scholarly journals Law of universal gravitation with finite velocity of gravity and mathematical model of motion of a finite number of material points

2020 ◽  
pp. 120-123
Author(s):  
Vasyl Slyusarchuk
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Finite Number ◽  
Ordinary Differential Equations ◽  
Classical Model ◽  
Universal Gravitation ◽  
Finite Velocity ◽  
Deviating Argument ◽  
The Law ◽  
Material Points

The law of universal gravitation is intro- duced taking into account the finiteness of the gravi- tational velocity. Based on this law, a mathematical model of the motion of a finite number of material points is constructed, a separate case of which is the classical model of the motion of points, which is de- scribed by a system of ordinary differential equations. The constructed model is a system of nonlinear dif- ferential equations with deviating argument and func- tional equations. It more accurately describes the dy- namics of a finite number of material points than the corresponding classical model. A mathematical model of the motion of two material points is also considered.

scholarly journals Memory effects on the proliferative function in the cycle-specific of chemotherapy

2021 ◽  
Author(s):  
Najma Ahmed ◽  
Dumitru Vieru ◽  
Fiazud Din Zaman
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Classical Model ◽  
Cell Mass ◽  
Time Derivative ◽  
Fractional Model ◽  
Fractional Differential ◽  
Mathcad Software

A generalized mathematical model of the breast and ovarian cancer is developed by considering the fractional differential equations with Caputo time-fractional derivatives. The use of the fractional model shows that the time-evolution of the proliferating cell mass, the quiescent cell mass, and the proliferative function are significantly influenced by their history. Even if the classical model, based on the derivative of integer order has been studied in many papers, its analytical solutions are presented in order to make the comparison between the classical model and the fractional model. Using the finite difference method, numerical schemes to the Caputo derivative operator and Riemann-Liouville fractional integral operator are obtained. Numerical solutions to the fractional differential equations of the generalized mathematical model are determined for the chemotherapy scheme based on the function of "on-off" type. Numerical results, obtained with the Mathcad software, are discussed and presented in graphical illustrations. The presence of the fractional order of the time-derivative as a parameter of solutions gives important information regarding the proliferative function, therefore, could give the possible rules for more efficient chemotherapy.

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.

Some applications of Hausdorff dimension inequalities for ordinary differential equations

1986 ◽  
Vol 104 (3-4) ◽  
pp. 235-259 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Hausdorff Dimension ◽  
Differential Equations ◽  
Finite Number ◽  
Ordinary Differential Equations ◽  
Critical Point ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Invariant Sets ◽  
Lorenz Equation ◽  
Independent Variable

SynopsisUpper bounds are obtained for the Hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable. From these are derived sufficient conditions for dissipative analytic n-dimensional ω-periodic differential equations to have only a finite number of ω-periodic solutions. For autonomous equations the same conditions ensure that each bounded semi-orbit converges to a critical point. These results yield some information about the Lorenz equation and the forced Duffing equation.

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