First-order Nonlinear Differential Equations Research Articles

Applications of the Differential Transformation Method and Multi-Step Differential Transformation Method to Solve a Rotavirus Epidemic Model

Epidemic models are essential in understanding the transmission dynamics of diseases. These models are often formulated using differential equations. A variety of methods, which includes approximate, exact and purely numerical, are often used to find the solutions of the differential equations. However, most of these methods are computationally intensive or require symbolic computations. This article presents the Differential Transformation Method (DTM) and Multi-Step Differential Transformation Method (MSDTM) to find the approximate series solutions of an SVIR rotavirus epidemic model. The SVIR model is formulated using the nonlinear first-order ordinary differential equations, where S; V; I and R are the susceptible, vaccinated, infected and recovered compartments. We begin by discussing the theoretical background and the mathematical operations of the DTM and MSDTM. Next, the DTM and MSDTM are applied to compute the solutions of the SVIR rotavirus epidemic model. Lastly, to investigate the efficiency and reliability of both methods, solutions obtained from the DTM and MSDTM are compared with the solutions from the Runge-Kutta Order 4 (RK4) method. The solutions from the DTM and MSDTM are in good agreement with the solutions from the RK4 method. However, the comparison results show that the MSDTM is more efficient and converges to the RK4 method than the DTM. The advantage of the DTM and MSDTM over other methods is that it does not require a perturbation parameter to work and does not generate secular terms. Therefore the application of both methods

Exact boundaries for the analytical approximate solution of a class of first-order nonlinear differential equations in the real domain

Дано решение одной из задач аналитического приближенного метода для одного класса нелинейных дифференциальных уравнений первого порядка с подвижными особыми точками в вещественной области. Рассматриваемое уравнение в общем случае не разрешимо в квадратурах и имеет подвижные особые точки алгебраического типа. Это обстоятельство требует решение ряда математических задач. Ранее авторами была решена задача влияния возмущения подвижной особой точки на аналитическое приближенное решение. Это решение основывалось на классическом подходе и, при этом, существенно уменьшилась область применения аналитического приближенного решения, по сравнению с областью, полученной в доказанной теореме существования и единственности решения. Поэтому в статье предлагается новая технология исследования, основанная на элементах дифференциального исчисления. Этот подход позволяет получить точные границы для аналитического приближенного решения в окрестности подвижной особой точки. Получены новые априорные оценки для аналитического приближенного решения рассматриваемого класса уравнений, хорошо согласующиеся с известными для общей области действия. При этом, представленные результаты дополняют ранее полученные, существенно расширена область применения аналитического приближенного решения в окрестности подвижной особой точки. Приведенные расчеты согласуются с теоретическими положениями, о чем свидетельствуют эксперименты, проведенные с нелинейным дифференциальным уравнением, обладающим точным решением. Дана технология оптимизации априорных оценок погрешности с помощью апостериорных оценок. В исследованиях применялись ряды с дробными отрицательными степенями.

Time-Delayed Nonlinear Integral Resonant Controller to Eliminate the Nonlinear Oscillations of a Parametrically Excited System

In this work, a nonlinear integral resonant controller is utilized for the first time to suppress the principal parametric excitation of a nonlinear dynamical system. The whole system is modeled as a second-order nonlinear differential equation (i.e., main system) coupled to a nonlinear first-order differential equation (i.e., controller). The control loop time-delays are included in the studied model. The multiple scales homotopy approach is employed to obtain an approximate solution for the proposed time-delayed dynamical system. The nonlinear algebraic equation that governs the steady-state oscillation amplitude has been extracted. The effects of the time-delays, control gain, and feedback gains on the performance of the suggested controller have been investigated. The obtained results indicated that the controller performance depends on the product of the control and feedback signal gains as well as the sum of the time-delays in the control loop. Accordingly, two simple objective functions have been derived to design the optimum values of the loop-delays, control gain, and feedback gains in such a way that enhances the efficiency of the proposed controller. The analytical and numerical simulations illustrated that the proposed controller could eliminate the system vibrations effectively at specific values of the control and feedback signal gains. In addition, the selection method of the loop-delays that either enhances the control performance or destabilizes the system motion has been explained in detail.

Data-based reduced-order modeling of nonlinear two-time-scale processes

This work focuses on data-based reduced-order modeling of nonlinear processes that exhibit time-scale multiplicity. Using time-series data from all the state variables of a nonlinear process, an approach that involves nonlinear principal component analysis and neural network function approximators is employed to identify the fast and slow process state variables as well as compute a nonlinear slow manifold function approximation in which the fast variables are “slaved” in terms of the slow variables. Subsequently, a nonlinear sparse identification approach is employed to calculate a dynamic model of nonlinear first-order ordinary differential equations that describe the temporal evolution of the slow process states. The method is applied to two chemical process examples, and the advantages of the proposed method over using only sparse identification for the original two-time-scale process are discussed. The approach can be thought of as a data-based analogue of the classical singular perturbation modeling of two-time-scale processes where explicit time-scale separation is assumed (and expressed in terms of a small positive parameter ε multiplying the time derivative of the fast states), and the reduced-order slow subsystem is calculated analytically.

HERK integration of finite-strain fully anisotropic plasticity models

For finite strain plasticity, we use the multiplicative decomposition of the deformation gradient to obtain a differential-algebraic system (DAE) in the semi-explicit form and solve it by a half-explicit algorithm. The terminology HERK is synonym of Half-Explicit Runge-Kutta method for DAE. The source is here the right Cauchy-Green tensor and an exact Jacobian of the second Piola-Kirchhoff stress is determined with respect to this source. The system is composed by a smooth nonlinear first-order differential equation and a non-smooth algebraic equation. The development of a half-explicit constitutive integrator is the content of this work. The integration makes use of an explicit Runge-Kutta method for the flow law complemented by the yield constraint. The flow law is a first-order differential equation and the yield constraint (including the loading/unloading conditions) is seen as the invariant of system. A half-explicit method is adopted to ensure satisfaction of the invariant. The resulting scalar equation is solved by the Newton-Raphson method to obtain the plastic multiplier. We make use of the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. Iso-error maps are presented for a combination of Neo-Hookean material using the Hill yield criterion and a associative flow law. Two complete numerical examples are presented.

Stabilizing Controllers for Landmark Navigation of Planar Robots in an Obstacle-Ridden Workspace

This paper essays a new solution to the landmark navigation problem of planar robots in the presence of randomly fixed obstacles through a new dynamic updating rule involving the orientation and steering angle parameters of a robot. The dynamic updating rule utilizes a first-order nonlinear ordinary differential equation for the changing of landmarks so that whenever a landmark is updated, the path followed by the robot remains continuous and smooth. This waypoints guidance is via specific landmarks selected from a new set of rules governing the robot’s field of view. The governing control laws guarantee asymptotic stability of the 2D point robot system. As an application, the landmark motion planning and control of a car-like mobile robot navigating in the presence of fixed elliptic-shaped obstacles are considered. The proposed control laws take into account the geometrical constraints imposed on steering angle and guarantee eventual uniform stability of the car-like system. Computer simulations, using Matlab software, are presented to illustrate the effectiveness of the proposed technique and its stabilizing algorithm.

Film flows with recirculation: hydrodynamics and heat transfer

Hydrodynamics and heat transfer in film flows along solid surfaces are analyzed by numerical simulation. The current version of the Kolmogorov-Prandtl turbulence model is used to obtain a compact mathematical description focused on engineering applications in power engineering and other heat technologies. Film flows with recirculation are studied as special modes of condensing (or evaporative) devices of power plants. Other possible applications are the problems of evaporative cooling, problems of hydraulic roughness of technical surfaces, technologies of thin-film materials and – in the field of large linear scales - modeling of some natural catastrophic phenomena, such as mudflows. The mathematical description is presented as a system of three ordinary first-order nonlinear differential equations - for distributions of turbulent energy, turbulent energy flux density, flow velocity. The phenomena of laminar – turbulent transition and special modes of “flooding” of the flow are diagnosed. Temperature distributions are calculated and heat fluxes are determined. The Reynolds number of the film and the Stanton number are presented as functions of two dimensionless determining parameters specifying the film thickness and the ratio of the friction stresses at the boundaries.

Null scrolls with prescribed curvatures in Lorentz-Minkowski 3-space

Abstract In Lorentz-Minkowski 3-space, null scrolls are ruled surfaces with a null base curve and null rulings. Their mean, as well as their Gaussian curvature, depends only on a parameter of a base curve. In the present paper, we obtain the first-order nonlinear differential equation (Riccati equation) which relates curvatures of a base curve to curvatures of a null scroll. Conditioned by this equation, we can determine a family of null scrolls with a given null base curve and prescribed curvatures, in particular, a family of minimal and constant mean curvature null scrolls.

Effects of sliding mode control antiretroviral drug on HIV-1 viral load

Human immunodeficiency virus (HIV) has devastating effects on human society. Researchers have proposed many models for the decay of CD4+T cells, the growth of infected cells, and viral load. In this paper, four first-order nonlinear coupled differential equations have been considered. Four variables are CD4+T cells, which are healthy, less infected cells,more infected cells capable of producing virus,and finally the viralload. Apart from the two drug therapies, protease inhibitor (PI) and reverse transcriptase inhibitor (RTI), which have already been considered in the literature, we have proposed antiretroviral drug (ARD) that works as sliding mode controller. We have used numerical methods to study the effect of RTI, PI, and ARD on healthy cells, infected cells, and the viral load. We have expressed our solutions in terms of log sigmoid functions and used memetic computing for the solution which is a hybridization of GA, a global optimizer, and sequential quadratic programming, a local optimizer. ARD, as a sliding mode controller, does help to improve the situation of the HIV patient, especially, in further reducing the viral load and also decreasing the infected cells which have the potential to produce virus. This helps in giving relief to the patient and an increase in life expectancy.

Hysteresis modeling and compensation of a pneumatic end-effector based on Gaussian process regression

This paper proposes a data-driven statistical learning method to identify the force-pressure hysteresis of a pneumatic end-effector based on Gaussian process regression (GPR). Given the actual complex characteristics of the hysteresis phenomena, GPR is used to establish the relationship between the input and output variables of the hysteresis as a first-order nonlinear differential equation ignoring the high-order dynamics without specifying any hyperparameters or considering the special features of hysteresis loops. The inverse hysteresis model can be derived directly. The parameters of the GPR are determined by choosing low-frequency triangle-wave pressure excitations as the training set, and then the prediction performance of the present model is tested under different types, amplitudes, and period conditions of pressure signals as the verification sets. Compared with two types of modified Prandtl-Ishlinskii model, the proposed model achieves better accuracy in both training and verification sets. Based on the inverse hysteresis feedforward compensator, comparative experiments of the force-tracking control are conducted in the form of the open-loop and closed-loop controllers, of which the results indicate the effectiveness and superiority of the proposed model.

Anti-eikonal equation of an eigenmirror.

We call a surface that appears undistorted when viewed in a curved mirror an eigensurface and the mirror an eigenmirror. Such pairs are described by a first-order nonlinear partial differential equation of the form a0+a1ux+a2uy+a3uxuy+a4ux2+a5uy2=0, where ai=ai(x,y,u), which we call the anti-eikonal equation. We give examples of symbolic and numerical solutions, including pairs that are geometrically congruent. Ray tracing simulations are included that visually confirm the unusual properties of these surfaces.

On the Computation of Approximate Solution to Ordinary Differential Equations by the Chebyshev Series Method and Estimation of Its Error

An approximate method for solving the Cauchy problem for nonlinear first-order ordinary differential equations is considered. The method is based on using the shifted Chebyshev series and a Markov quadrature formula. Some approaches are given to estimate the error of an approximate solution expressed by a partial sum of a certain order series. The error is estimated using the second approximation of the solution expressed by a partial sum of a higher order series. An algorithm for partitioning the integration interval into elementary subintervals to ensure the computation of the solution with a prescribed accuracy is constructed on the basis of the proposed approaches to error estimation.

Nonlinear dynamics of the six-pole rotor-AMB system under two different control configurations

The nonlinear dynamics of the six-pole rotor active magnetic bearings system is studied in this article for the first time. Two control configurations based on a proportional-derivative feedback current controller are proposed to mitigate lateral vibrations of the considered system. The first configuration is designed in such a way that only four electromagnetic poles are responsible for the system vibration control in the $$ Y $$ -direction, while all six poles control the lateral vibration in the $$ X $$ -direction. A second configuration is proposed in which the same four poles of the first configuration control the system vibration in the $$ Y $$ -direction, while the other two poles are responsible for controlling the system vibration in the $$ X $$ -direction. According to the suggested control methods, a mathematical model is derived that simulates lateral oscillations of the system. Using the perturbation analysis, four autonomous and coupled first-order nonlinear differential equations that govern the system oscillation amplitudes and the corresponding phase angles in both $$ X $$ and $$ Y $$ -directions are extracted. Various bifurcation diagrams are obtained using rotor spinning-speed and disk eccentricity as a bifurcation control parameter. The conditions at which the system can whirl either forward or backward are investigated. Numerical validations for the obtained bifurcation diagrams are introduced which illustrate excellent agreement with the analytical results. Based on the acquired analytical and numerical results, it is found that the first control configuration has the best dynamical behavior for controlling the vibration of such systems.

Optimal stopping problems for running minima with positive discounting rates

We present analytic solutions to some optimal stopping problems for the running minimum of a geometric Brownian motion with exponential positive discounting rates. The proof is based on the reduction of the original problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the minimal solutions of certain first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual dual American lookback options with fixed and floating strikes in the Black–Merton–Scholes model from the point of view of short sellers.

Topology optimization of structural systems based on a nonlinear beam finite element model

A method for the topology optimization of structures composed of nonlinear beam elements based on a hysteretic finite element modeling approach is presented. In the context of the optimal design of structures composed of truss or beam elements, studies reported in the literature have mostly considered linear elastic material behavior. However, certain applications require consideration of the nonlinear response of the structural system to the given external forces. Particular to the methodology suggested in this paper is a hysteretic beam finite element model in which the inelastic deformations are governed by principles of mechanics in conjunction with first-order nonlinear ordinary differential equations. The nonlinear ordinary differential equations determine the onset of inelastic deformations and the approximation of the signum function in the differential equation with the hyperbolic tangent function permits the derivation of analytical sensitivities. The objective of the optimization problem is to minimize the volume of the system such that a system-level displacement satisfies a specified constraint value. This design problem is analogous to that of seismic design where inelastic deformations are permitted, yet sufficient stiffness is required to limit the overall displacement of the system to a specified threshold. The utility of the method is demonstrated through numerical examples for the design of two structural frames and a comparison with the solution from the topology optimization assuming linear elastic material behavior. The comparisons show that the nonlinear design is either comprised of a lower volume for a given level of performance, or offers better performance for a given volume in comparison with optimized linear design.

Об оценке погрешности приближенного решения обыкновенных дифференциальных уравнений, определенного с помощью рядов Чебышёва

Рассматривается приближенный метод решения задачи Коши для нелинейных обыкновенных дифференциальных уравнений первого порядка, основанный на применении смещенных рядов Чебышёва и квадратурной формулы Маркова. Приведены способы оценки погрешности приближенного решения, выраженного в виде частичной суммы ряда некоторого порядка. Погрешность оценивается с помощью второго приближенного решения, вычисленного специальным образом и представленного частичной суммой ряда более высокого порядка. На основе предложенных способов оценки погрешности построен алгоритм автоматического разбиения промежутка интегрирования на элементарные сегменты, делающие возможным вычисление приближенного решения с наперед заданной точностью. Работа метода проиллюстрирована примерами, в том числе примером из небесной механики. An approximate method of solving the Cauchy problem for nonlinear first-order ordinary differential equations is considered. The method is based on using the shifted Chebyshev series and a Markov quadrature formula. Some approaches are given to estimate the error of an approximate solution expressed by a partial sum of a certain order series. The error is estimated using the second approximation of the solution expressed by a partial sum of a higher order series. An algorithm of partitioning the integration interval into elementary subintervals to ensure the computation of the solution with a prescribed accuracy is discussed on the basis of the proposed approaches to error estimation.

Long internal ring waves in a two-layer fluid with an upper-layer current

We consider a two-layer fluid with a depth-dependent upper-layer current (e.g. a river inflow, an exchange flow in a strait, or a wind-generated current). In the rigid-lid approximation, we find the necessary singular solution of the nonlinear first-order ordinary differential equation responsible for the adjustment of the speed of the long interfacial ring wave in different directions in terms of the hypergeometric function. This allows us to obtain an analytical description of the wavefronts and vertical structure of the ring waves for a large family of the current profiles and to illustrate their dependence on the density jump and the type and the strength of the current. In the limiting case of a constant upper-layer current we obtain a 2D ring waves' analogue of the long-wave instability criterion for plane interfacial waves. On physical level, the presence of instability for a sufficiently strong current manifests itself already in the stable regime in the squeezing of the wavefront of the interfacial ring wave in the direction of the current. We show that similar phenomenon can also take place for other, depth-dependent currents in the family.

A Sturm\u2013Liouville Problem Arising in the Atmospheric Boundary-Layer Dynamics

We present an approach that facilitates the generation of explicit solutions to atmospheric Ekman flows with a height-dependent eddy viscosity. The approach relies on applying to the governing equations, of Sturm–Liouville type, a suitable Liouville substitution and then reducing the outcome to a nonlinear first-order differential equation of Riccati type.

On the nonlinear dynamics of constant stiffness coefficients 16-pole rotor active magnetic bearings system

The present study is devoted to investigate the oscillatory behaviors of the 16-pole rotor active magnetic bearing system. A controllable electromagnetic force generated via a conventional proportional-derivative controller is utilized to stabilize the system lateral oscillations that excited by the rotating disk eccentricity when the spinning speed (Ω) is close to or equal the system linear natural frequency (ω). The nonlinear dynamical equations governing the controlled system lateral vibrations at constant stiffness coefficients are derived in this article for the first time. Then, four nonlinear autonomous first-order differential equations to describe the considered system oscillation amplitudes and the corresponding phase angles are obtained applying the asymptotic analysis. Bifurcation behavior of the system periodic motions under varying the different control parameters is explored. The main acquired results confirm that the 16-pole rotor-AMB system at constant stiffness coefficients can exhibit one of three oscillatory motions that are periodic, quasiperiodic, or chaotic motions depending on the derivative gain coefficient. Moreover, the system may respond with one-stable solution, bi-stable solutions, tri-stable solutions, or quadri-stable solutions depending on the proportional gain coefficient. Numerical simulations for different system motions are validated via the system time response, Poincare map, orbit plot, and frequency spectrum that are showed an excellent agreement with the obtained analytical results.

Z-Control on COVID-19-Exposed Patients in Quarantine

In this paper, a mathematical model for diabetic or hypertensive patients exposed to COVID-19 is formulated along with a set of first-order nonlinear differential equations. The system is said to exhibit two equilibria, namely, exposure-free and endemic points. The reproduction number is obtained for each equilibrium point. Local stability conditions are derived for both equilibria, and global stability is studied for the endemic equilibrium point. This model is investigated along with Z-control in order to eliminate chaos and oscillation epidemiologically showing the importance of quarantine in the COVID-19 environment.