Time-dependent Nonlinear Equations Research Articles

Large-amplitude nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes

The present study discusses the nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes (FG-CNTs). Mechanical properties of the plate have been continuously changed along the thickness of the plate by considering four different distribution patterns of CNTs. Based on the first-order shear deformation theory (FSDT) and von Ḱarḿan’s geometrical nonlinearity, the governing equations of the plate are derived using Hamilton’s principle. Galerkin discretization technique is employed to convert the partial differential equations of motion to ordinary differential equations. Afterward, the nonlinear time-dependent equations of the plate were analytically solved using the multiple time scales method. Eventually, the effect of CNT parameters, the geometrical ratios and rotational speed on the nonlinear frequency of the plate have been studied. For verification purposes, the present numerical results have been compared with those available in the literature with evident agreement. The present research may give benchmark solutions and some guidelines for designing FG-CNTs rotating plates.

Discrete-time zeroing neural network with quintic error mode for time-dependent nonlinear equation and its application to robot arms

Time-dependent nonlinear equation (TDNE) arises in numerous engineering applications. Recently, zeroing neural network (ZNN) has been proven to be an effective alternative for solving the TDNE. In this paper, we present a new solution to the TDNE by using the discrete-time ZNN (DTZNN). Specifically, a special difference formula is first constructed via Taylor series expansion. Then, by utilizing such a formula to discretize the existing continuous-time ZNN model, the new DTZNN model is proposed to determine the TDNE solution. Theoretical analysis and numerical results further indicate the validity and superiority of the proposed DTZNN model in comparison with the previous models. Finally, the DTZNN practicality is presented by the simulation and experiment on the DOBOT arm using the proposed model.

Riemann problem for a general variable coefficient Burgers equation with time-dependent damping

In this paper, we study the Riemann problem for a general variable coefficient Burgers equation with time-dependent damping. We use a nonlinear time-dependent viscosity equation with a similarity variable. Thus, when the viscosity goes to zero, we obtain Riemann solutions to the general variable coefficient Burgers equation with time-dependent damping. Moreover, we use the Lax–Friedrichs scheme to obtain numerical evidence of the Riemann solutions.

TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method

Physics-informed neural networks (PINNs) have become an attractive machine learning framework for obtaining solutions to partial differential equations (PDEs). PINNs embed initial, boundary, and PDE constraints into the loss function. The performance of PINNs is generally affected by both training and sampling. Specifically, training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs, and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished. However, a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category, namely, time-dependent PDEs, where temporal information plays a key role in the algorithms used. There is one method, called Causal PINN, that considers temporal causality at the training level but not special temporal utilization at the sampling level. Incorporating temporal knowledge into sampling remains to be studied. To fill this gap, we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality. By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain, we provide a practical solution by incorporating temporal information into sampling. Numerical experiments of several nonlinear time-dependent PDEs, including the Cahn–Hilliard, Korteweg–de Vries, Allen–Cahn and wave equations, show that our proposed sampling method can improve the performance. We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods, especially when points are limited.

A Neurodynamic Approach for Solving Time-Dependent Nonlinear Equation System: A Distributed Optimization Perspective

A Neurodynamic Approach for Solving Time-Dependent Nonlinear Equation System: A Distributed Optimization Perspective

Quantitative Aeroelastic Stability Prediction of Wings Exhibiting Nonlinear Restoring Forces

In engineering practice, eigen-solution is used to assess the stability of linear dynamical systems. However, the linearity assumption in dynamical systems sometimes implies simplifications, particularly when strong nonlinearities exist. In this case, eigen-analysis requires linerisation of the problem and hence fails to provide a direct stability estimation. For this reason, a more reliable tool should be implemented to predict nonlinear phenomena such as chaos or limit cycle oscillations. One method to overcome this difficulty is the Lyapunov Characteristic Exponents (LCEs), which provide quantitative indications of the stability characteristics of dynamical systems governed by nonlinear time-dependent differential equations. Stability prediction using Lyapunov Characteristic Exponents is compatible with the eigen-solution when the problem is linear. Moreover, LCE estimations do not need a steady or equilibrium solution and they can be calculated as the system response evolves in time. Hence, they provide a generalization of traditional stability analysis using eigenvalues. These properties of Lyapunov Exponents are very useful in aeroelastic problems possessing nonlinear characteristics, which may significantly alter the aeroelastic characteristics, and result in chaotic and limit cycle behaviour. A very common nonlinearity in flexible systems is the nonlinear restoring force such as cubic stiffness, which would substantially benefit from using LCEs in stability assessment. This work presents the quantitative evaluation of aeroelastic stability indicators in the presence of nonlinear restoring force. The method is demonstrated on a two-dimensional aeroelastic problem by comparing the system behaviour and estimated Lyapunov Exponents.

General analytical solution expressions for analyzing Langmuir-type kinetics of sonochemical degradation of nonvolatile organic contaminants in water

Detailed kinetic studies of the ultrasonic decomposition of contaminants in water are scarce. Most of the work has used a pseudo-first order kinetics law, which is unrealistic. The model based on a Langmuir-type mechanism has been shown to fit the sonolytic decomposition data well, especially by using the non-linear technique. To avoid unrealistic assumptions, general analytical solutions to a time-dependent non-linear Langmuir-type equation may be the appropriate method. In this work, the sonolytic oxidation of organic contaminants, i.e., naphthol blue black and furosemide, in water was analyzed using two general analytical solution expressions of the Langmuir-type kinetics model, which describe the pollutant concentration in water. The validity of the two general analytical solution expressions was tested under a diversity of operating conditions, such as initial substrate concentration and varying ultrasonication frequency and intensity. As the initial substrate concentration increased, the sonolytic oxidation kinetics decreased, while the initial ultrasonic decomposition rate increased and then plateaued. Consequently, a heterogeneous kinetics equation based on a Langmuir-type mechanism can be used to simulate the sono-decomposition process. The decomposition yield increased with increasing sonication intensity and decreasing frequency. The two analytical solution expressions seem to be in excellent agreement with the experimental results of the sonochemical decomposition of the nonvolatile organic contaminants tested for the different operating conditions examined. These expressions provide a valuable tool for the analysis and simulation of advanced sonochemical oxidation processes under various experimental conditions.

On the superposition of multipolar spherical and cylindrical oscillations in swirling rigid flow

It is shown that a superposition of an arbitrary number of multipolar spherical and cylindrical oscillations in a cylindrical rigid flow with swirl is an exact solution to the time-dependent nonlinear vorticity equation. The oscillations are normal modes whose fundamental frequency and radial wavenumber are given by the angular speed and scaled inverse pitch, respectively, of the rigid flow.

Cyclic voltammetric response of homogeneous catalysis of electrochemical reaction. Part 3: A theoretical and numerical approach for one-electron two-step reaction scheme

• Homogeneous molecular catalysis of electrochemical reactions for the one-electron two-step reaction scheme is discussed. • The exact expression of species concentrations, current-voltage and turnover frequency is derived. • The effect of the intrinsic and operational parameters on current and turnover frequency is analysed. A theoretical model for homogeneous redox catalysis of electrochemical reactions for the one-electron two-step reaction scheme is discussed. This model is described by the system of nonlinear time dependent partial differential equations for which the nonlinear term is related to a homogeneous reaction. The species concentration, current-voltage and turnover frequency curves for the transient condition have been calculated by solving the equations using a new analytical method. The current response and turnover frequency predicted under steady-state conditions when t → ∞ are in good agreement with earlier results, proving the validity of the mathematical analysis. The effect of two rate constants and initial concentration of substrate on current and turn over frequency is discussed. Our analytical results for concentration are compared with numerical (MATLAB) results, and good agreement is noted.

Codebase release 0.5 for HarmonicBalance.jl

HarmonicBalance.jl is a publicly available Julia package designed to simplify and solve systems of periodic time-dependent nonlinear ordinary differential equations. Time dependence of the system parameters is treated with the harmonic balance method, which approximates the system’s behaviour as a set of harmonic terms with slowly-varying amplitudes. Under this approximation, the set of all possible steady-state responses follows from the solution of a polynomial system. In HarmonicBalance.jl, we combine harmonic balance with contemporary implementations of symbolic algebra and the homotopy continuation method to numerically determine all steady-state solutions and their associated fluctuation dynamics. For the exploration of involved steady-state topologies, we provide a simple graphical user interface, allowing for arbitrary solution observables and phase diagrams. HarmonicBalance.jl is a free software available at https://github.com/NonlinearOscillations/HarmonicBalance.jl.

HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance

HarmonicBalance.jl is a publicly available Julia package designed to simplify and solve systems of periodic time-dependent nonlinear ordinary differential equations. Time dependence of the system parameters is treated with the harmonic balance method, which approximates the system’s behaviour as a set of harmonic terms with slowly-varying amplitudes. Under this approximation, the set of all possible steady-state responses follows from the solution of a polynomial system. In HarmonicBalance.jl, we combine harmonic balance with contemporary implementations of symbolic algebra and the homotopy continuation method to numerically determine all steady-state solutions and their associated fluctuation dynamics. For the exploration of involved steady-state topologies, we provide a simple graphical user interface, allowing for arbitrary solution observables and phase diagrams. HarmonicBalance.jl is a free software available at https://github.com/NonlinearOscillations/HarmonicBalance.jl.

High order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations

In this paper, we will develop a class of high order asymptotic preserving (AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray radiative transfer equations (GRTEs). Inspired by the works in [6,55], we propose to penalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term. A hyperbolic, namely Δt=O(h) in the transport regime where Δt and h are the time step and mesh size respectively, and unconditional stability instead of parabolic time step restriction Δt=O(h2) in the diffusive regime are obtained, which are also free from the photon mean free path. We further employ a Picard iteration with a predictor-corrector procedure, to decouple the resulting global nonlinear system to a linear system with local nonlinear algebraic equations within each outer iterative loop. For the resulting scheme, only an implicit system for the macroscopic variable needs to be solved, while the microscopic variable can be updated explicitly. Besides, the nonlinear implicit system is decoupled to a linear positive definite system followed by nonlinear algebraic equations due to the Picard iteration. Namely, high dimension, implicit treatment and nonlinearity are all decoupled. Our scheme is shown to be AP and asymptotically accurate (AA). Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is high order accurate, effective and efficient.

Investigation of Nonlinear Thermo-Elastic Behavior of Fluid Conveying Piezoelectric Microtube Reinforced by Functionally Distributed Carbon Nanotubes on Viscoelastic-Hetenyi Foundation

In this paper, nonlinear and nonlocal thermo-elastic behavior of a microtube reinforced by Functionally Distributed Carbon Nanotubes, with internal and external piezoelectric layers, in the presence of nonlinear viscoelastic-Hetenyi foundation, and axial fluid flow inside the microtube is studied. Nonlinear partial differential equations governing the system are derived using Reddy’s third-order shear deformations theory along with the Von-Karman theory including the effect of fluid viscosity. Then, the equations are converted to time-dependent ordinary nonlinear equations using the Galerkin method. Afterward, the governing equations of the microtube’s lateral displacements are solved using the multiple scales method. The analysis of the piezoelectric’s parametric resonance is performed by obtaining trivial and nontrivial stationary solutions and plotting characteristic curves of the frequency response and voltage response. At the end, the effect of different parameters including the flow velocity, excitation voltage, parameters of the foundation, viscosity parameter, thermal loading and nanotubes’ volume fraction index on the nonlinear behavior of the system, under parametric resonance condition, is investigated.

Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

In this article, we have considered a nonlinear nonlocal time dependent fourth order equation demonstrating the deformation of a thin and narrow rectangular plate. We propose \(C^1\) conforming virtual element method (VEM) of arbitrary order, \(k\ge 2\), to approximate the model problem numerically. We employ VEM to discretize the space variable and fully implicit scheme for temporal variable. Well-posedness of the fully discrete scheme is proved under certain conditions on the physical parameters, and we derive optimal order of convergence in both space and time variable. Finally, numerical experiments are presented to illustrate the behaviour of the proposed numerical scheme.

On the solution of hyperbolic equations using the peridynamic differential operator

Numerical solution of hyperbolic differential equations, such as the advection equation, poses challenges. Classically, this issue has been addressed by using a scheme known as the upwind scheme. It simply invokes more points from the upwind side of the flow stream when calculating derivatives. This study presents a generalized upwind scheme, referred to as directional nonlocality, for the numerical solution of linear and nonlinear hyperbolic Partial Differential Equations (PDEs) using the peridynamic differential operator (PDDO). The PDDO provides the nonlocal form of the differential equations by introducing an internal length parameter (horizon) and a weight function. The weight function controls the degree of interaction among the points within the horizon. A modification to the weight function, i.e., upwinded-weight function, accounts for directional nonlocality along which information travels. This modification results in a stable PDDO discretization of hyperbolic PDEs. Solutions are constructed in a consistent manner without special treatments through simple discretization. The capability of this approach is demonstrated by considering time dependent linear and nonlinear hyperbolic equations as well as the time invariant Eikonal equation. Numerical stability is ensured for the linear advection equation and the PD solutions compare well with the analytical/reference solutions.

Nonlinear Free Vibration Analysis of In-plane Bi-directional Functionally Graded Plate with Porosities Resting on Elastic Foundations

This paper deals with the nonlinear free vibration analysis of in-plane bi-directional functionally graded (IBFG) rectangular plate with porosities which are resting on Winkler–Pasternak elastic foundations. The material properties of the IBFG plate are assumed to be graded along the length and width of the plate according to the power-law distribution, as well as, even and uneven types are taken into account for porosity distributions. Equations of motion are developed by means of Hamilton’s principle and von Karman nonlinearity strain–displacement relations based on classical plate theory (CPT). Afterward, the time-dependent nonlinear equations are derived by applying the Galerkin procedure. The nonlinear frequency is determined by using modified Poincare–Lindstedt method (MPLM). Numerical results are obtained in tabular and graphical form to examine the effects of some system key parameters such as porosity coefficients, distribution patterns, gradient indices, elastic foundation coefficients, aspect ratio and vibration amplitude on the nonlinear frequency of the porous IBFG plate. To validate the analysis, the results of this paper have been compared to the published data and good agreements have been found.

Nonlinear dynamic analysis of FG carbon nanotube/epoxy nanocomposite cylinder with large strains assuming particle/matrix interphase using MLPG method

In this paper, a first known study on the large deformation dynamic behavior of functionally graded carbon nanotube-reinforced (CNTR) nanocomposite cylinder considering the particle/matrix interphase property is presented. The geometrically nonlinear dynamic analysis is carried out using the meshless local Petrov–Galerkin (MLPG) method based on the total Lagrangian approach. The obtained nonlinear time dependent equations are solved using the incremental-iterative Newmark/Newton–Raphson technique. The effective properties of CNTR cylinder is evaluated by employing the three-phase Halpin–Tsai model and rule of mixture. In these models, three constituent phases including nanoparticle, interphase and matrix is considered. It is assumed that CNTs is dispersed in the resin matrix with four different function along the radial direction to obtain the optimum grading pattern. The effects of interphase properties, CNT's weight fraction, CNT distribution pattern and volume fraction exponent on the dynamic characteristics of the cylinder are discussed in details. Numerical results demonstrated the high performance of the proposed MLPG method for geometrically nonlinear dynamic analysis of functionally graded carbon nanotube/epoxy nanocomposites due to its advantage in eliminating the mesh distortion and high capability in FG material modeling. Moreover, it is found that the interphase properties may dramatically affect the dynamic behavior of nanocomposite cylinder especially at the higher CNT volume fractions. So, it is important to consider the interphase properties to correctly model the nanocomposite structures.

An efficient hybrid computational technique for the time dependent Lane-Emden equation of arbitrary order

The study of dynamic behaviour of nonlinear models that arise in ocean engineering play a vital role in our daily life. There are many examples of ocean water waves which are nonlinear in nature. In shallow water, the linearization of the equations imposes severe conditions on wave amplitude than it does in deep water, and the strong nonlinear effects are observed. In this paper, q-homotopy analysis Laplace transform scheme is used to inspect time dependent nonlinear Lane-Emden type equation of arbitrary order. It offers the solution in a fast converging series. The uniqueness and convergence analysis of the considered model is presented. The given examples confirm the competency as well as accuracy of the presented scheme. The behavior of obtained solution for distinct orders of fractional derivative is discussed through graphs. The auxiliary parameter ħ offers a suitable mode of handling the region of convergence. The outcomes reveal that the q-HATM is attractive, reliable, efficient and very effective.

EXACT SOLUTION OF VAN DER POL NONLINEAR OSCILLATORS ON FINITE DOMAIN BY PADE APPROXIMANT AND ADOMIAN DECOMPOSITION METHODS

This paper is concerned with a thorough investigation in achieving exact analytical solution for the Van der Pol (VDP) nonlinear oscillators models via Adomian decomposition method (ADM). The models are nonlinear time dependent second order ordinary differential equations. ADM has already been applied, in existing literatures, to obtain approximate results. But, we adapt the method by adjusting the source term; a procedure that is base on the asymptotic Taylor's series expansion on the term that would have resulted to proliferation of terms during the invertible process. Then, the rational Pade Approximant is applied to clarify and get a better understanding of the uniqueness and convergence of our findings. Two models were used as illustrations and their result pictured to indicate their behaviour in the given domains. And, we found that the adaptation on the models yielded exact results which were further displayed in constructed tables.

The nonlinear thermo-hyperelasticity wave propagation analysis of near-incompressible functionally graded medium under mechanical and thermal loadings

This article presents a non-Fourier thermo-hyperelastic model to investigate thermal and stress wave propagation phenomenon in a near-incompressible functionally graded medium (FGM) for various thermal and mechanical boundary conditions. A strain energy function is chosen to modify FGM’s hyperelastic equations considering the coupling effects of mechanical and thermal terms. By switching the strain tensor's invariants, equations are developed to estimate a near-incompressible model for rubber. The rubber is characterized by a gradual variation in the longitudinal direction. Therefore, the material properties of rubber mainly depend on coordinates in through an exponential function. The nonlinear governing equations are derived from the large displacement approach using Finite Strain Theory. To find an acceptable solution of nonlinear time-dependent thermo-hyperelastic equations, Newmark's time integration process and a nonlinear Hermitian finite element algorithm are employed. The final system’s responses to different boundary conditions such as input surface traction, heat flux and variable material properties are described schematically, and their influence on the wave propagation is calculated. It is shown that the amplitude of oscillation in a functionally graded hyperelastic medium is less than that of a medium with constant properties. The results also show that the wave travels through the medium faster than the FGM. Moreover, the modified Fourier law of heat conduction is applied and the impact of enhanced heat conduction model on the thermo-hyperelastic responses is discussed.