scholarly journals Sensitivity, Equilibria, and Lyapunov Stability Analysis in Droop’s Nonlinear Differential Equation System for Batch Operation Mode of Microalgae Culture Systems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2192
Author(s):  
Abraham Guzmán-Palomino ◽  
Luciano Aguilera-Vázquez ◽  
Héctor Hernández-Escoto ◽  
Pedro Martin García-Vite
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Value Added ◽  
Biofuel Production ◽  
State Variables ◽  
Minimal Cell ◽  
Cell Quota ◽  
Depth Analysis ◽  
Number Of Equilibria

Microalgae-based biomass has been extensively studied because of its potential to produce several important biochemicals, such as lipids, proteins, carbohydrates, and pigments, for the manufacturing of value-added products, such as vitamins, bioactive compounds, and antioxidants, as well as for its applications in carbon dioxide sequestration, amongst others. There is also increasing interest in microalgae as renewable feedstock for biofuel production, inspiring a new focus on future biorefineries. This paper is dedicated to an in-depth analysis of the equilibria, stability, and sensitivity of a microalgal growth model developed by Droop (1974) for nutrient-limited batch cultivation. Two equilibrium points were found: the long-term biomass production equilibrium was found to be stable, whereas the equilibrium in the absence of biomass was found to be unstable. Simulations of estimated parameters and initial conditions using literature data were performed to relate the found results to a physical context. In conclusion, an examination of the found equilibria showed that the system does not have isolated fixed points but rather has an infinite number of equilibria, depending on the values of the minimal cell quota and initial conditions of the state variables of the model. The numerical solutions of the sensitivity functions indicate that the model outputs were more sensitive, in particular, to variations in the parameters of the half saturation constant and minimal cell quota than to variations in the maximum inorganic nutrient absorption rate and maximum growth rate.

scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

scholarly journals A Rational Numerical Method for Simulation of Drop-Impact Dynamics of Oleo-Pneumatic Landing Gear

2021 ◽  
Vol 11 (9) ◽  
pp. 4136
Author(s):  
Rosario Pecora
Keyword(s):  
Numerical Method ◽  
Initial Conditions ◽  
Impact Load ◽  
Numerical Models ◽  
Landing Gear ◽  
Design Stage ◽  
Impact Dynamics ◽  
State Variables ◽  
Adopted Model ◽  
The Impact

Oleo-pneumatic landing gear is a complex mechanical system conceived to efficiently absorb and dissipate an aircraft’s kinetic energy at touchdown, thus reducing the impact load and acceleration transmitted to the airframe. Due to its significant influence on ground loads, this system is generally designed in parallel with the main structural components of the aircraft, such as the fuselage and wings. Robust numerical models for simulating landing gear impact dynamics are essential from the preliminary design stage in order to properly assess aircraft configuration and structural arrangements. Finite element (FE) analysis is a viable solution for supporting the design. However, regarding the oleo-pneumatic struts, FE-based simulation may become unpractical, since detailed models are required to obtain reliable results. Moreover, FE models could not be very versatile for accommodating the many design updates that usually occur at the beginning of the landing gear project or during the layout optimization process. In this work, a numerical method for simulating oleo-pneumatic landing gear drop dynamics is presented. To effectively support both the preliminary and advanced design of landing gear units, the proposed simulation approach rationally balances the level of sophistication of the adopted model with the need for accurate results. Although based on a formulation assuming only four state variables for the description of landing gear dynamics, the approach successfully accounts for all the relevant forces that arise during the drop and their influence on landing gear motion. A set of intercommunicating routines was implemented in MATLAB® environment to integrate the dynamic impact equations, starting from user-defined initial conditions and general parameters related to the geometric and structural configuration of the landing gear. The tool was then used to simulate a drop test of a reference landing gear, and the obtained results were successfully validated against available experimental data.

Biohydrogen production from used diapers: Evaluation of effect of temperature and substrate conditioning

2017 ◽  
Vol 35 (3) ◽  
pp. 267-275 ◽  
Author(s):  
PX Sotelo-Navarro ◽  
HM Poggi-Varaldo ◽  
SJ Turpin-Marion ◽  
A Vázquez-Morillas ◽  
M Beltrán-Villavicencio ◽  
...  
Keyword(s):  
Value Added ◽  
Clean Energy ◽  
Solid Substrate ◽  
Biofuel Production ◽  
Biohydrogen Production ◽  
Effect Of Temperature ◽  
Batch Reactors ◽  
Disposable Diapers ◽  
Batch Bioreactors ◽  
H2 Yield

This research assessed the viability to use disposable diapers as a substrate for the production of biohydrogen, a valuable clean-energy source. The important content of cellulose of disposable diapers indicates that this waste could be an attractive substrate for biofuel production. Two incubation temperatures (35 °C and 55 °C) and three diaper conditioning methods (whole diapers with faeces, urine, and plastics, WD; diapers without plastic components, with urine and faeces, DWP; diapers with urine but without faeces and plastic, MSD) were tested in batch bioreactors. The bioreactors were operated in the solid substrate anaerobic hydrogenogenic fermentation with intermittent venting mode (SSAHF-IV). The batch reactors were loaded with the substrate at ca. 25% of total solids and 10% w/w inoculum. The average cumulative bioH2 production followed the order WD > MSD > DWP. The bio-H2 production using MSD was unexpectedly higher than DWP; the presence of plastics in the first was expected to be associated to lower degradability and H2 yield. BioH2 production at 55 °C was superior to that of 35 °C, probably owing to a more rapid microbial metabolism in the thermophilic regime. The results of this work showed low yields in the production of H2 at both temperatures compared with those reported in the literature for municipal and agricultural organic waste. The studied process could improve the ability to dispose of this residue with H2 generation as the value-added product. Research is ongoing to increase the yield of biohydrogen production from waste disposable diapers.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

scholarly journals Numerical Analysis of the Transport and Fate of Nitrate in the Soil and Nitrate Leaching to Drains

2001 ◽  
Vol 1 ◽  
pp. 170-180 ◽  
Author(s):  
Alaa El-Sadek ◽  
Mona Radwan ◽  
Jan Feyen
Keyword(s):  
Nitrate Leaching ◽  
Initial Conditions ◽  
Continuous Cropping ◽  
Nitrogen Transport ◽  
State Variables ◽  
Nitrogen Application ◽  
Transport And Transformation ◽  
Bottom Boundary Condition ◽  
Rainfall Depth ◽  
Simulation Results

In this study, the transport and fate of nitrate within the soil profile and nitrate leaching to drains were analyzed by comparing historic field data with the simulation results of the DRAINMOD model. The nitrogen version of DRAINMOD was used to simulate the performance of the nitrogen transport and transformation of the Hooibeekhoeve experiment, situated in the sandy region of the Kempen (Belgium) and conducted for a 30-year (1969–1998) period. In the analysis, a continuous cropping with maize was assumed. Comparisons between experimentally measured and simulated state variables indicate that the nitrate concentrations in the soil and nitrate leaching to drains are controlled by the fertilizer practice, the initial conditions, and the rainfall depth and distribution. Furthermore, the study reveals that the model used gives a fair description of the nitrogen dynamics in the soil and subsurface drainage at field scale. From the comparative analysis between experimental data and simulation results it can also be concluded that the model after calibration is a useful tool to optimize as a function of the combination “climate-crop-soil-bottom boundary condition” the nitrogen application strategy resulting in an acceptable level of nitrate leaching for the environment.

Oscillatory flow past a circular cylinder in a rotating frame

1997 ◽  
Vol 345 ◽  
pp. 101-131
Author(s):  
M. D. KUNKA ◽  
M. R. FOSTER
Keyword(s):  
Circular Cylinder ◽  
Steady Flow ◽  
Stagnation Point ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Temporal Integration ◽  
Oncoming Flow ◽  
Flow Past ◽  
Rear Stagnation Point

Because of the importance of oscillatory components in the oncoming flow at certain oceanic topographic features, we investigate the oscillatory flow past a circular cylinder in an homogeneous rotating fluid. When the oncoming flow is non-reversing, and for relatively low-frequency oscillations, the modifications to the equivalent steady flow arise principally in the ‘quarter layer’ on the surface of the cylinder. An incipient-separation criterion is found as a limitation on the magnitude of the Rossby number, as in the steady-flow case. We present exact solutions for a number of asymptotic cases, at both large frequency and small nonlinearity. We also report numerical solutions of the nonlinear quarter-layer equation for a range of parameters, obtained by a temporal integration. Near the rear stagnation point of the cylinder, we find a generalized velocity ‘plateau’ similar to that of the steady-flow problem, in which all harmonics of the free-stream oscillation may be present. Further, we determine that, for certain initial conditions, the boundary-layer flow develops a finite-time singularity in the neighbourhood of the rear stagnation point.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

scholarly journals Numerical daemons of hydrological models are summoned by extreme precipitation

2021 ◽  
Author(s):  
Peter T. La Follette ◽  
Adriaan J. Teuling ◽  
Nans Addor ◽  
Martyn Clark ◽  
Koen Jansen ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Extreme Precipitation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Computational Cost ◽  
Numerical Error ◽  
Hydrological Models ◽  
Multiple Model ◽  
Numerical Techniques ◽  
Modeling Framework

Abstract. Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on approximate numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation like that observed during hurricanes Harvey and Katrina impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation events. In this experiment, a large number of hydrographs is generated with the modular modeling framework FUSE, using eight numerical techniques across a variety of forcing datasets. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational expense and numerical error associated with each hydrograph were recorded. It was found that numerical error (root mean square error) usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both low numerical error and low computational cost. A basic literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be sub-optimal. We conclude that relatively large numerical errors might be common in current models, and because these will likely become larger as the climate changes, we advocate for the use of low cost, low error numerical methods.

Difference Synchronization of Identical and Nonidentical Chaotic and Hyperchaotic Systems of Different Orders Using Active Backstepping Design

2018 ◽  
Vol 13 (5) ◽  
Author(s):  
Eric Donald Dongmo ◽  
Kayode Stephen Ojo ◽  
Paul Woafo ◽  
Abdulahi Ndzi Njah
Keyword(s):  
Chaotic System ◽  
Initial Conditions ◽  
Lyapunov Stability Theory ◽  
The State ◽  
State Variables ◽  
State Variable ◽  
New Type ◽  
The Difference ◽  
Synchronization Scheme ◽  
Hyperchaotic Systems

This paper introduces a new type of synchronization scheme, referred to as difference synchronization scheme, wherein the difference between the state variables of two master [slave] systems synchronizes with the state variable of a single slave [master] system. Using the Lyapunov stability theory and the active backstepping technique, controllers are derived to achieve the difference synchronization of three identical hyperchaotic Liu systems evolving from different initial conditions, as well as the difference synchronization of three nonidentical systems of different orders, comprising the 3D Lorenz chaotic system, 3D Chen chaotic system, and the 4D hyperchaotic Liu system. Numerical simulations are presented to demonstrate the validity and feasibility of the theoretical analysis. The development of difference synchronization scheme has increases the number of existing chaos synchronization scheme.

Effects of geometric similarity ratio on the disturbance amplitude of shock-wave front in viscosity measurement

2021 ◽  
pp. 2150330
Author(s):  
Kai Yang ◽  
Quan-Yu Xu ◽  
Xiao Wu ◽  
Xiao-Juan Ma
Keyword(s):  
Shock Wave ◽  
Phase Shift ◽  
Wave Front ◽  
Shock Wave Front ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Viscosity Measurement ◽  
Geometric Similarity ◽  
Similarity Ratio ◽  
Simulation Results

Geometric similarity ratio is one of the important factors that affects the disturbance amplitude of shock-wave front in viscosity measurement. In this paper, the Euler difference scheme of two-dimensional (2D) equations of viscous fluid mechanics is used to simulate the disturbance amplitude damping curves under different geometric similarity ratios, and the corresponding numerical solutions are shown. The samples of aluminum shocked to 80 GPa are taken as an example. The simulation results show that the initial conditions, material viscosity, wavelength, and sample geometric similarity ratio affect the evolution of the shock front sine wave disturbance. For flyer-impact flow field, the phase shift increases from 0 to a certain value with the viscosity coefficient for sample with wavelength [Formula: see text] mm and geometric similarity ratio [Formula: see text], 0.1. So, the geometric similarity method can be used to measure the viscosity of material. But it is found that the phase shift is sensitive to the geometric similarity ratio, which should be considered in Zaidel’s equation. So, some flyer-impact experiments will be carried out to determine the simulation results, and find the quantity relation of phase shift and viscosity of material in the future investigation.

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