a piston of a control valve is proposed.

Interaction of spherical gas bubbles in liquid

2007 ◽  
Vol 5 ◽  
pp. 66-72
Author(s):  
A.A. Aganin ◽  
A.I. Davletshin ◽  
V.G. Malakhov
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Gas Bubbles ◽  
Second Order

A mathematical model of interaction of two spherical gas bubbles in liquid is proposed. This model is a system of ordinary differential equations of the second order in the radii of bubbles and the spatial positions of their centers. It differs from the analogues known in the literature in that it allows much smaller distances between interacting bubbles.

SELF-SIMILAR PROBLEMS FOR MODELING THE SURFACE CHEMICAL REACTIONS WITH THE GRAVITATION

1998 ◽  
Vol 3 (1) ◽  
pp. 25-32
Author(s):  
Jânis Cepîtis ◽  
Harijs Kalis
Keyword(s):  
Boundary Layer ◽  
Mathematical Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Chemical Reactions ◽  
Glass Fibre ◽  
Surface Chemical ◽  
Self Similar ◽  
Surface Chemical Reactions ◽  
The Mathematical Model

The mathematical model of a chemical reaction which takes place on the surface of the uniformly moving vertically imbedded glass fibre material is considered. The effect of gravitation is taken into account. Boussinesq's and boundary layer fittings allow to derive boundary value problems for self‐similar systems of ordinary differential equations.

scholarly journals The evolution of pebble size and shape in space and time

2012 ◽  
Vol 468 (2146) ◽  
pp. 3059-3079 ◽  
Author(s):  
G. Domokos ◽  
G. W. Gibbons
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Markov Process ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Direct Simulation ◽  
Size And Shape ◽  
Segregation Process ◽  
Strong Segregation

We propose a mathematical model which suggests that the two main geological observations about shingle beaches, i.e. the emergence of predominant pebble size ratios and strong segregation by size, are interrelated. Our model is based on a system of ordinary differential equations (ODEs) called the box equations that describe the evolution of pebble ratios. We derive these ODEs as a heuristic approximation of Bloore's partial differential equation (PDE) describing collisional abrasion and verify them by simple experiments and by direct simulation of the PDE. Although representing a radical simplification of the latter, our system admits the inclusion of additional terms related to frictional abrasion. We show that non-trivial attractors (corresponding to predominant pebble size ratios) only exist in the presence of friction. By interpreting our equations as a Markov process, we illustrate by direct simulation that these attractors may only be stabilized by the ongoing segregation process.

scholarly journals The amyloid cascade hypothesis and Alzheimer’s disease: A mathematical model

2020 ◽  
pp. 1-20 ◽  
Author(s):  
M. BERTSCH ◽  
B. FRANCHI ◽  
L. MEACCI ◽  
M. PRIMICERIO ◽  
M.C. TESI
Keyword(s):  
Alzheimer’S Disease ◽  
Mathematical Model ◽  
Alzheimer's Disease ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Senile Plaques ◽  
Total Production ◽  
Amyloid Cascade Hypothesis ◽  
Amyloid Cascade ◽  
The Brain

The paper presents a conceptual mathematical model for Alzheimer’s disease (AD). According to the so-called amyloid cascade hypothesis, we assume that the progression of AD is associated with the presence of soluble toxic oligomers of beta-amyloid. Monomers of this protein are produced normally throughout life, but a change in the metabolism may increase their total production and, through aggregation, ultimately results in a large quantity of highly toxic polymers. The evolution from monomeric amyloid produced by the neurons to senile plaques (long and insoluble polymeric amyloid chains) is modelled by a system of ordinary differential equations (ODEs), in the spirit of the Smoluchowski equation. The basic assumptions of the model are that, at the scale of suitably small representative elementary volumes (REVs) of the brain, the production of monomers depends on the average degradation of the neurons and in turn, at a much slower timescale, the degradation is caused by the number of toxic oligomers. To mimic prion-like diffusion of the disease in the brain, we introduce an interaction among adjacent REVs that can be assumed to be isotropic or to follow given preferential patterns. We display the results of numerical simulations which are obtained under some simplifying assumptions. For instance, the amyloid cascade is modelled by just three ordinary differential equations (ODEs), and the simulations refer to abstract 2D domains, simplifications which can be easily avoided at the price of some additional computational costs. Since the model is suitably flexible to incorporate other mechanisms and geometries, we believe that it can be generalised to describe more realistic situations.

scholarly journals A Mathematical Model of Distributing New Information in Society

2020 ◽  
Vol 9 (1) ◽  
pp. 5-17
Author(s):  
Sergey Timofeev
Keyword(s):  
Mathematical Model ◽  
Mass Media ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Social Groups ◽  
Geometric Interpretation ◽  
Stationary Points ◽  
New Information ◽  
Proposed Model ◽  
Conceptual Meaning

Diversification of any new idea or concept in the society often begins with the emergence in the mass media of information that can win the attention of particular social groups or individuals. The author analyzes and interprets the results obtained by the means of constructing and studying a basic mathematical model of distribution of new information. The article describes factors and regularities that make the basis for the model, comments on all the relators that form the structure of the model, and describes the parameters of this structure. The proposed model is represented by a system of four ordinary differential equations. The stationary points of the system are found, and their conceptual meaning is described. The stationary points have cardinally different properties, depending on different relators. This provides for several possible scenarios of distributing new information in the society, whose description and geometric interpretation is given in the article.

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