Nonlinear Equation Model Research Articles

Effects of heavy metals on Antarctic bacterial cell growth kinetics and degradation of waste canola oil

Aim: The aim of the present study was to study the effect of heavy metals on growth kinetics of Antarctic bacterial in degradation of waste canola oil. Methodology: The BS14 Antarctic bacterial community was introduced in the minimal salt media containing 1 ppm of heavy metals (Cd, Cr, Al, Zn, Ni, As and Co) with 1% waste canola oil, and the effects of heavy metals on biodegradation of waste canola oil was analysed by gravimetric analysis. The turbidity of bacteria was obtained through UV-visible spectroscopy at 600 nm of wavelength for every 24 hr within seven days of incubation period, and the data were regressed with linear and non-linear kinetic equations. Results: The results demonstrated that Co was the most active metal that led to 4.217% increase in waste canola oil and the least active metal in biodegradation of oil was zinc, as it degraded the waste canola oil only to 29.26%. Overall, the bacterial growth was inhibited in increasing order of Al > Cd> As> Zn> Ni> Cr> Co whereas the waste canola oil biodegradation was inhibited in the order of Zn> Cr> Ni> Al> Cd> As> Co. The best fitted-regression model was determined by comparing the kinetic parameters estimated between linear and non-linear model equations, where the R2 value for non-linear regression was highest at 0.8421, and low sy.x at 0.324 for Ni with a maximum growth rate (0.01131 hr-1) of the Antarctic bacterial in degrading waste canola oil, meantime best-fitted in the linear regression model was Zn with high R2 and growth constant values (0.9082 and 0.2075 hr-1, respectively) as well as low value of statistical error, which was 0.2075. Interpretation: The presence of heavy metals in Antarctic bacterial community could suppress the ability of bacteria to degrade waste canola oil, and this can slower the rate of bacterial growth in the kinetics studies. Hence, this work would be helpful in actual bioremediation operations by understanding and manipulating the process of the kinetics parameters.

Stable solutions to the nonlinear RLC transmission line equation and the Sinh–Poisson equation arising in mathematical physics

AbstractThe Sinh–Poisson equation and the RLC transmission line equation are important nonlinear model equations in the field of engineering and power transmission. The modified simple equation (MSE) procedure is a realistic, competent and efficient mathematical scheme to ascertain the analytic soliton solutions to nonlinear evolution equations (NLEEs). In the present article, the MSE approach is put forward and exploited to establish wave solutions to the previously referred NLEEs and accomplish analytical broad-ranging solutions associated with parameters. Whenever parameters are assigned definite values, diverse types of solitons originated from the general wave solutions. The solitons are explained by sketching three-dimensional and two-dimensional graphs, and their physical significance is clearly stated. The profiles of the attained solutions assimilate compacton, bell-shaped soliton, peakon, kink, singular periodic, periodic soliton and singular kink-type soliton. The outcomes assert that the MSE scheme is an advance, convincing and rigorous scheme to bring out soliton solutions. The solutions obtained may significantly contribute to the areas of science and engineering.

Simulating waves on a horizontal liquid film entrained by a gas flow

Nonlinear waves on a horizontal liquid film surface are considered. The effect of adjacent gas flow is taken into account through the data on shear stress at the film-gas interface obtained by the Boussinesq model of turbulence. A model nonlinear equation for the film thickness deviation from the undisturbed level is used to simulate nonlinear wave modes. Weakly nonlinear steady-state travelling solutions of this equation with wave numbers located in the vicinity of neutral wave numbers are constructed analytically. The evolution of periodic perturbations with wave numbers lying in the depth of the linear instability region is also considered numerically. Several typical scenarios of their evolution have been identified.

Nonlinear hybrid continuum–discrete dynamic model of influence of hydrogen concentration on strength of materials

A new nonlinear lattice model for an influence of the hydrogen concentration on the elastic constants of the lattice model of a material is developed. A weakly nonlinear long-wavelength continuum model is considered, and a model nonlinear equation for the dynamics of concentration of hydrogen is obtained asymptotically. The model predicts a decrease in the stiffness coefficient as well as a light local moving increase due to the localized nonlinear concentration wave propagation.

A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations

This work aims to propose a new analyzing tool, called the fractional iteration algorithm I for finding numerical solutions of nonlinear time fractional-order Cauchy reaction-diffusion model equations. The key property of the suggested technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The proposed approach can be utilized without the use of any transformation, Adomian polynomials, small perturbation, discretization or linearization. The main feature of the fractional iteration algorithm-I is the improvement of an auxiliary parameter that can ensure a rapid convergence. To check the stability, accuracy and speed of the method, obtained results are compared numerically and graphically with the exact solutions and results available in the latest literature. In addition, numerical results are displayed graphically for various cases of the fractional-order α. These results demonstrate the viability of the proposed technique and show that this technique is exceptionally powerful and suitable for solving fractional PDEs.

Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order

With the help of Maple, the precise traveling wave solutions of three fractal-order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended (G′/G)-expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional-order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.

Working limit of the circular saws in the cutting of high density fiberboard (HDF) through a nonlinear model and differential equations

The wood industries focus their efforts on improving the productivity and efficiency of their operations to increase their competitiveness. There is a notable interest in the model development to assist autonomous control systems within the operations of the wood industry. For this, it is very important to know the behavior of the different outputs in the production process, such as the power consumption during cutting, since it provides vital information on the state of the wear of the cutting tools even when they are in operation. This allows knowing when to change the cutting tool in advance to avoid any negative effects on the products. In this article, a non-linear fit composed of a modeled function with quasi-logarithmic behavior and an exponential part of the power consumption based on a cutting length in circular saws in high density fiberboard (HDF) is presented. The working limit of the cutting tools was determined by means of differential equations identifying the inflection point in the function. The theoretical model was experimentally corroborated in a multi-objective way by relating the results of power consumption, the surface and the dimensional quality of the boards, the cutting length and the calculated inflection point. It can be affirmed that the developed non-linear model presents a good approximation towards the final part of the function (region of practical interest), the moment when it is necessary to change the cutting tools. By applying the suggested methodology, it is possible to avoid all the described negative consequences of an uncontrolled machining process in an adaptive way to the natural variability of a real industrial production.

Traveling and localized solitary wave solutions of the nonlinear electrical transmission line model equation

Analytical solutions describing the kink–antikink type, hyperbolic, trigonometric, doubly periodic, localized bright and dark soliton like, breather and rational periodic train solutions to the discrete nonlinear electrical transmission line in $$(2+1)$$ -dimensions have been investigated with the Kudryashov method, the fractional linear transform method, and the simplest equation method with its modified and extended versions. Consequently, many new and more general explicit forms of traveling waves are retrieved under parametric conditions. The 3D, 2D, and density profile of some of the obtained results are numerically simulated by selecting suitable values of the various parameters of the NLTL model equation. Furthermore, all the obtained results were checked and verified by putting back into the considered nonlinear transmission line model equation with Maple and Mathematica software. The acquired results show the validity and reliability of the proposed schemes to the studied nonlinear electrical transmission line model.

Nonlinear forced vibration of the magnetic gear system with the internal resonance

Considering the nonlinear magnetic coupling stiffness among components of the magnetic gear with intersecting axes and a straight SP (MGIASS), the nonlinear dynamic model and differential equations of the MGIASS system are established. Then, the expression of the forced response is obtained when the excited frequency is close to the torsional mode frequency of the input rotor. The forced responses analysis of the example system shows that: the response frequencies include the excited frequency and the double frequency. Multiple proximity between the natural frequencies directly affects the stability and resonance amplitude.

Continuous-wave stability and multipulse structures in a universal complex Ginzburg–Landau model for passively mode-locked lasers with a saturable absorber

The stability of continuous waves and the dynamics of multipulse structures are studied theoretically for a generic model of passively mode-locked fiber lasers with arbitrary nonlinearity. The model is characterized by a complex Ginzburg–Landau equation with a nonlinear term I m / ( 1 + Γ I ) n , where I is the field intensity, m and n are real numbers, and Γ is the saturation power. Fixed-point solutions of the governing equations reveal an interesting loci of singular points in the amplitude-frequency plane consisting of zero, one, or two fixed points, depending on the values of m and n . The continuous-wave stability is analyzed within the framework of the modulational-instability theory, and the results demonstrate a bifurcation in the continuous-wave amplitude growth rate and propagation constant characteristic of multiperiodic wave structures. In the full nonlinear regime, these multiperiodic wave structures turn out to be multipulse trains, unveiled via numerical simulations of the model nonlinear equation, the rich variety of which is highlighted by considering different combinations of values for the pair ( m , n ). The results are consistent with previous analyses of the dynamics of multipulse structures in several contexts of passively mode-locked lasers with a saturable absorber, as well as with predictions about the existence of multipulse structures and bound-state solitons in optical fibers with strong optical nonlinearity, such as cubic-quintic and saturable nonlinearities.

Accurate method for PV solar cells and modules parameters extraction using I–V curves

The main contribution of this paper is proposing a new approach for retrieving the five parameters of the single diode equivalent model (SDM) of photovoltaic cells/modules including the series and shunt resistances. The least square method is used as an error minimization technique for fitting the non-linear transcendental model equation of the solar panel to the measured I–V characteristics. Newton Raphson method is applied to solve the system of five non-linear equations which represent the error of each parameter. Initial guess values are calculated with an optimised algorithm depending on information extracted from the same measured data. MATLAB programming script was used in all implementation steps. This approach is useful for a wide variety of applications where the five SDM parameters have to be determined from the measured curves, particularly, for self-fabricated cells/modules or in case of no available datasheet. One of the strengths of this method is the higher level of accuracy because of the absence of mathematical simplifications and physical assumptions. The method was validated on different types of PV devices, including a crystalline silicon cell, a polycrystalline module, and an amorphous module by using measurement data obtained at a wide range of solar irradiance conditions and temperatures.

Effect of Cattaneo-Christov Heat Flux on Radiative Hydromagnetic Nanofluid Flow between Parallel Plates using Spectral Quasilinearization Method

In this paper, we numerically solve the equations for hydromagnetic nanofluid flow past semi-infinite parallel plates where thermal radiation and a chemical reaction are assumed to be present and significant. The objective is to give insights on the important mechanisms that influence the flow of an electrically conducting nanofluid between parallel plates, subject to a homogeneous chemical reaction and thermal radiation. These flows have great significance in industrial and engineering applications. The reduced nonlinear model equations are solved using a Newton based spectral quasilinearization method. The accuracy and convergence of the method is established using error analysis. The changes in the fluid properties with various parameters of interest is demonstrated and discussed. The spectral quasilinearization method was found to be rapidly convergent and accuracy is shown through the computation of solution errors.

Research on SOC balance strategy of STEKF battery based on MMC-BESS model

Under the condition of the battery current fluctuation and the uncertain initial value of the battery SOC in the MMC(Modular Multilevel Converter) battery energy storage system, there are relatively big errors in the prediction of SOC estimation values and there are poor results of the battery SOC balance, by using the ampere-hour integration method and the open-circuit voltage method. The method of model prediction was adopted to study the SOC balance strategy. Firstly, the power transfer function between each port is derived and the second-order Thevenin equivalent nonlinear model of the battery is established. On this foundation, the more accurate estimation results of the battery SOC are obtained employing the strong tracking extended Kalman filter algorithm, under the condition of the large fluctuation of the battery current and the uncertainty initial value of the battery SOC in the MMC-BESS model. In the end of the paper, the validity of the method introduced in the paper way was confirmed according to the simulation results of the simulation calculation based on the power transfer relationship between the battery SOC and each port, using the three-level SOC balance strategy to balance the battery SOC.

Blow-up instability in non-linear wave models with distributed parameters

We consider two model non-linear equations describing electric oscillations in systems with distributed parameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations for classical solutions of the Cauchy problem and the first and second initial-boundary value problems for the original equations in the half-space . Using the contraction mapping principle, we prove the local-in-time solubility of these problems. For one of these equations, we use the Pokhozhaev method of non-linear capacity to deduce a priori bounds giving rise to finite-time blow-up results and obtain upper bounds for the blow-up time. For the other, we use a modification of Levine’s method to obtain sufficient conditions for blow-up in the case of sufficiently large initial data and give a lower bound for the order of growth of a functional with the meaning of energy. We also obtain an upper bound for the blow-up time.

Wilton Ripples in Weakly Nonlinear Dispersive Models of Water Waves: Existence and Analyticity of Solution Branches

Traveling waves on the surface of the ocean play an important role in many oceanographic processes which necessitates a detailed quantitative understanding of their properties. The water wave equations, which govern the free-surface evolution of an ideal fluid, are the most successful model for this phenomena, but are exceedingly difficult to analyze due to their strongly nonlinear character and the fact that they are posed on a domain with moving boundary. For this reason, weakly nonlinear dispersive models are an essential tool for practitioners, and in this contribution, we study traveling wave solutions of a broad class of such models. The simplest family of traveling wave solutions are the Stokes waves which can be characterized as simple bifurcation (one-dimensional null space of the linearized operator) from the trivial (flat-water) branch of solutions. We focus our analysis on the much less studied non-simple case of Wilton ripples which have linear behavior characterized by two co-propagating harmonics (a two-dimensional null space of the linearized operator). More specifically, we show that such branches of solutions exist for a class of nonlinear dispersive model equations, and that they are analytic with respect to a natural wave height/slope parameter.

Bi-stability behavior in electrostatically actuated non-contact based micro-actuator

The nonlinear mechanical response of an electrostatically actuated doubly-clamped micro-actuator assuming a non-contact based actuation arrangement is examined. The electrostatic actuating force is numerically approximated through solving a nonlinear electrical field problem using a finite-elements based analysis. An electro-mechanical model is established within the framework of the Euler–Bernoulli continuous nonlinear beam theory, where both the nonlinear geometric mid-plane stretching and the nonlinear electric forcing effects have been both taken into consideration. Then, assuming a Galerkin’s based modal expansion technique, the structural behavior of the micro-actuator under the effect of the resultant electric out-of-plane fields is solved numerically. The resultant nonlinear reduced-order model equations are examined to get the deflection of the microbeam against an assumed bias voltage for different vertical initial gap sizes. The snap-through and pull-out voltages of the micro-system are calculated for different assumed initial vertical gap separation. Frequency diagrams are analyzed against any applied DC bias voltage through solving a linearized eigenvalue problem. Thorough parametric simulations indicate a prospect of such bi-stable design of the micro-actuator with large stroke and high fundamental frequency enabling it to be potentially useful for MEMS based high frequency energy harvester devices with bi-stable capabilities.

Solitary wave solutions to some nonlinear fractional evolution equations in mathematical physics

The objective of this article is to construct new and further general analytical wave solutions to some nonlinear evolution equations of fractional order in the sense of the modified Riemann-Liouville derivative relating to mathematical physics, namely, the space-time fractional Fokas equation, the time fractional nonlinear model equation and the space-time fractional (2 + 1)-dimensional breaking soliton equation by exerting a rather new mechanism (G'/G,1/G) -expansion method. We use the fractional complex transformation and associate the fractional differential equations to the solvable integer order differential equations. A comprehensive class of new and broad-ranging exact traveling and solitary wave solutions are revealed in terms of trigonometric, rational and hyperbolic functions. The attained wave solutions are sketched graphically by using Mathematica and make a comparison to the results attained by the presented technique with other techniques in a comprehensive manner. It is notable that the method can be considered as a reduction of the reputed (G'/G) -expansion method commenced by Wang et al. It is noticeable that, the two variable (G'/G,1/G) -expansion method appears to be more reliable, straightforward, computerized and user-friendly.

Enhanced performance of PID controllers for unstable time delay systems using direct synthesis method

ABSTRACT Design proportional–integral–derivative (PID) controllers for unstable first-order plus time delay (FOPTD) and unstable second-order plus time delay (SOPTD) systems with/without a zero is proposed by direct synthesis method. For unstable FOPTD system, a series PID controller is considered. The derivative time of series PID controller is made equal to 0.5 times time delay to cancel the numerator zero with the pole of the loop transfer function (product of process transfer function and controller transfer function). The reduced characteristic equation is compared with the desired characteristic equation to get PI settings. The conventional PID controller parameters are obtained from series PID controller. For unstable SOPTD systems, PID controller with a first-order lead/lag compensator is considered. The controllers designed by the proposed method are implemented on various transfer function models, nonlinear model equations of bioreactor and on inverted pendulum (IP) set-up to show the efficiency of the proposed method. The performance of the proposed controllers is compared with the controllers reported in the recent literature in terms of IAE. The robustness of the controller is determined by the maximum magnitude of sensitivity function, phase margin and gain margin.

Position-Estimation Deviation-Suppression Technology of PMSM Combining Phase Self-Compensation SMO and Feed-Forward PLL

The sensorless drive method of the permanent magnet synchronous motor (PMSM) has attracted wide attention for its low cost and high reliability. As a critical technology, a fast and high-precision rotor-position estimation is essential. This article addresses the position-estimation deviation issue of the sensorless drive method based on the sliding-mode observer (SMO) and the phase-locked loop (PLL). A nonlinear equivalent model of the SMO is established to analyze and compensate for the position-estimation deviation caused by the SMO, and a feed-forward PLL is employed to suppress the steady-state position tracking error under variable speed operation. First, the phase-frequency characteristic of the SMO is obtained by studying the SMO and the switching functions in detail. Then, the analysis of the conventional PLL is carried out in terms of the error-transfer function. In addition, the position-estimation performance of the feed-forward PLL is discussed with the dynamic error coefficient method. Theoretical analysis and experimental evaluation validated the effectiveness of the proposed position-estimation deviation-suppression technology of the PMSM, combining the phase self-compensation SMO and the feed-forward PLL.

УЗАГАЛЬНЕНИЙ РОЗВ'ЯЗОК ЗАДАЧІ ДІРІХЛЕ ДЛЯ МОДЕЛЬНОГО АНІЗОТРОПНОГО ВАГОВОГО РІВНЯННЯ

The paper deals with the Dirichlet problem for a model nonlinear degenerate anisotropic elliptic second-order equation. Anisotropy and degeneration (with respect to the independent variables) is characterized by the presence of different exponents q1 , q2 and weighted functions |x|^q1 та |x|^q2 in the left side of the equation. The main result of the paper is theorem on the existence of the generalized solution of the Dirichlet problem under consideration.