Strong Nonlinear Terms Research Articles

A Semi-Analytical Solution to a Generalized Nonlinear Van Der Pol Equation in Plasma By MsDTM

This article examines the effectiveness of the multistage differential transform method (MsDTM) in solving equations with a very strong nonlinear term. It introduces MsDTM as a method for solving the generalized nonlinear Van der Pol equation, which features strong nonlinearity. The generalized nonlinear Van der Pol equation arises in plasma and describes the propagation of various nonlinear phenomena, such as wave propagation in astrophysical plasma. MsDTM demonstrates greater accuracy compared to other analytical and numerical methods, such as the 4th-order Runge-Kutta Method (4thRKM), due to its ability to enhance accuracy through two factors: the number of iterations and the time step size. Most numerical methods rely solely on reducing the time step size to improve accuracy, but for some types of equations, this requires an impractically small time step size, causing the method to fail. In contrast, MsDTM offers an additional means of improving accuracy by increasing the number of iterations. The paper successfully applies MsDTM to solve the Van der Pol equation and presents the results, demonstrating that the method is highly effective for equations with very strong nonlinearity.

A novel dimensionality reduction iterative method for the unknown coefficient vectors in TGFECN solutions of unsaturated soil water flow problem

The main purpose of this paper is to reduce the dimensionality of unknown coefficient vectors in finite element (FE) solutions of two-grid FE Crank-Nicolson (CN) (TGFECN) format for the unsaturated soil water flow (USWF) problem with two strong nonlinear terms by using proper orthogonal decomposition (POD). For this purpose, our first step involves designing a time semi-discrete CN (TSDCN) scheme for the USWF problem and demonstrate the existence, boundedness, and error estimations of TSDCN solutions. Thereafter, we discretize the TSDCN scheme using the two-grid FE method to create a new TGFECN format and prove the existence, boundedness, and error estimations of TGFECN solutions. The primary focus should be on reducing the dimension of unknown coefficient vectors of TGFECN solutions through the utilization of the POD technique in creating a novel format, referred to as dimension reduction iterative TGFECN (DRITGFECN) format, while establishing the existence, boundedness, and error estimations for DRITGFECN solutions. Lastly, we use two sets of numerical tests to exhibit the advantage of the DRITGFECN format. Due to the presence of two highly nonlinear terms in the unsaturated soil flow problem, the development and analysis of DRITGFECN format pose greater challenges and necessitates more advanced technical skills compared to previous studies. However, the significance and broad applications of this research make it a valuable subject for investigation.

Local existence of compressible magnetohydrodynamic equations without initial compatibility conditions

In this paper, we study the initial‐boundary value problem of three‐dimensional viscous, compressible, and heat conductive magnetohydrodynamic equations. Local existence and uniqueness of strong solutions are established with any such initial data that the initial compatibility conditions do not be required. The analysis is based on some suitable prior estimates for the strong coupling term and strong nonlinear term . Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

Nonlinear processes in tsunami simulations for the Peruvian coast with focus on Lima and Callao

Abstract. This investigation addresses the tsunami inundation in Lima and Callao caused by the massive 1746 earthquake (Mw 9.0) along the Peruvian coast. Numerical modeling of the tsunami inundation processes in the nearshore includes strong nonlinear numerical terms. In a comparative analysis of the calculation of the tsunami wave effect, two numerical codes are used, Tsunami-HySEA and TsunAWI, which both solve the shallow water (SW) equations but with different spatial approximations. The comparison primarily evaluates the flow velocity fields in inundated areas. The relative importance of the various parts of the SW equations is determined, focusing on the nonlinear terms. Particular attention is paid to the contribution of momentum advection, bottom friction, and volume conservation. The influence of the nonlinearity on the degree and volume of inundation, flow velocity, and small-scale fluctuations is determined. The sensitivity of the solution concerning the bottom friction parameter is also investigated, showing the effects of nonlinearity processes in the inundated areas, wave heights, current velocity, and the spatial structure variations shown in tsunami inundation maps.

Magnetohydrodynamic effects on a pathological vessel: An Euler–Lagrange approach

For numerically studying blood flow in a pathological vessel under the influence of a magnetic field, it is necessary to develop an approach that tracks the moving tissue and accounts for interactions between the fluid, the arterial wall, and the magnetic field. The current study discusses a mathematical approach of the fluid's motion under the influence of a magnetic field using fluid mechanics principles. A mixed Euler–Lagrange formulation is introduced to mathematically describe the blood flow in the aneurysm during the entire cardiac cycle. Blood is considered a Newtonian, incompressible, and electrically conducting fluid, subjected to a static and uniform magnetic field. Generalized curvilinear coordinates are used to transform the transport equations into body-fitted geometries and provide a manageable form of equations. The system of equations related to motion consists of a coupled and nonlinear system of partial differential equations (PDEs). The discretization of the PDEs is performed using the finite volume method. The addition of the Lorentz force in the momentum PDEs describes the applied uniform magnetic field in the blood flow. Due to strong coupling and nonlinear terms, a simultaneous solution approach is applied. The results show that the magnetic field strongly influences blood flow, reducing the velocity field q¯ and increasing the pressure drop, Δp.

Global solutions and blow-up for Klein–Gordon equation with damping and logarithmic terms

In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown.

Asymptotic stability of rarefaction wave for a blood flow model

This paper is concerned with nonlinear stability of rarefaction wave to the Cauchy problem for a blood flow model, which describes the motion of blood through axi‐symmetric compliant vessels. Inspired by the stability analysis of classical ‐system, we show the solution of this typical model tends time‐asymptotically toward the rarefaction wave under some suitably small conditions and there are more difficulties in the proof due to the appearance of strong nonlinear terms regarding second‐order derivative of with respect to the spatial variable . The main result is proved by employing the elementary energy methods. This is the first result regarding nonlinear stability of some nontrivial profiles (i.e., non‐constant function patterns) for the blood flow model.

The higher-order modified Korteweg-de Vries equation: Its soliton, breather and approximate solutions

In this study, the fifth-order modified Korteweg-de Vries (F-MKdV) equation is first addressed using Hirota’s bilinear method. Thereafter, the exact and approximative solutions of the generalized form of the F-MKdV equation are investigated using the modified Kudryashov method, the Riccati equation and its Backlund transformation method, the solitary wave ansatz method, and the homotopy perturbation transform method (HPTM). As a result, solitons, breather, and solitary wave solutions are derived from these methods. In particular, we obtain some new solutions such as the dark soliton, bright soliton, singular soliton, periodic trigonometric, exponential, hyperbolic, and rational solutions. The constraint conditions associated with the resulting solutions are also discussed in detail. The HPTM is employed to construct approximate solutions to the aforementioned generalized model due to its strong nonlinear terms and only a few terms are required to obtain accurate solutions. These findings may help to understand complex nonlinear phenomena.

On a high-order iteration technique for a wave equation with nonlinear viscoelastic term

In this paper, a high-order iterative technique for solving a wave equation with strong damping and nonlinear viscoelastic term is constructed. For this purpose, we adapt the high-order iterative method used in the earlier works and establish the existence theorem of a recurrent sequence associated with the proposed problem. Thereafter, we prove a N-order convergence of the obtained sequence to a unique solution of the problem.

Analysis of Frank-Kamenetskii thermal explosion system of non-class a geometry based on Boltzmann method

Exothermic reaction systems of non-class A geometries are very common, with an endless rectangular rod typical. As a strong nonlinear source word is included in the governing equation, which is sensitive to the frank-kamenetskii parameter, there is no analytical solution. Many methods were previously suggested. However, with them are often non-physical solutions obtained. In this paper, the lattice Boltzmann process provides us with complete physical and precise solutions. We also analysed the sensitivity of the strong nonlinear source term and suggested advice for similar numerical calculations and experiments with thermal explosion.

Construction of analytical solution for Hirota–Satsuma coupled KdV equation according to time via new approach: Residual power series

In this work, a modern and novel approach method called the residual power series technique has been applied to find an analytical solution for an important equation in optical fibers called the Hirota–Satsuma coupled KdV equation with time as a series solution. Comparison of the analytical approximate solution with the exact solution concluded that the present method is an important addition for analyzing a system of partial differential equations that have a strong nonlinear term. We also represented graphically and discussed the effect of initial condition parameters and reaction of time on the model.

DATA-DRIVEN SPARSE IDENTIFICATION OF GOVERNING EQUATIONS FOR FLUID DYNAMICS

It is a challenging and important issue to establish a nonlinear dynamic model of system by use of limited data. The data-driven sparse identification method is an effective method developed recently to identify the governing equations of the dynamic system from data developed in recent years. In this paper, governing equations for different flows are identified by data-driven sparse identification methods. Partial differential equation functional identification of nonlinear dynamics (PDE-FIND) scheme and least absolute shrinkage and selection operator (LASSO) scheme are used to identify the governing equations of two-dimensional flow past a circular cylinder, liddriven cavity flow, Rayleigh-Benard convection and three-dimensional turbulent channel flow. An over-complete candidate library is constructed by direct numerical simulation flow field data in the process of identification. Variables in the library are retained up to second order, variable derivatives are retained up to second order, and nonlinear terms are retained up to fourth order. By comparing the results from the two methods, we find both methods show good performance in identifying governing equation with no nonlinear terms, i.e., vorticity transport equation, heat transport equation and continuity equation. PDE-FIND scheme correctly identified the governing equations and Rayleigh number and Reynolds number for the flow field. But LASSO scheme failed to identify the governing equations which contain strong nonlinear terms, i.e., Navier-Stokes equations. This is because grouping effect may occur among the items in the candidate library and only one item in the group is chosen in such case in LASSO scheme. So PDE-FIND scheme is more effective than LASSO scheme in sparse identification of strongly nonlinear partial differential equation. It is also found that selecting data from regions with abundant flow structures can improve the accuracy of data-driven sparse identification results.

Global solutions and exponential decay to a Klein–Gordon equation of Kirchhoff-Carrier type with strong damping and nonlinear logarithmic source term

Abstract This paper is concerned with the existence and exponential stability of the global solution to a Klein–Gordon equation of Kirchhoff-Carrier type with strong damping − Δ u t and logarithmic source term u l n | u | R 2 . We apply the potential well corresponding to the logarithmic nonlinearity. We prove the global weak solutions and the exponential stability for initial data in the set of stability created from the Nehari Manifold.

Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term

In this paper, we investigate the orbital stability of solitary waves for the following generalized long-short wave resonance equations of Hamiltonian form: 0.1{iut+uxx=αuv+γ|u|2u+δ|u|4u,vt+β|u|x2=0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} iu_{t}+u_{{xx}}=\\alpha uv+\\gamma \\vert u \\vert ^{2}u+\\delta \\vert u \\vert ^{4}u, \\\\ v_{t}+\\beta \\vert u \\vert ^{2}_{x}=0. \\end{cases} $$\\end{document} We first obtain explicit exact solitary waves for Eqs. (0.1). Second, by applying the extended version of the classical orbital stability theory presented by Grillakis et al., the approach proposed by Bona et al., and spectral analysis, we obtain general results to judge orbital stability of solitary waves. We finally discuss the explicit expression of det (d^{prime prime }) in three cases and provide specific orbital stability results for solitary waves. Especially, we can get the results obtained by Guo and Chen with parameters alpha =1, beta =-1, and delta =0. Moreover, we can obtain the orbital stability of solitary waves for the classical long-short wave equation with gamma =delta =0 and the orbital instability results for the nonlinear Schrödinger equation with beta =0.

Initial boundary value problem for generalized Zakharov equations with nonlinear function terms

In this paper, we consider the initial boundary value problem for generalized Zakharov equations. Firstly, we prove the existence and uniqueness of the global smooth solution to the problem by a priori integral estimates, the Galerkin method, and compactness theory. Furthermore, we discuss the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero.

An implicit-explicit Runge-Kutta-Chebyshev finite element method for the nonlinear Lithium-ion battery equations

We introduce a numerical method to integrate the nonlinear system of equations that model the physical-chemical laws of the Lithium-ion batteries. The mathematical model, formulated in Doyle et al. J. Electrochem. Soc. 140 (1993) 1526–1533, is a system of strongly coupled nonlinear parabolic and elliptic equations. Our numerical method combines a second order implicit–explicit Runge–Kutta–Chebyshev scheme for time discretization of the parabolic equations governing the dynamics of the variables ce and cs, with linear finite element space discretization of the system. The implicit-explicit numerical formulation of the parabolic equations allows us to decouple the strong nonlinear reaction terms, which are treated implicitly, from the linear diffusion terms, treated explicitly. This approach is computationally efficient because (a) it has an extended stability region, and (b) the coupled system of algebraic equations becomes a single variable nonlinear equation per mesh point, which is easily solved by Newton method.

Iterative Analytical Solutions for Nonlinear Two-Phase Flow with Gas Solubility in Shale Gas Reservoirs

The governing equations of a two-phase flow have a strong nonlinear term due to the interactions between gas and water such as capillary pressure, water saturation, and gas solubility. This nonlinearity is usually ignored or approximated in order to obtain analytical solutions. The impact of such ignorance on the accuracy of solutions has not been clear so far. This study seeks analytical solutions without ignoring this nonlinear term. Firstly, a nonlinear mathematical model is developed for the two-phase flow of gas and water during shale gas production. This model also considers the effects of gas solubility in water. Then, iterative analytical solutions for pore pressures and production rates of gas and water are derived by the combination of travelling wave and variational iteration methods. Thirdly, the convergence and accuracy of the solutions are checked through history matching of two sets of gas production data: a China shale gas reservoir and a horizontal Barnett shale well. Finally, the effects of the nonlinear term, shale gas solubility, and entry capillary pressure on the shale gas production rate are investigated. It is found that these iterative analytical solutions can be convergent within 2-3 iterations. The solutions can well describe the production rates of both gas and water. The nonlinear term can significantly affect the forecast of shale gas production in both the short term and the long term. Entry capillary pressure and shale gas solubility in water can also affect shale gas production rates of shale gas and water. These analytical solutions can be used for the fast calculation of the production rates of both shale gas and water in the two-phase flow stage.

Modelling and numerical simulation of vibrations induced by mixed faults of a rotor system immersed in an incompressible viscous fluid

Extraction of features of a specific signal in fluid–structure interaction is among the hottest problems in the field of mechanics. Yet, a comprehensive study of such problems remains a challenge due to their high nonlinearity and multidisciplinary nature. The study presented in this article is focused on a particular engineering application of fluid–structure interaction. The governing equation of a spinning rotor submerged in an incompressible viscous fluid is modelled by means of well-established dissipative energy principle, yielding a highly coupled 3-degree-of-freedom system with strong nonlinear terms. A two-dimensional model of the Navier–Stokes equations for the incompressible flow is developed for the viscous fluid motion around the spinning rotor under high fluctuations induced by unbalance, rotor–stator rub and a crack. The extracted features through frequency spectrum, orbit patterns and rotor-coupled deflection revealed that the performances of rotor systems are highly impacted by the hydrodynamic terms which are the sources of multiple frequency response. The results showed that the complex fluid–rotor model yields good analysis of fault diagnosis, and responses at more than one parametric resonance appear and reach a point of complex feature extractions when more than one fault coexists in the system. Furthermore, a nonlinear denoising by thresholding the wavelet coefficients is performed to overcome the complexity of discretization and for effective multiple fault diagnoses.

A harmonic balance approach with alternating frequency/time domain progress for piezoelectric mechanical systems

Abstract A semi-analytical approach based on harmonic balance (HB) method is proposed to predict steady-state responses of nonlinear piezoelectric mechanical systems. In order to handle complicated non-polynomial forces of the systems, the alternating frequency/time domain (AFT) progress is applied to establish a set of implicit algebraic equations of HB method. The stabilities of periodic responses are analyzed via Floquet theory. The proposed approach is successfully implemented a nonlinear piezoelectric energy harvester with a magnetic oscillator and a Duffing energy harvester with strong nonlinear electric terms. The HB solutions are well supported by the simulation results of the Runge-Kutta method. Besides, an experiment piezoelectric device is developed by a circular laminated plate, and an experimental model is established by system identifications. The model includes both the nonlinear stiffness and nonlinear damping which are characterized by the combined functions of polynomials and hyperbolic tangents. The model is analyzed via the proposed approach. The numerical simulations under different excitations show good agreements with the harmonic solutions. The frequency sweeps experimentally pursue the system has both softening and hardening nonlinearities which are verified by HB results. The proposed approach requires less analysis during the solving processes, and shows high accuracy to the strongly nonlinear piezoelectric mechanical systems, even the system with complicated nonlinearities.

Nonparametric Identification of Nonlinear Piezoelectric Mechanical Systems

Two novel nonparametric identification approaches are proposed for piezoelectric mechanical systems. The novelty of the approaches is using not only mechanical signals but also electric signals. The expressions for unknown mechanical and electric terms are given based on the Hilbert transform. The signals are decomposed and re-assembled to obtain smooth stiffness and damping curves. The current mapping approach is developed to identify accurately a piezoelectric mechanical system with strongly nonlinear electric terms. The developed identification approaches are successfully implemented to simulate signals obtained from different nonlinear piezoelectric mechanical systems, including Duffing nonlinearity, softening and hardening nonlinearity, and Duffing nonlinearity with strong nonlinear electric terms. The proposed approaches are successfully applied to experimental signals of a circular laminated plate device in order to identify the nonlinear stiffness functions, damping functions, electromechanical coupling functions, and equivalent capacitance functions. The results show both softening and hardening nonlinearity in the stiffness characteristic and weak nonlinearity in electric characteristics. The results of the Hilbert transform based approach and the current mapping approach are compared, and the outcomes show good agreements.