Order System Of Differential Equations Research Articles

Computational Investigation of Brownian Motion and Thermophoresis Effect on Blood-based Casson Nanofluid on a Non-linearly Stretching Sheet

The present article investigates numerically the impacts of thermophoresis and Brownian motion on boundary layer nanofluid flow across a porous, non-linearly stretching sheet at a fixed temperature of the surface and under partial slip. Using the similarity variables, non-linear ordinary differential equations are obtained from the dictating partial differential equations, and then these non-linear ordinary differential equations are converted to a first order system of differential equations. The obtained first order system is then executed using bvp4c module in MATLAB along with the shooting technique. The numerical solutions obtained are scrutinized through graphs. It is observed that, as Nt and Nb levels rise, the temperature rises, and the thermal boundary layer thickens. Additionally, it has been found that the energy distribution expands, and nanoparticle concentration falls with rising values of the thermophoresis parameter.

Hepatitis-B disease modelling of fractional order and parameter calibration using real data from the USA

<p>In this paper, a new mathematical model of Hepatitis B is studied to investigate the transmission dynamics of the Hepatitis B virus (HBV). Many diseases can start from the womb and find us humans throughout our lives. These diseases are specific abnormal conditions that negatively affect the structure or function of all or part of an organism and do not suddenly occur in any region due to external injury. In this study, we focus on HBV, and we state the graphics, interpretations, and detailed information about the disease and the newly established mathematical model of the disease. A fractional order differential equation system with a memory effect is used to model anomalous processes and to understand the effect of past infection events on the future spread dynamics of the system. In the model, susceptible, latent, acute, carrier, and recovered populations are taken into account by considering vertical transmission, which provides information about the inter-generational course of the disease. However, the migration effect is also used in the model due to the risk of disease transmission and increased migration in recent years. The course of the disease is examined using real data from the USA. Moreover, the model's positivity and boundedness are studied, and the equilibrium points are calculated. Additionally, the stability conditions for the disease-free equilibrium (DFE) are stated. A parameter calibration technique is used to determine the most accurate parameter values in the model. Finally, we provide numerical results and their biological interpretations to estimate the future course of the disease. The paper addresses the current migration problem with the migration parameter in the model. These differences from the literature can be regarded as important novelties of the paper.</p>

Stability Properties of Multi-Order Fractional Differential Systems in 3D

This paper is devoted to studying three-dimensional non-commensurate fractional order differential equation systems with Caputo derivatives. Necessary and sufficient conditions for the asymptotic stability of such systems are obtained.

Dynamic perturbation analysis of fractional order differential quasiperiodic Mathieu equation

The paper investigates the influence of parameters on the stability of fractional order differential quasiperiodic Mathieu equations. First, we use the perturbation method to obtain approximate expressions (i.e., transition curves) for the stability and unstable region boundaries of the equation. After obtaining the approximate expression of the transition curve, we use Lyapunov's first method to analyze the stability of the fractional order differential quasiperiodic Mathieu system, thereby obtaining the conditions for the stability of the fractional order differential quasiperiodic Mathieu equation system. Second, by comparing the approximate expressions of the transition curve of the steady-state periodic solution of the quasiperiodic Mathieu oscillator under different parameter conditions, we obtained the conclusion that the fractional order differential term exists in the form of equivalent stiffness and equivalent damping in the fractional order differential quasiperiodic Mathieu system. By comparison, we have summarized the general forms of equivalent linear damping and equivalent stiffness of the system. Through this general form, we can define an approximate expression for the thickness of unstable regions to better study the characteristics of fractional order differential quasiperiodic Mathieu systems. Finally, the influence of the parameters of the fractional order differential quasiperiodic Mathieu equation on the transition curve of the equation was intuitively analyzed through numerical simulation, to analyze the stability changes in the equation.

СКЛАДАННЯ МАТЕМАТИЧНОЇ МОДЕЛІ ВІБРАЦІЙНОГО СТОЛУ

In this article, a mathematical model is obtained, which combines the design and technological parameters of the technological set of equipment for the concrete goods production (vibrating table), in which the vibrating exciter is fixed on the lever vertically in the center under the plate compactor. This equipment is used for the manufacture of small-sized concrete products. The mathematical model is composedwith the help of Lagrange equations of the second kind, which are the most common method for solving problems about the motion of a mechanical system. Methods of mathematical physics and physical and mathematical modeling by methods of applied mechanics were used in the compilation.To determine the position and describe the free motions of the material bodies that make up the mechanical system under consideration, an orthogonal vi-brational reference system of three coordinate systems was used. Analyzing the kinematic diagram of the vibration table, it is determined that the position in space of all material bodies of the mechanical system, which simulates the specified vibrating table, can be uniquely set by the following independent parameters: Cartesian coordinates, vibration angles and the angle of rotation of debalance.Thus, the mechanical system in question has seven degrees of freedom with seven generalized coordi-nates.As a result, a mathematical model of a vibrating table for concrete products manufacturing in the form of seven second order differential equations system, which describes the vibrating table spatial motion surface, is obtained.The theoretical values of the vibrating table vibrations amplitude at variable lever length were also determined and compared with experi-mental data at the same initial parameters. The difference was 15%, which confirms the adequacy of the created mathematical model to the real technological process on the vibrating table, which is being studied.Analyzing the change in the vibration oscillations amplitude on the vibrating table working surface from the change in the lever length on which the vibrating exciter is fixed, it was found that the lever fixation allows an almost linear increase in the amplitude of vibration oscillations due to an increase in the lever length. This, in turn, makes it possible to reduce energy consumption when compacting the concrete mixture.

A fractional-order mathematical model based on vaccinated and infected compartments of SARS-CoV-2 with a real case study during the last stages of the epidemiological event

In 2020 the world faced with a pandemic spread that affected almost everything of humans’ social and health life. Regulations to decrease the epidemiological spread and studies to produce the vaccine of SARS-CoV-2 were on one side a hope to return back to the regular life, but on the other side there were also notable criticism about the vaccines itself. In this study, we established a fractional order differential equations system incorporating the vaccinated and re-infected compartments to a SIR frame to consider the expanded and detailed form as an SVIIvR model. We considered in the model some essential parameters, such as the protection rate of the vaccines, the vaccination rate, and the vaccine’s lost efficacy after a certain period. We obtained the local stability of the disease-free and co-existing equilibrium points under specific conditions using the Routh–Hurwitz Criterion and the global stability in using a suitable Lyapunov function. For the numerical solutions we applied the Euler’s method. The data for the simulations were taken from the World Health Organization (WHO) to illustrate numerically some scenarios that happened.

3D hygro-elastic shell model for the analysis of composite and sandwich structures

The present work proposes the study of the hygrometric loading effects in the static analysis of multilayered composite and sandwich plates and shells. The employed model is based on a general exact 3D shell theory valid for one-layered and multilayered sandwich and composite plates, cylinders and cylindrical/spherical shell panels. The employed 3D equilibrium equations are developed in an orthogonal mixed curvilinear coordinate system for spherical shells in order to obtain the other simpler geometries as particular cases. A layer-wise approach is employed and equilibrium equations are solved in the thickness direction via the exponential matrix method. The partial derivatives in the in-plane directions are exactly calculated by assuming simply-supported edges and harmonic forms for the variables. The presented 3D shell model is extended to hygro-elastic cases by considering moisture content amplitudes imposed at the external surfaces of the structures in steady-state conditions. The moisture content profile through the thickness of the structures can be included in the 3D shell model using three different methodologies: it can be calculated by solving the steady-state 3D or 1D version of the Fick diffusion law, or it can be “a priori” assumed as linear through the thickness. Therefore, the developed system includes three non-homogeneous second order differential equilibrium equations that are solved after the introduction of opportune mathematical layers to consider the curvature effects through the thickness. The reduction to a first order differential equation system is obtained by redoubling the number of variables. The exponential matrix method is employed to accurately define both general and particular solutions. The implemented 3D hygro-elastic shell model is firstly validated and then it is employed with confidence for new benchmarks where the effects of thickness ratio, geometry, lamination scheme, material, thickness layer and imposed moisture content values at the external surfaces are investigated for composite and sandwich plates and shells. Showed results demonstrate the importance of an accurate developing of the elastic part of the 3D shell model combined with an appropriate evaluation of the moisture content profile through the thickness. Assumed linear moisture content profiles are correct only for thin one-layered isotropic structures, the use of the 1D Fick diffusion law allows only the correct definition of the material layer effect, the use of its 3D version allows the correct definition of both material and thickness layer effects.

Numerical solution of MHD free convective Jeffrey fluid of variable viscosity with chemical reaction and heat source

The study on heat and mass transfer effects on convective Jeffrey fluid on a stretched sheet is carried out with the presence of heat source, chemical reaction and variable viscosity parameter. The generated partial order differential equations system is made dimensional with suitable variables and the obtained non-dimensional system is numerically solved using ND solve of Mathematica software. All the results of the pertaining parameters are depicted in graphical mode. One of the major results of the numerical study is that the Hartmann number parameter plays a wide role in both velocity and temperature profiles.

Singular points classification of first order differential equations system not solved for derivatives

The article considers the singular system points’ classification of first-order differential equations that are not resolved with respect to derivatives. By displaying using some ratios that considered integral curves locations by special point in space taking into account the mutually located integral parabolas we obtain the corresponding pictures of the integral curves location by the singular system point. The types of singular equation points can be classified according the characteristic equation roots type, which we call a saddle, node, focus, center, or saddle focus

Revitalisasi Danau Limboto dengan Pengerukan Endapan di Danau: Pemodelan, Analisis, dan Simulasinya

Limboto lake is one of assets of Province of Gorontalo that provides many benefits to the surrounding society. The main problem of Limboto lake is the silting of the lake due to sedimentation caused by forest erosion, household waste, water hyacinth, and fish farming which is not environmentally friendly. In this article, a mathematical approach is used to modeling the Limboto lake siltation by including the revitalization solution namely the lake dredging. Mathematical modeling begins by building and limiting assumptions, constructing variables and parameters in mathematical symbols, and forming them into a first order differential equation system deterministically. Furthermore, we study the dynamics of the model such as identifying the existence of equilibrium points and their stability conditions. We also give a numerical simulations to show the conditions based on the stability requirements in previous analytical results.

3D shell model for the thermo-mechanical analysis of FGM structures via imposed and calculated temperature profiles

The exact three-dimensional (3D) shell model proposed in the present paper is able to perform the thermal stress analysis of simply-supported Functionally Graded Material (FGM) spherical and cylindrical shells, cylinders and plates. The model is based on the 3D equilibrium equations for spherical shells developed using an orthogonal mixed curvilinear coordinate system. The use of this reference system allows the investigation of cylindrical shells, cylinders and plates as particular cases of spherical shells by means of simple considerations on the radii of curvature. The 3D shell model uses a layer-wise approach and the exponential matrix method to calculate the general and the particular solutions through the thickness direction z. The system of second order differential equations in z is not homogeneous because of the thermal terms which are externally defined. The system is reduced to a group of first order differential equations in z simply redoubling the number of variables. The solution is in closed form in the in-plane directions α and β because of the hypotheses of simply-supported boundary conditions, harmonic forms for displacement and temperature fields, and isotropic behavior in the in-plane directions for functionally graded materials. In order to define the equivalent thermal load, the temperature profile through the thickness is separately defined by means of three possible ways. Using the hypothesis of temperature amplitudes imposed at the top and bottom external surfaces in steady-state conditions, the temperature profile can be: imposed as linear through the entire thickness direction, calculated by solving the 1D version of the Fourier heat conduction equation, or calculated by solving the 3D version of the Fourier heat conduction equation. The effects of different temperature profiles on the displacement and stress analyses of FGM plates and shells are here remarked. The first order differential equation system in z has not constant coefficients because of the presence of radii of curvature for shells and through-the-thickness variable elastic and thermal coefficients for the FGM layers. An appropriate number of mathematical layers is introduced to calculate the curvature influence for shells and the elastic and thermal material coefficients for FGM layers. Therefore, the system can be considered as differential equations with constant coefficients. The proposed results allow the evaluation of thickness ratio, geometry, lamination scheme, thickness material law and temperature profile effects in the related thermal stress analysis of single-layered and sandwich FGM plates, cylinders, spherical and cylindrical shells.

Chaos in integer order and fractional order financial systems and their synchronization

In this paper, we use integer order and fractional order differential equation systems to model a financial system. Based on the interaction among several financial factors, a model is constructed. Both mathematical analyses and numerical simulations are carried out to illustrate the characteristic of the model. We find that the system displays a variety of rich dynamic behaviours including chaos over a wide range of system parameters. Our investigation indicates that the interplay among several financial factors lead to chaos under some circumstances. We then design control laws to synchronization two integer order financial systems and two fractional order financial systems. Numerical simulations are presented to verify the effectiveness of the designed control laws.

Thermo-elastic analysis of multilayered plates and shells based on 1D and 3D heat conduction problems

Abstract The present work shows a generic 3D exact shell solution for the thermo-mechanical analysis of a heterogeneous group of one- and multi-layered isotropic, composite and sandwich structures. Plates, cylinders, cylindrical and spherical shells can be investigated using orthogonal mixed curvilinear coordinates. The 3D equilibrium equations for spherical shells automatically degenerate in those for simpler geometries. The elastic part of the proposed 3D model is based on a consolidated layer-wise exact solution which uses the exponential matrix method to solve the equilibrium differential equations through the thickness direction. The closed-form solution is obtained assuming simply-supported boundary conditions and harmonic forms for displacement and temperature fields. The temperature amplitudes are imposed at the top and bottom external surfaces in steady-state conditions. Therefore, the temperature profile can be evaluated through the thickness direction in three different ways: – calculation of the temperature profile via the steady-state 3D Fourier heat conduction equation; – evaluation of the temperature profile using the steady-state simplified 1D version of the Fourier heat conduction equation; – a priori assumed linear temperature profile through the entire thickness direction ranging from the bottom temperature value to the top temperature value. Once the temperature profile is defined at each thickness coordinate, it is considered as a known term in the 3D differential equilibrium equations. The obtained system consists in a set of non-homogeneous second order differential equilibrium equations which can be solved introducing appropriate mathematical layers. After a reduction to a first order differential equation system, the exponential matrix method is used to calculate both the general and the particular solutions. The effects of the temperature field on the static response of plates and shells are evaluated in terms of displacements and stresses. The proposed solution will be validated using reference results available in the literature. Then, new analyses will be presented for different thickness ratios, geometries, lamination schemes, materials and temperature values at the external surfaces. Results will demonstrate the importance in the 3D shell model of both the correct definition of the elastic part and the appropriate evaluation of the temperature profile through the thickness of the structure. .

Exact transfer- and stiffness matrix for the composite beam-column with Refined Zigzag kinematics

The Refined Zigzag Theory (RZT), developed by Tessler/Di Sciuva/Gherlone is among the most promising approaches for analyzing shear-elastic composite structures today. Since its appearance many contributions have been published dealing with finite elements for laminated structures based on the efficient kinematic of RZT. For composite beam-columns C0-elements of different orders as well as p-type approximations are formulated and assessed.In this work a different approach is given. After establishing the governing equations in a first order differential equation system, the transfer matrix is obtained by a matrix series solution and by similarity transformation. The transfer matrix approach, in principle suited for 1D-structural elements such as beams, disks, circular plates and rotational shells, has been successfully applied in the past. Sometimes this approach exhibits numerical instabilities. The well-known relations between the transfer- and stiffness matrix are invoked to circumvent this drawback. The dynamic stiffness matrix and the load vector are obtained by reordering and partially inverting the submatrices of the transfer matrix. The results, which are obtained by one finite element only, are in agreement with available analytical and numerical solutions.

Stability of weakly damped MDGKN-systems: The role of velocity proportional terms

Many problems in mechanical engineering, in the linearized case, can be modeled as a second order system of differential equations of type MDGKN, where the matrices correspond to inertia, damping, gyroscopic, stiffness, and circulatory forces. The latter may lead to self-excited vibrations which in general are unwanted and sometimes dangerous. It is well known that circulatory systems are very sensitive to damping and their stability behavior may strongly depend on the structure of the damping matrix. Moreover, it has been known for a long time, that the addition of (even infinitesimally small) damping may also destabilize such systems. The present note studies in more detail some of the effects of velocity proportional terms on the stability of mechanical systems of this type. The aim is to extend the findings recently presented by HAGEDORN et al. Therefore, the analytically derived stability boundary of an MDGKN-system with two degrees of freedom is analyzed with regard to infinitesimally small, incomplete, and indefinite damping matrices as well as the role of gyroscopic terms and the spacing of the eigenfrequencies.

Closed form numerical approach for a kinetic interpretation of high-cis polybutadiene rubber vulcanization with sulphur

A novel mathematical approach to predict the vulcanization degree of high-cis polybutadiene rubber vulcanized with sulphur is presented. The model has kinetic base, it is constituted by four reactions occurring in series and parallel and takes contemporarily into consideration, within a simplified but reliable scheme, the actual reactions occurring during polybutadiene sulphur curing, namely primary crosslinking and possible de-vulcanization. The first order differential equation system obtained is suitably rearranged and a closed form expression for the vulcanization degree is derived, depending the four kinetic constants characterizing the chemistry describing reactions. Instead of using classic least-squares optimization routines to characterize kinetic constants on experimental data, a simplified but reliable approach is proposed, where a system of four non-linear equations is solved with a recursive strategy, allowing estimating kinetic constants that proved to fit well normalized experimental data. The procedure is fast and its reliability is tested on a number of experimental data available, relying into a high-cis polybutadiene rubber cured under different temperatures and accelerators concentrations. Very good approximations of experimental data are obtained, also in comparison with a heuristic numerical approach where optimization is obtained interactively.

Interlacing of Hurwitz series

ABSTRACTIn this paper, we introduce the notion of interlacing of Hurwitz series. We begin by reviewing some important properties of the ring of Hurwitz series over a commutative ring A of arbitrary characteristic, and we introduce and investigate properties of the maps exp and log. We show that solutions of linear homogeneous differential equations with constant coefficients from the ring A can be described simply as interlacings of solutions of a first order system of differential equations. We give several examples to illustrate this result, and we conclude by defining and investigating properties of trigonometric functions using interlacings of Hurwitz series.

A Class of Integer Order and Fractional Order Hyperchaotic Systems via the Chen System

In this article, we investigate the generation of a class of hyperchaotic systems via the Chen chaotic system using both integer order and fractional order differential equation systems. Based on the Chen chaotic system, we designed a system with four nonlinear ordinary differential equations. For different parameter sets, the trajectory of the system may diverge or display a hyperchaotic attractor with double wings. By linearizing the ordinary differential equation system with divergent trajectory and designing proper switching controls, we obtain a chaotic attractor. Similar phenomenon has also been observed in linearizing the hyperchaotic system. The corresponding fractional order systems are also considered. Our investigation indicates that, switching control can be applied to either linearized chaotic or nonchaotic differential equation systems to create chaotic attractor.

A gradient elasticity model of Bernoulli–Euler nanobeams in non-isothermal environments

Abstract The Bernoulli–Euler beam formulation is extended by means of the nonlocal strain gradient theory in the nonhomogenous temperature field setting. Starting from the nonlocal continuum mechanics, a thermodynamically consistent model is obtained. The governing higher order system of differential equations for axial and transverse displacements is presented. Utilization of boundary conditions is demonstrated on four examples. It can be concluded that the nonhomogenous temperature field has a profound influence on the nanobeam mechanics. Some conclusions are drawn at the end of the paper.

Exact solution of bending problem of clamped orthotropic rectangular thin plates

The generalized integral transform technique (GITT) is employed to obtain an exact solution for the bending problem of fully clamped orthotropic rectangular thin plates. The use of the GITT approach in the analysis of the transverse deflection equation leads to a coupled system of fourth order differential equations (ODEs) in the dimensionless longitudinal spatial variable. The resulting transformed ODE system is then numerically solved by making use of the subroutine DBVPFD from IMSL Library. Numerical results with automatic global accuracy control are obtained for different values of aspect ratio. Critical comparisons with previously reported numerical results are performed with excellent agreement. A set of reference results for clamped orthotropic rectangular plates is also provided for future covalidation purposes, which are verified by those obtained from the general-purpose finite element software package ABAQUS.