Form Of Nonlinear Differential Equations Research Articles

Coupled energy and mass transport for non-Newtonian nanofluid flow through non-parallel vertical enclosure

Several phenomena that occur in nature and engineering are expressed mathematically in large part by nonlinear differential equations. Therefore, heat and mass transport for buoyancy driven nanofluid flow through converging/diverging channel are modelled using non-Newtonian constitutive model in form of non-linear differential equations. It is worth mentioning that such models are frequently involved in blood flow through arteries and flow through nozzles, etc. Utilizing non-dimensionless transformations, leading partial differential equations are converted to a system of nonlinear ordinary differential equations. The Runge-Kutta Fehlberg algorithm with shooting technique is exercised to tackle reduced system numerically. The computed results for dimensionless velocity, temperature, nanoparticle concentration, skin-friction coefficient, local Nusselt and Sherwood numbers are presented for both buoyancy-assisting and opposing flow. Detailed analysis revealed that temperature field is higher at opening of flow in converging channel. Furthermore, with growing buoyancy force, the skin friction coefficient decreases while local Nusselt number increases.

Theoretical investigation of injection-locked differential oscillator

A preliminary analysis of published works on this topic showed that at present there is no sufficiently substantiated theory of such devices, and the approximate approaches used are rough and do not always meet the requirements of practice. The proposed transition from a differential self-oscillator to an equivalent single-circuit oscillator has not received a convincing justification.
 This article presents a methodology for studying a synchronized differential oscillator using rigorous methods. A mathematical model of such oscillator is presented in the form of nonlinear differential equations obtained on the basis of Kirchhoff's laws. Their analysis made it possible to substantiate the transition to the model of a single circuit LC oscillator, equivalent to a differential one. A technique for such a transition is proposed, including the procedure for determining the nonlinear characteristics of the amplifying element of this self-oscillator, based on the nonlinear characteristics of two amplifying elements of the differential oscillator.
 The mathematical model of an equivalent oscillator is represented by a non-linear differential Van der Pol equation in a dimensionless form, it is simple and accurate. This form of representation made it possible to single out a small parameter and estimate its value. In the case of small values of the small parameter, as is known, traditional methods can be used for its analysis. Thus, the task of studying the synchronization process of a differential oscillator is reduced to the study of the synchronization process of a Van der Pol oscillator. The presented results can be useful in the development of various devices based on synchronized differential oscillators.

The modeling of dehydroalkilation of methylcyclohexane via methanol

Based on the experimental data on the oxidizing dehydroalkilation of methylcyclohexane via methanol, a model of process statics presented in the form of non-linear differential equations has been developed. It is shown that these equations in the partial derivatives with variable coefficients allow to calculate the constants of complicated reactions with the staging mechanism in the presence of hydrocarbons.

Thermal-Magnetic-Elastic Effect Analysis of a Thin Current-Carrying Conical Frustum Shell*

A thermal-magnetic-elastic problem for a thin current-carrying conical frustum shell in a magnetic field is studied. The normal Cauchy form of nonlinear differential equations, which include in total eight basic unknown variables, are obtained by the variable replacement method. Using Newmark’s stable finite equidifferent formulas and the quasi-linearization method, the nonlinear partial differential equations are reduced to a sequence of quasi-linear differential equations, which can be solved by the discrete-orthogonalization method. The temperature field in a thin conical frustum shell and the integral eigenvalues are derived after considering Joule’s heat effect in an electromagnetic field and the thermal equilibrium equation. The change of stresses, displacements, and temperatures in the thin current-carrying conical frustum shell with variation of the electromagnetic parameters is discussed. It is proved that the stresses, strains, and temperatures in thin shells can be controlled by changing the electromagnetic and mechanical parameters by considering a specific example. These results are expected to be a theoretical reference for further analysis of this case.

Impact of thermal radiation on electrical MHD rotating flow of Carbon nanotubes over a stretching sheet

The main objective of this article is to study the inventive conception of the electrical Magneto hydrodynamics (MHD) rotational flow of Single and Multi-Walled Carbon nanotubes (SWCNTs/MWCNTs) base on the fluids (water, engine oil, ethylene glycol and kerosene oil). The thermal radiation impact is taken to be varying the purpose, to see the concentration as well as the temperature modifications between the nanofluid and the surfaces. Kerosene oil is taken as based nanofluids because of its unique attention due to their advanced thermal conductivities, exclusive features and applications. The fluid flow is assumed in steady state. The basic Navier Stocks equations have been transformed through similarity variables in the form of nonlinear differential equations. The solution of the problem has been obtained through Homotopy Analysis Method (HAM). Results obtained for single and multi-wall carbon nanotubes are compared. Plots have been presented in order to examine how the velocities and temperature profile get affected by various flow parameters. The numerical outputs of the physical properties are shown trough tables. The impact of Skin fraction coefficient and Nusselt number are shown in tables.

Intercomparison of Warm-Rain Bulk Microphysics Schemes using Asymptotics

Abstract Clouds are important components of the atmosphere. As it is usually not possible to treat them as ensembles of huge numbers of particles, parameterizations on the basis of averaged quantities (mass and/or number concentration) must be derived. Since no first-principles derivations of such averaged schemes are available today, many alternative approximating schemes of cloud processes exist. Most of these come in the form of nonlinear differential equations. It is unclear whether these different cloud schemes behave similarly under controlled local conditions, and much less so when they are embedded dynamically in a full atmospheric flow model. We use mathematical methods from the theory of dynamical systems and asymptotic analysis to compare two operational cloud schemes and one research scheme qualitatively in a simplified context in which the moist dynamics is reduced to a system of ODEs. It turns out that these schemes behave qualitatively differently on shorter time scales, whereas at least their long time behavior is similar under certain conditions. These results show that the quality of computational forecasts of moist atmospheric flows will generally depend strongly on the formulation of the cloud schemes used.

Analyses of Bio-Based Nano-PCM filled Concentric Cylindrical Energy Storage System in Vertical Orientation

Abstract In this work, we have investigated the detailed melting process of a nano-PCM inside a concentric cylindrical thermal energy storage (C-TES) system. The C-TES system is made of two concentric cylinders where inner cylinder acts as a thermal source having a constant surface temperature. The space between two cylinders is filled with a bio-based nano-PCM. Copper oxide (CuO) nanoparticles are dispersed in coconut oil bio-based PCM to form the nano-PCM in this research. Initially, a mathematical model is developed for the proposed C-TES system in the form of non-linear differential equations. The modelled differential equations are solved numerically to visualize the melting process, track the interface between the solid and liquid phase of nano-PCM, calculate the transient heat transfer rate, and determine the melt fraction. The effect of nanoparticles volume fraction and size of the cylinders on the melting and heat transfer performance are investigated extensively. A comprehensive experimental setup is developed to execute the visualization experiment and compare the melting process of PCM and nano-PCM. Both numerical and experimental results exhibit that the melting rate is faster in nano-CPM when compared to the base-PCM.

Reconstruction of normal forms by learning informed observation geometries from data

The discovery of physical laws consistent with empirical observations is at the heart of (applied) science and engineering. These laws typically take the form of nonlinear differential equations depending on parameters; dynamical systems theory provides, through the appropriate normal forms, an "intrinsic" prototypical characterization of the types of dynamical regimes accessible to a given model. Using an implementation of data-informed geometry learning, we directly reconstruct the relevant "normal forms": a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics. Interestingly, the state variables and the parameters of these realizations are inferred from the empirical observations; without prior knowledge or understanding, they parametrize the dynamics intrinsically without explicit reference to fundamental physical quantities.

СЖИМАЕМОСТЬ ДИЗЕЛЬНОГО ТОПЛИВА В ТРУБОПРОВОДАХ ВЫСОКОГО ДАВЛЕНИЯ АВТОМОБИЛЬНЫХ ДВС

According to the mathematical model in the form of non-linear differential equations investigated the influence of compressibility factors and dynamic viscosity of diesel fuel by changing the pressure in the fuel system of a diesel engine with output.

Real-Time Lumped Parameter Modeling of Cardiovascular Dynamics Using Electrocardiogram Signals: Toward Virtual Cardiovascular Instruments

We present an approach to deriving a real-time, lumped parameter cardiovascular dynamics model that uses features extracted from online electrocardiogram (ECG) signal recordings to generate certain surrogate hemodynamic signals. The model represents the coupled dynamics of the heart chambers, valves, and pulmonary and systemic blood circulation loops in the form of nonlinear differential equations. The features extracted from ECG signals were used to estimate the timings and amplitudes of the atrioventricular activation input functions as well as other model parameters that capture the effect of cardiac morphological and physiological characteristics. The model was tested using hemodynamic signals from the PhysioNet MGH/MF Waveform database. The results suggest that the model can capture the salient time and frequency patterns of the measured central venous pressure, pulmonary arterial pressure, and respiratory impedance signals (R(2) > 0.65). We have developed a method based on Anderson-Darling statistic and Kullback-Leibler divergence to compare the clinical measures (i.e., systolic and diastolic pressures) estimated from model waveform-extrema with those from actual measurements. The test statistics of the model waveform-extrema were statistically indistinguishable from the measured values with beat-to-beat rejection rates of 10%. The results indicate the potential of a virtual instrument that uses the model-derived signals for clinical diagnosis in lieu of expensive instrumentation.

DYNAMIC ANALYSES OF A FLEXIBLE VEHICLE MOVING ALONG A FLEXIBLE GUIDEWAY CONSIDERING THE TIP-OFF EFFECT

A semi-analytical solution for the tip-off response of a vehicle moving along a guideway is obtained, considering the dynamic interaction between the two subsystems. The guideway is modeled as an inclined simply-supported uniform flexible beam, and the vehicle as a flexible free-free beam under a pre-specified thrust force. The equations of motion for the vehicle and guideway are developed using the Lagrangian approach and the assumed mode method based on the Euler–Bernoulli hypothesis. In the form of nonlinear differential equations, they are solved by the Petzold-Gear backward differentiation formula (BDF) method. The solutions obtained are validated by comparing them with the published results for the models with a rigid vehicle running over a rigid guideway or a flexible guideway. Comparisons of the present solutions with the existing ones for the vehicle and guideway reveal the advantages of the approach proposed herein. Other effects on the tip-off responses of the vehicle that are investigated include the length of the guideway, distance between the shoes of the vehicle, and mass and rigidity ratios of the vehicle to the guideway. The results presented herein provide valuable information for the design of the vehicle launch system.

Analysis of the equilibrium positions of nonlinear dynamical systems in the presence of coarse-graining disturbance in space

Loosely speaking, a coarse-grained space is a space in which the generic point is not infinitely thin, but rather has a thickness; and here this feature is modelled as a space in which the generic increment is not dx, but rather (dx) α , 0<α<1. The purpose of the article is to analyze the non-linearity so induced by this coarse-graining effect. This approach via (dx) α leads us to the use of fractional analysis which so provides models in the form of nonlinear differential equations of fractional order. The purpose of the paper is to examine how the coarse-graining effect affects the equilibrium position of dynamical systems, and to this end, one uses the linearization technique on the one hand, and the Lyapunov function approach on the other hand in the framework of fractional analysis. It is shown that in some instances, the coarse-graining phenomenon can convert the initial stable system into an unstable system.

An approach via fractional analysis to non-linearity induced by coarse-graining in space

Loosely speaking, a coarse-grained space is a space in which the generic point is not infinitely thin, but rather has a thickness; and here this feature is modelled as a space in which the generic increment is not d x , but rather ( d x ) α , 0 < α < 1 . The purpose of the article is to analyse the non-linearity induced by this coarse-graining effect. This approach via ( d x ) α leads us to the use of fractional analysis which thus provides models in the form of nonlinear differential equations of fractional order. Two illustrative examples are considered. In the first one, one shows that a particle which has a Gaussian white noise in a coarse-grained spaces exhibits a trajectory which looks like a generalised fractional Brownian motion. The second example shows how a simple one-dimensional linear dynamics is converted into a non-linear system of fractional order.

Non-steady Flow Method for Measuring the Micro Flow at Constant Micropressure

In order to measure the micro flow in micro channel precisely and easily, a new measuring method based on the idea of utilizing non-steady flow state to be equivalent to steady flow state used in traditional measurement is proposed. The applicable condition of pressure head for the new measuring method is derived. Based on Lagrange integral principle in fluid dynamics, a mathematic model in form of non-linear differential equation is built and analyzed numerically by using MATLAB software, so the theoretical error of the proposed method is obtained. The numerical results reveal that the difference between measurement result under the non-steady flow state and that under the steady flow state reduces with the decreasing of the ratio of the areas of the micro channel and the tube in upper stream when the pressure loss in the tube is omitted. Then, an apparatus for measuring the micro flow is established according to the proposed method. The measured results from the apparatus agree well with the theoretical calculation results. Both results show that the measuring error is lower than 2.27% when the applicable condition of pressure head is satisfied and the ratio of cross sectional areas of the micro channel and the tube in upper stream is smaller than 3.422%.

Modeling and control of MEMS with high speed synchronous micromotors and controllers/drivers-on-VLSI-chip ICs

This articles addresses and solves the modeling, analysis and control problems for microelectromechanical systems (MEMS) that integrate high speed permanent magnet synchronous micromotors (motion microdevices) and controllers/drivers-on-VLSI-chip ICs. High performance micromotors (2, 4 and 6 mm diameter) with and without Hall effect microsensors are fabricated using high aspect ratio and surface micromachining techniques, while the ICs are made using biCMOS technology. In addition to fabrication, a wide spectrum of fundamental problems in MEMS synthesis, modeling, analysis and control are solved. The MEMS synthesis is performed using basic principles with the ultimate goal to guarantee operationability, functionality, compatibility, integrity, controllability and synergy. Synchronous micromotors are designed by applying the MEMS Synthesis and Classification Solver. Signal processing, filtering, computing, interfacing and amplification are performed by the controllers/drivers-on-VLSI-chip ICs that control the micromotors by properly applying the phase voltages to the micromotor windings. Analysis, control and optimization are based on mathematical models. Mathematical modeling is done by applying basic principles and the fundamental laws of electromechanics. The derived lumped parameter mathematical model, which is given in the form of nonlinear differential equations, allows one to perform the data intensive analysis, heterogeneous simulations with outcome prediction and controller design. Based on micromotor electromagnetics–electromechanics and microelectronics capabilities, we derive control algorithms. An innovative synthesis procedure is reported to design soft switching continuous controllers with nonlinear switching surfaces. These control laws, which ensure robust tracking and disturbance rejection, are implemented. The algorithms designed are different from the existing variable structure control laws. Compared with hard switching discontinuous controllers, the advantages of the soft switching sliding mode control paradigm are that the singularity and sensitivity problems are avoided, robustness and stability are improved, chattering (high frequency switching) is eliminated, etc. These features expand the MEMS operating envelopes, enhance robustness and controllability, as well as improve efficiency and reliability. Experimental verification to assess the fundamental results is performed. Integrated MEMS and their components are fabricated and tested. We study the MEMS performance, verify mathematical models, analyze applicability and effectiveness of soft switching algorithms for micromotors controlled by ICs, etc.

Solid film at the anode influences the process of polyvalent metal electrorefining: theoretical bifurcation analysis of the problem

A theoretical study of polyvalent metal anodic electrorefining has been carried out, and the decisive influence of surface film on the long-term process has been shown. A mathematical model of the long-term continuous electrorefining process in the form of non-linear differential equations is proposed. The developed model was investigated by the methods of the bifurcation theory. The model predicts the following types of electrolytic system: 1. The low-valence compound of the anodic film does not form. 2. A steady-state process with a stationary thickness of the anodic deposit. 3. An unlimited accumulation of low-valence compounds at the anode. 4. An unstable steady state: a bifurcation with a transition to case 3 or to the regime of undamped oscillations. The model provides satisfactory explanations of the experimental results related to long-term continuous electrorefining processes in molten salts.

Selectivity and Yield of Exothermic Consecutive Reactions in Catalytically Active Porous Membrane Reactor.

This paper analyzes the performance of a membrane reactor, in which the inside porous structure is impregnated with catalysts, under nonisothermal conditions for exothermic complex consecutive reactions. The mathematical model based on an extended Fick’s model, which incorporates both diffusive and convective fluxes inside the membrane pores, was developed in the form of nonlinear differential equations to be solved numerically. The selectivity and yield of the desired intermediate product were calculated as a function of critical dimensionless parameter group’s such as the mass and heat Peclet groups and the Thiele modulus. It was clarified with the calculated and experimental results that the selectivity and yield of intermediates are significantly affected by convective flow inside the macro-pores.

Neural networks for solving linear inequality systems

In this paper a neural network approach to the on-line solution of linear inequality systems is considered. Three different techniques are discussed and for each technique a novel neural network implementation is proposed. The first technique is a standard penalty method implemented as an analog neural network. The second technique is based on the transformation of inequality constraints into equality constraints with simple bounds on the variables. The transformed problem is then solved using least squares (LS) and least absolute values (LAV) optimisation criteria. The third technique makes use of the regularised total least squares criterion (RTLS). For each technique a suitable neural network architecture and associated algorithm in the form of nonlinear differential equations has been developed. The validity and performance of the proposed algorithms has been verified by computer simulation experiments. The analog neural networks are deemed to be particularly well suited for high throughput, real time applications.

NONLINEAR MATHEMATICAL MODELS OF HORMONAL SYSTEMS

The paper considers various approaches to mathematical modelling of endocrine systems. The functional and operational complexity of hormonal activities turns out to be the result of the cooperation of three factors: global feedback structures on the level of glands, subtle feedback and regulatory mechanisms on the level of single cells and molecules (including messengers, receptors and functional proteins like G-proteins) and finally, coupling to other organs (predominantly to the brain, e.g. via hypothalamus). To date, it is practically impossible to construct a mathematical model comprising together all these aspects. The paper aims at providing some major building bricks to such an endeavor. In the first part we summarize some of our recent models on the gobal structure of hormonal systems, in the form of nonlinear differential equations containing delay terms; oscillatory input from the brain is taken into account. Solutions of the equations display nearly all kinds of dynamical behaviour as stable limit cycles, phase locking, quasi-periodic and chaotic motions. Special emphasis is put on developing a mathematical model for the fine-tuned sequence of hormone-induced transmembrane signalling, where agonist couples to some cellular effector via transfer-proteins — this principle is widely spread among the hormone-targeted cells and crucially involved in regulating cells' behaviour towards external stimuli, e.g. their ability to desensitize as a reaction to sustained hormonal input.

Collective control without explicit coding: The case of communal nest excavation

The nests of social insects result from a succession of stimulus responses steps involving the environment, the workers, and the by-product of their activities (which modify their environment). In this way social insects can build without any reference to a blueprint. In this paper we explore the link between individual building behavior and the characteristics (form, size, location, etc.) of the structures produced. We show with a mathematical model (in the form of nonlinear differential equations) that social insects using behavioral mechanisms, which do not require an explicit measure of the nest and the colony size, can nevertheless effectively regulate, at the level of the colony, the size of the nest in response to changes in the size of the colony population. In addition, even though individual workers do not directly compare environmental characteristics, the colony can expand the nest “preferentially” in the most favorable zone. The models used show how such regulations and decision making can be a by-product of an amplifying communication between the builders and their work and how different patterns of building through time can be generated tuning the same basic rules.