First-order Nonlinear Differential Equations Research Articles

Enhanced resolution in solving first-order nonlinear differential equations with integral condition: a high-order wavelet approach

Enhanced resolution in solving first-order nonlinear differential equations with integral condition: a high-order wavelet approach

Non-autonomous Volterra series expansion of the Variable Phase Approximation and its application to the nucleon-nucleon inverse scattering problem

Abstract In this paper, the non-linear Volterra series expansion is extended and used to describe certain types of non-autonomous differential equations related to the inverse scattering problem in nuclear physics. The non-autonomous Volterra series expansion lets us determine a dynamic, polynomial approximation of the variable phase approximation (VPA), which is used to determine the phase shifts from nuclear potentials through first-order non-linear differential equations. By using the first-order Volterra expansion, a robust approximation is formulated to the inverse scattering problem for weak potentials and/or high energies. The method is then extended with the help of radial basis function neural networks by applying a nonlinear transformation on the measured phase shifts to be able to model the scattering system with a linear approximation given by the first order Volterra expansion. The method is applied to describe the 1S0 NN potentials in neutron+proton scattering below 200 MeV laboratory kinetic energies, giving physically sensible potentials and below $1\%$ averaged relative error between the re-calculated and the measured phase shifts.

Exact solutions to differential equations with different arguments

Various linear and non-linear first-order differential equations with different arguments are considered. Exact solutions to these equations are provided. Systems of two coupled linear first-order differential equations are also solved explicitly and exactly.

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.

New oscillation results for first-order nonlinear difference equations with retarded arguments

New oscillation results for first-order nonlinear difference equations with retarded arguments

THEORETICAL ANALYSIS AND NUMERICAL SOLUTION OF LINEAR AND NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

In this paper, a collocation method based on the Haar wavelet is presented for the solution of both linear and nonlinear first-order neutral delay differential equations. The Haar functions are used to approximate the first-order derivative, and the approximate solution is obtained by using initial condition and integration. Some examples from the literature are used to test the suggested method efficiency and applicability. A comparison of exact and approximate solutions is given in figures for different numbers of collocation points. The root mean square and maximum absolute errors are calculated for different numbers of collocation points. The rate of convergence is calculated which is approximately equal to 2. The comparison of the present method with the other numerical methods is also given. The results demonstrate that the Haar wavelet collocation method is simple and effective for solving first-order linear and nonlinear neutral delay differential equations.

Influence of Raman gain on dynamics of spatiotemporal chaos in optical ring microresonators

We theoretically investigate on the evolution of spatiotemporal chaos under the influence of Raman gain, in an optical ring microresonator. The dynamics of the system is shown to be governed by the Lugiator–Lefever equation, with an additional real and imaginary Raman gain term. In the absence of pump field intensity and Raman gain, we observe an increase in the amplitude and frequency of the traveling optical field as the pump detuning frequency increases. However this field gradually vanishes as time increases, owing to the intra-cavity loss. By exploring an appropriate ansatz which incorporate a chirping parameter, the modified Lugiator–Lefever equation is transformed to a set of four coupled first-order nonlinear ordinary differential equations. Fixed point stability analysis strongly suggest that variation in Raman gain has the propensity of inducing the co-existence of bright and dark solitons in the anomalous dispersion regime. Results of numerical simulations underscore that the structural profiles of the optical field amplitude are greatly modified by the Raman effects. Variation in Raman gain equally leads to a bifurcation analysis of the system, in which the Lyapunov exponent diagrams reveal regions of period doubling and chaos. We finally observe that for a fixed value of the real part of Raman gain coefficient, an increase in the effects of soliton self-frequency shift generally drives the system to chaotic regime.

SPREADING PATTERN OF INFECTIOUS DISEASES: SUSCEPTIBLE INFECTED RECOVERED MODEL WITH VACCINATION AND DRUG-RESISTANT CASES (APPLICATION ON TB DATA IN INDONESIA)

Mycobacterium tuberculosis is the causative agent of the infectious illness tuberculosis (TB). Indonesia is the world's third-highest TB burden country. TB transmission is prevented by the BCG vaccination. A directly observed treatment, short-course (DOTS) treatment approach can cure TB illness. Recurrent TB may occur due to either relapse or reinfection with drug-resistant bacteria. The goals of this article are formulating the SVITR model with relapse and drug-resistant cases, applying the model to the TB data in Indonesia, determining the model accuracy, determining the spreading pattern and interpreting the result, and simulating the parameters. Literature study and application methods are used in this research. The SVITR model with relapse and drug-resistant cases is a first-order nonlinear differential equation system. The model is applied to TB in Indonesia based on annual data from Indonesian Health Profile, World Bank, and WHO. The model is solved by the fourth-order Runge-Kutta method. The model is accurate enough to explain the spread of TB in Indonesia with a MAPE value of 15,5%. The spreading pattern of tuberculosis infection is upward from 2010 to 2050. In 2050, there are still 8.115.976 TB cases in Indonesia. Hence in 2050, Indonesia's free of TB target has yet to be achieved. Simulation is conducted by increasing BCG vaccination to 95%, reducing contact with TB patients to 5%, increasing treatment to 95%, and lowering relapses and drug-resistant cases to 0.00005%, so the Indonesia free of TB target in 2050 can be achieved from 2042.

Physics-informed neural network for solution of forward and inverse kinematic wave problems

The one-dimensional kinematic wave (KW) model (KWM) is widely used in surface water and water quality hydrology as well as for modeling the movement of traffic on long highways. The KW model involves a first-order nonlinear partial differential equation which is solved numerically for realistic geometric representation, input, and initial and boundary conditions. This paper developed a method to solve the partial differential equation using a neural network structure, called the physics-informed neural network (PINN). The PINN was applied to solve the one-dimensional nonlinear KWM for forward and inverse problems. To enhance the accuracy and convergence and to suppress the spurious oscillation at the steep front, a technique with redistributed collocation points as an improved algorithm of PINN was employed. To deal with the fractional order of the nonlinear advection term, a stop-gradient technique was proposed. For validation, the KWM for steady and unsteady overland flow, and open channel flow without lateral inflow was analyzed for the forward problem. For the inverse problem, the unknown Manning coefficient was correctly computed, demonstrating the capability of the proposed algorithm for the identification of KWM parameters.

Feasibility and Stability Analysis for Basic Measles Model Using Fuzzy Parameter

In this article, we investigate the behaviors of the measles viral pandemic using fuzzy susceptible-infectiousrecovered (SIR) model. To examine the effects of various compartment phases, we analyze disease-free equilibrium points along with basic reproduction number. The measles model is stated to be globally asymptotically stable at the disease-free equilibrium point. In order to mathematically simulate the measles, we use a first-order nonlinear differential equation. The numerical solution is computed using the Runge-Kutta method, and the model’s feasibility is also covered.

Theoretical investigation of the free fall of a spherical particle in a viscous fluid using continuous piecewise linearization method

This article presents a theoretical investigation of the problem of free fall of a spherical particle in a viscous fluid. The classic Boussinesq-Basset-Oseen (BBO) model for particle motion in laminar flow was modified for generalized flow by using a drag law that is applicable for 0<Re≤2.0×105. By assuming that the acceleration in the Basset force integral is constant, the Basset force effect was approximated to form an integrated added mass coefficient. Consequently, the integro-differential equation of the BBO model was transformed to a first-order nonlinear ordinary differential equation that accounts for the Basset force effect and was solved using the continuous piecewise linearization method (CPLM). The CPLM algorithm was developed based on the jerk-velocity relationship and is applicable to zero and non-zero initial conditions, steady motion, increasing or decreasing velocities and the corresponding acceleration and jerk responses. The CPLM algorithm was shown to predict published experimental results accurately and compared very well with numerical solutions and existing analytical solutions. Examination of the fall response under varying parameters showed that the fall distance, fall time and terminal velocity depend strongly on the sphere diameter, sphere density, and the density and viscosity of the fluid medium. Also, an analytical solution for the power dissipated in the fluid medium as the sphere falls to reach its terminal velocity was derived. The power dissipated was found to increase exponentially as the initial velocity deviates positively from the terminal velocity.

Assessment of Numerical Performance of Some Runge-Kutta Methods and New Iteration Method on First Order Differential Problems

This research focuses on the assessment of the numerical performance of some Runge-Kutta methods and New Iteration Method “NIM” for solving first-order differential problems. The assessment is conducted through extensive numerical experiments and comparative analyses. Accuracy, efficiency, and stability are among the key factors considered in evaluating the performance of the methods. A range of first-order differential problems with diverse characteristics and complexity levels is employed to thoroughly examine the methods' capabilities and limitations. The numerical investigation that is defined in the study as well as the results that are stated in the Tables, demonstrates that all the approaches produce extremely accurate results. However, the “NIM” was shown to be the most effective of the three methods used in this study. Conclusively, the “NIM” should be employed to solve first-order nonlinear and linear ordinary differential equations in place of Runge-Kutta Fourth order method (RK4M) and Butcher Runge-Kutta Fifth order method (BRK5M). In addition, BRK5M is more applicable and efficient than RK4M when solving first order ordinary differential problems.

The Charging and Discharging of Reverse Biased Schottky Diode Voltage-Dependent Capacitor

&#x0D; Transient analysis of RC circuit with voltage step source, constant resistor and nonlinear voltage-dependent capacitors with the capacitance given by C(vc) = Co/√(1+vc/Vbi), where Co and Vbi are constant and vc is the voltage across the capacitor, was carried out. The analysis was restricted to this class of nonlinear capacitors, because they represent with acceptable approximation the capacitances of reverse-biased Schottky diodes, which are very important and usually used in several static power converters practical applications. The charging and discharging are described by explicitly solvable nonlinear first-order differential equations. The theoretical analysis results are verified by numerical examples with computer simulation and also by experimentation in the laboratory.&#x0D; &#x0D;

New Numerical Iteration Schemes Based on Perturbation Iteration Algorithms

Perturbation–Iteration algorithms (PIA) have been developed recently to solve differential equations analytically. A continuous solution in terms of closed form functions as an approximation of the original equation can be found using the method. The method has been implemented to algebraic equations, ordinary and partial differential equations successfully. Based on the formalism developed previously, in this work, purely numerical versions of the perturbation–iteration algorithms are proposed for the first time for first-order nonlinear ordinary differential equations. The new algorithms are called as the Numerical Perturbation–Iteration Algorithms (NPIA) to distinguish them from the continuous analytical ones (PIA) and the method in general will be called the Numerical Perturbation–Iteration Method (NPIM). Iteration schemes based on one correction term in the perturbation solution and first-order derivatives in the Taylor series expansions (NPIA(1,1)), and one correction term in the perturbations and second-order derivatives in the series expansions (NPIA(1,2)) are derived. The numerical results are compared with the exact solutions and a very good match is observed. NPIA(1,2) produced slightly better results compared to NPIA(1,1) with total errors being at least [Formula: see text], [Formula: see text] being the step size for both methods. An improvement suggestion for NPIA(1,1) is proposed in the last section to eliminate unrealistic blow-up solutions which are encountered for some specific equations.

Asymptotic Stability of Periodic Solutions for a Nonlinear Neutral First-order Differential Equation with Functional Delay

We consider a nonlinear neutral differential equation and use the contraction mapping principle to show the asymptotic stability of the zero solution provided that 𝑄(𝑡, 0) = 𝑓(0) = 0.

DC operating points of Mott neuristor circuits

In 1960 Hewitt Crane conceptualized neuristors as electronic logic networks that could mimic action potential generation by biological neurons. In 2012 Mott memristors were presented as nano-scale electronic devices for physically constructing neuristor networks. Both the original Mott memristor model and the simplified model presented in 2020 are stiff nonlinear first-order ordinary differential equations (ODEs) that describe the voltage controlled threshold switching and current controlled negative differential resistance, attributed to a Mott insulator-to-metal phase transition. By design, a Mott neuristor contains two identical Mott memristors plus linear resistors and capacitors (no inductors), which enables alternating threshold switching by the memristors to generate periodic (AC) output voltage spikes in response to a DC current or voltage input. This paper presents a steady-state analysis of dynamic neuristor models, including four-, five-, and six-state variables (each of which is a system of first-order ODEs involving the states) based on the original phenomenological model of the Mott memristor. Specifically, it reveals the unique stable operating points that occur for applied currents or voltages below the amounts needed to induce resistive switching. These DC operating points represent the physically meaningful “OFF” states for the circuit when the neuristor does not output voltage spikes. Jacques Hadamard's mathematical criteria for a well-posed physical model (existence, uniqueness, continuity, stability) motivated the search for these DC operating points (OFF states), which are needed for efficiently initializing neuristor models before conducting dynamic neuristor simulations. The key result of our analysis is the neuristor steady-state eq. (29) which is useful for understanding neuristor OFF states as well as the role of negative-differential resistance (NDR) in the operation of the Mott memristor model and neuristor circuits. In addition to the steady-state analysis, three numerical examples of dynamic neuristor circuit operation are also given and MATLAB code for the dynamic four-state model is available for interested readers.

Sensitivity Analysis-Based Validation of the Modified NERA Model for Improved Performance

Marijuana is an illicit narcotic with multiple negative health effects as it continues to pose a severe danger to the health of people in emerging regions. Marijuana is spread through interactions between users and non-users. This study utilizes first-order non-linear ordinary differential equations to modify the Non-user, Experimental, Recreational, and Addicted (NERA) users’ model by adding a new class known as the hospitalized class of marijuana smokers. Moreover, this paper focuses on validating the NERA model using the sensitivity analysis concept. The NERA model presented in a previous study for marijuana usage among the target population consists of four distinct stages: the first stage is known as the non-user stage and is represented by the letter N; the second stage is known as the experimental stage of marijuana smoking and is represented by the letter E; the third stage is known as the recreational stage of marijuana smoking and is represented by the letter R; the fourth stage is represented by the letter A; and this last stage is known as the addicted stage of marijuana smoking. The suggested mathematical method achieved the reproduction number () for marijuana use. The results of the sensitivity tests indicate how crucial a variety of variables are to the spread of marijuana. The parameters that have a key impact on marijuana consumption were discovered based on sensitivity analysis. All these procedures, including the reproductive number and invariant region, using the sensitivity analysis concept were utilized to validate the updated model. In this study, the NERA model is modified by utilizing first-order non-linear ordinary differential equations. Furthermore, as demonstrated by the graphical and technique sections, the proposed modified models were successfully validated for improved performance.

Dynamical Modelling of a Centrifugal Fan Driven by an Induction Motor and Experimental Validation

The application of electrical drives in the control of centrifugal fans and pumps is a well-established area that leads to substantial energy savings. It requires electrical and automation engineers to have some knowledge relevant to drives about fan and pump modelling since ignoring or oversimplifying fan/pump modelling for the intended drive design compromises the required control quality. This paper improves the existing dynamical models of fans and pumps integrated with induction motors via neural network estimation of overall fan/pump efficiency; this estimation is based on the voltage frequency of the driving induction motor, pressure, and flow rate, followed by the separation of the fan’s or pump’s own efficiency for the motor load torque computation, enhancing the accuracy due to the correct reflection of the power balance. The model is developed with a view to being convenient for control design applications. For the first time for dynamical models, it is verified experimentally, justifying the necessity of the first-order nonlinear differential equation for the flow rate. The validation includes a direct approach based on analysis of transient behaviour caused by a small-step perturbation of the frequency of the motor voltages and an indirect approach based on the introduced concept of the dynamical flow rate and pressure estimators.

APPLICATION OF NON-STANDARD FINITE DIFFERENCE METHOD ON COVID-19 MATHEMATICAL MODEL WITH FEAR OF INFECTION

This study presents a novel application of Non -Standard Finite Difference (NSFD) Method to solve a COVID-19 epidemic mathematical model with the impact of fear due to infection. The mathematical model is governed by a system of first-order non-linear ordinary differential equations and is shown to possess a unique positive solution that is bounded. The proposed numerical scheme is used to obtain an approximate solution for the COVID-19 model. Graphical results were displayed to show that the solution obtained by NSFD agrees well with those obtained by the Runge-Kutta-Fehlberg method built-in Maple 18.

Extended Runge-Kutta Scheme and Neural Network Approach for SEIR Epidemic Model with Convex Incidence Rate

For solving first-order linear and nonlinear differential equations, a new two-stage implicit–explicit approach is given. The scheme’s first stage, or predictor stage, is implicit, while the scheme’s second stage is explicit. The first stage of the proposed scheme is an extended form of the existing Runge–Kutta scheme. The scheme’s stability and consistency are also offered. In two phases, the technique achieves third-order accuracy. The method is applied to the SEIR epidemic model with a convex incidence rate. The local stability is also examined. The technique is evaluated compared to existing Euler and nonstandard finite difference methods. In terms of accuracy, the produced plots show that the suggested scheme outperforms the existing Euler and nonstandard finite difference methods. Furthermore, a neural network technique is being considered to map the relationship between time and the amount of susceptible, exposed, and infected people.