scholarly journals Investigation of numerical solutions for chaotic Lorenz system using Mathcad software

2022 ◽  
Keyword(s):  
Lorenz System ◽  
Numerical Solutions ◽  
Mathcad Software

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.

A novel approach to obtain analytical-numerical solutions of nonlinear Lorenz system

2013 ◽  
Vol 67 (1) ◽  
pp. 93-107
Author(s):  
Gilberto González-Parra ◽  
Luis Acedo ◽  
Abraham J. Arenas
Keyword(s):  
Lorenz System ◽  
Numerical Solutions ◽  
Novel Approach

A Simple Numerical Simulation Method for Lorenz System Families

2013 ◽  
Vol 444-445 ◽  
pp. 786-790
Author(s):  
Cheng Li Zhang ◽  
Yun Zeng
Keyword(s):  
Analytical Solutions ◽  
Lorenz System ◽  
Numerical Solutions ◽  
Simulation Method ◽  
Chen System ◽  
Similar System ◽  
Recursion Method ◽  
Small Segment ◽  
Approximate Analytical Solutions ◽  
Recursion Formulas

Lorenz system families contain Lorenz system, Chen system and Lu system, their accurate analytical solutions are not yet obtained now. The segmenting recursion method was put forward in this paper, the equations of Lorenz system families were reasonably linearized within small segment, the recursion formulas were obtained by solving the approximate analytical solutions within small segment, and all numerical solutions were got by the recursion formulas. The chaotic motion of Lorenz system families were numerically simulated by means of the segmenting recursion method, the simulation results were compared with Runge-Kutta method. The comparative results show that the segmenting recursion method is very effective to numerically simulate Lorenz system families, not only method is simple, programming is easy, but result is accurate. this method is a universal new method to numerically simulate similar system.

scholarly journals Equations of motion of masses of the Chelomey pendulum model

2021 ◽  
Author(s):  
N. Kryshchuk ◽  
A. Tsybenko ◽  
Y. Lavrenko ◽  
A. Oleshchuk A.
Keyword(s):  
Software Package ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Newtonian Mechanics ◽  
Inertial Forces ◽  
Pendulum Model ◽  
Unit Work ◽  
Conservation Of Momentum ◽  
Mathcad Software ◽  
The Masses

Abstract. To verify the provisions stated by V.I. Bogomolov, B.I. Puzanov. and Linevich E.I. about the possibility of performing over-unit work by inertial forces, a closed mechanical system in the form of kinematically connected rotating masses is proposed for consideration. The research aimed, within the framework of Newtonian mechanics, to study the fulfillment of the laws of conservation of momentum, angular momentum and energy, to establish the possibility of performing work by inertial forces (centrifugal and Coriolis), to assess the change in kinetic parameters using the example of the Chelomey pendulum model. For the complex radial-circular motion of the masses of the Chelomey pendulum model, resolving equations are obtained. To verify the analytical calculations, algorithms for numerical solutions of the above problems have been developed and implemented in the MathCAD software package.

scholarly journals Stability and Stabilizing of Fractional Complex Lorenz Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Stability Theory ◽  
Lorenz System ◽  
Numerical Solutions ◽  
Fractional Order Systems ◽  
Caputo Derivatives ◽  
Stability And Stabilization ◽  
The Stability

We study the stability and stabilization of complex fractional Lorenz system. The fractional calculus are taken in sense of the Caputo derivatives. The technique is based on stability theory of fractional-order systems. Numerical solutions are imposed.

scholarly journals Numerical Counterexamples of Lorenz System in Implicit Time Scheme

2018 ◽  
Author(s):  
Xuan Chen
Keyword(s):  
Lorenz System ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Explicit Scheme ◽  
Implicit Time ◽  
Lorenz Equations ◽  
Consistent System ◽  
Runge Kutta Method ◽  
Self Consistent ◽  
The One

In nonlinear self-consistent system, Lorenz system (Lorenz equations) is a classic case with chaos solutions which are sensitively dependent on the initial conditions. As it is difficult to get the analytical solution, the numerical methods and qualitative analytical methods are widely used in many studies. In these papers, Runge-Kutta method is the one most often used to solve these differential equations. However, this method is still a method based on explicit time scheme, which would be the main reason for the chaotic solutions to Lorenz system. In this work, numerical experiments based on implicit time scheme and explicit scheme are setup for comparison, the results show that: in implicit time scheme, the numerical solutions (counterexamples) are without chaos; for an original volume, the volume shrinks exponentially fast to 0 in common.

scholarly journals Equations of motion of masses of the Chelomey pendulum model

2021 ◽  
Vol 5 (3) ◽  
Author(s):  
N. Kryshchuk
Keyword(s):  
Software Package ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Newtonian Mechanics ◽  
Inertial Forces ◽  
Pendulum Model ◽  
Unit Work ◽  
Conservation Of Momentum ◽  
Mathcad Software ◽  
The Masses

To verify the provisions stated by V.I. Bogomolov, B.I. Puzanov. and Linevich E.I. about the possibility of performing over-unit work by inertial forces, a closed mechanical system in the form of kinematically connected rotating masses is proposed for consideration. The research aimed, within the framework of Newtonian mechanics, to study the fulfillment of the laws of conservation of momentum, angular momentum and energy, to establish the possibility of performing work by inertial forces (centrifugal and Coriolis), to assess the change in kinetic parameters using the example of the Chelomey pendulum model. For the complex radial-circular motion of the masses of the Chelomey pendulum model, resolving equations are obtained. To verify the analytical calculations, algorithms for numerical solutions of the above problems have been developed and implemented in the MathCAD software package

Atomic Imaging of Crystals using Large-Angle Electron Scattering in STEM

1990 ◽  
Vol 48 (1) ◽  
pp. 74-75
Author(s):  
D.E. Jesson ◽  
S. J. Pennycook
Keyword(s):  
Wave Function ◽  
Electron Scattering ◽  
Numerical Solutions ◽  
Large Angle ◽  
Real Space ◽  
High Angle ◽  
Analytical Treatment ◽  
Order Zero ◽  
Experimental Conditions ◽  
Ewald Sphere

It is well known that conventional atomic resolution electron microscopy is a coherent imaging process best interpreted in reciprocal space using contrast transfer function theory. This is because the equivalent real space interpretation involving a convolution between the exit face wave function and the instrumental response is difficult to visualize. Furthermore, the crystal wave function is not simply related to the projected crystal potential, except under a very restrictive set of experimental conditions, making image simulation an essential part of image interpretation. In this paper we present a different conceptual approach to the atomic imaging of crystals based on incoherent imaging theory. Using a real-space analysis of electron scattering to a high-angle annular detector, it is shown how the STEM imaging process can be partitioned into components parallel and perpendicular to the relevant low index zone-axis.It has become customary to describe STEM imaging using the analytical treatment developed by Cowley. However, the convenient assumption of a phase object (which neglects the curvature of the Ewald sphere) fails rapidly for large scattering angles, even in very thin crystals. Thus, to avoid unpredictive numerical solutions, it would seem more appropriate to apply pseudo-kinematic theory to the treatment of the weak high angle signal. Diffraction to medium order zero-layer reflections is most important compared with thermal diffuse scattering in very thin crystals (<5nm). The electron wave function ψ(R,z) at a depth z and transverse coordinate R due to a phase aberrated surface probe function P(R-RO) located at RO is then well described by the channeling approximation;

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