Discrete Nonlinear Equation Research Articles

Intelligent Planning of Tourist Routes Based on Cloud Computing and Marching Algorithm

In order to improve the effect of modern tourism route planning, this article combines the cloud computing advancement algorithm to construct a tourism route intelligent planning system and combines the current people's tourism needs to carry out intelligent tourism route planning. Based on the fast forward algorithm of the program function equation, this article uses the fast forward algorithm to solve the discrete nonlinear program function equation and directly obtain the cloud computing data. In addition, the self-intersection phenomenon of the wave front must be considered when describing the characteristics of the cloud computing information field by using the process of the equivalent cloud computing information front expanding to the surrounding. Finally, this article constructs the intelligent system structure. The research results show that the intelligent travel route planning system based on cloud computing and marching algorithm proposed in this article can play an important role in modern smart tourism.

Continuous Limit, Rational Solutions, and Asymptotic State Analysis for the Generalized Toda Lattice Equation Associated with 3 × 3 Lax Pair

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to 3×3 matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known 3×3 Lax pair of this equation, the discrete generalized (m,3N−m)-fold Darboux transformation was constructed for the first time and extended from the 2×2 Lax pair to the 3×3 Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants.

Dynamic Response of Some Noncarbon Nanomaterials Using Multiscale Modeling Involving Material and Geometric Nonlinearities

Abstract Nonlinear dynamic response of some noncarbon nanomaterials, involving material and geometric nonlinearities under different types of dynamic loads, is investigated using computationally efficient multiscale modeling. Multiscale-based finite element model is developed in the framework of the Cauchy–Born rule, which couples the deformation at the atomic scale to deformation at the continuum scale. The Tersoff–Brenner type interatomic potential is employed to model the atomic interactions. The governing finite elemental equations are derived through Hamilton's principle for a dynamic system. The linearization of nonlinear discrete equations is done using Newton–Raphson method and are solved using Newmark's time integration technique. The effects of material and geometric nonlinearities, inherent damping, different types of dynamic loads, and initial strain on the transient response of noncarbon nanosheets with clamped boundary conditions are reported in detail. The present results obtained from the multiscale-based finite element method are compared with those obtained from molecular dynamics (MD) simulation for the free vibration analysis, and the results are found to be in good agreement. The present results are also compared with the results of those obtained from Kirchhoff plate model for some cases.

An upwind generalized finite difference method for meshless solution of two-phase porous flow equations

This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation. Combining with moving least square approximation and local Taylor expansion, spatial derivatives of oil-phase pressure at a node are approximated by generalized difference operators in the local influence domain of the node. By introducing the first-order upwind scheme of phase relative permeability, and combining the discrete boundary conditions, fully-implicit GFDM-based nonlinear discrete equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, and points out the significant effect of the symmetry or uniformity of the node allocation in the node influence domain on the accuracy of generalized difference operators, and the radius of the node influence domain should be small to achieve high calculation accuracy, which is a significant difference between the studied hyperbolic two-phase porous flow problem and the elliptic problems when GFDM is applied. In all, the upwind GFDM with the fully implicit nonlinear solver and related analysis about computational performances given in this work may provide a critical reference for developing a general-purpose meshless numerical simulator for porous flow problems.

KAZE-SAR: SAR Image Registration Using KAZE Detector and Modified SURF Descriptor for Tackling Speckle Noise

Synthetic aperture radar (SAR) image registration is still a challenging task in remote sensing. The scale-invariant feature transform (SIFT) method and its extensions are most widely used feature detectors in SAR images. However, due to the presence of nonlinear speckle noise, some wrong features are chosen which directly influence the feature matching process. In this article, for the first time, the KAZE algorithm with a modified version of speeded-up robust features (SURF) descriptor is used for SAR image registration. It uses discrete stochastic second order nonlinear partial differential equations (PDEs) to model the SAR image’s edge structure. KAZE uses nonlinear diffusion filtering to build up the scale levels of the SIFT descriptor. It preserves the edges while simultaneously decreasing the speckle noise in smoothing the image. Therefore, it captures the exact location of features in down-sampling and filtering stages of SIFT. Experimental results show that our proposed method has increased the localization accuracy and distinctiveness of feature extraction through the use of KAZE detector. Our method is also able to handle challenging cases in different image sources, different view angles, different image times, complex affine mapping, presence of blurring, and noise.

PERIODIC DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS WITH PERTURBED AND SUB-LINEAR TERMS

In this paper, we study a class of perturbed discrete nonlinear Schrödinger equations with sub-linear nonlinearities at infinity and obtain the existence of solitons for this class of equations by using a generalized saddle point theorem. To the best of our knowledge, there is no published result focusing on this class of perturbed discrete nonlinear equations by this method.

Asymptotic properties of solutions to discrete Volterra type equations

We study higher order nonlinear discrete Volterra equations of the form We discuss the problem of the existence of solutions with prescribed asymptotic behavior. In the proof of the main theorem we use the iterated remainder operator and the Schauder's type fixed point theorem. This approach allows us to control the degree of approximation. Our results are also new for the special cases of the above equation such as linear Volterra equation, ordinary difference equation and discrete Emden–Fowler type equation.

An efficient family of Steffensen‐type methods with memory for solving systems of nonlinear equations

The article discusses derivative-free algorithms with and without memory for solving numerically nonlinear systems. We proposed a family of fifth- and sixth-order schemes and extended them to algorithms with memory. We further discuss the convergence and computational efficiency of these algorithms. Numerical examples of mixed Hammerstein integral equation, discrete nonlinear ordinary differential equation, and Fisher's partial differential equation with Neumann's boundary conditions are discussed to demonstrate the convergence and efficiency of these schemes. Finally, some numerical results are included to examine the performance of the developed methods.

Nonlinear vibration of a moving flexible printed electron web under multiphysics dynamics

Flexible electronic printing technology is a scientific technology that uses an “ink” material with conductive, dielectric, or semiconductor properties printed on a flexible web substrate to achieve precise preparation of flexible electronic devices, which are widely used in information, energy, medical, and military fields. In the preparation of the printing process of flexible printed electron webs under complex working conditions, the moving web will experience substantial unstable nonlinear dynamic behavior, such as divergence, flutter, bifurcation, and chaos. Accordingly, because of the coupling effects of the complex working conditions of the magnetic field, air and nonlinear electrostatic field forces, it is indispensable to explore the nonlinear dynamic equation of the flexible printed electron web in motion. The theory of multiphysics dynamics establishes a nonlinear vibration equation for the flexible printed electron web under multiphysics conditions. The discrete nonlinear vibration equation of state space equation was obtained by the Bubnov–Galerkin method. Utilizing the Runge–Kutta technique of the fourth-order, Poincaré maps, phase-plane diagrams, power spectra, bifurcation graphs, and time history diagrams of the moving flexible printed electron web were obtained. The influences of the velocity, electrostatic field, magnetic induction intensity, and follower force on the flexible printed electron web were analyzed. In addition, the Ansoft Maxwell finite element simulation software was used to simulate the magnetic field distribution of the moving web during roll-to-roll transmission. This paper determines the stable working range of the moving flexible printed electron web, which provides a theoretical basis for the preparation of flexible printed electronic webs.

Asymptotic Properties of Solutions to Discrete Volterra Monotone Type Equations

We investigate the higher order nonlinear discrete Volterra equations. We study solutions with prescribed asymptotic behavior. For example, we establish sufficient conditions for the existence of asymptotically polynomial, asymptotically periodic or asymptotically symmetric solutions. On the other hand, we are dealing with the problem of approximation of solutions. Among others, we present conditions under which any bounded solution is asymptotically periodic. Using our techniques, based on the iterated remainder operator, we can control the degree of approximation. In this paper we choose a positive non-increasing sequence u and use o(un) as a measure of approximation.

Solitons in nerve axons

An axon or nerve fiber is a long, slender projection of a nerve cell, or neuron, in vertebrates that typically conducts electrical impulse known as action potentials away from the nerve cell body. Axons are the cable-like protrusions of neurons which wire up the nervous system. These delicate structures frequently need to survive for a lifetime. Parallel polar, bundles of microtubules run beside axons to form their structural backbones and are acting as highways for life-sustaining transportation. In this paper, we investigate the excitations of soliton along axons that are governed by the Nerve Soliton (NS) model. The NS model of nerve axon is governed by the discrete nonlinear equation of motion derived using the finite difference method. We done modulational instability (MI) analysis for the discrete equation of motion with effect of damping force by using Rotating wave approximation (RWA). The inherent intrinsic localized wave modes were generated which is responsible for energy localization in the nerve axons.

Hamiltonian systems, Toda lattices, solitons, Lax pairs on weighted Z-graded graphs

We consider discrete one-dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions that allow one to lift one-dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons for static potentials on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph, the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.

Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form Δ∗β[k](t)=wt+β,k(t+β),k(λ(t+β)), with condition k(0)=p[k] for t∈N1−β, 0<β≤1, λ∈(0,1) and investigate the properties of asymptotic stability of solutions. We will prove the main results by the aid of Krasnoselskii’s and generalized Banach fixed point theorems. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Nonlinear vibration analysis of a flexible rotor shaft with a longitudinally dispositioned unbalanced rigid disc

This study aims to investigate the nonlinear dynamics of a rotor system of a flexible rotor shaft carrying an unbalanced rigid disc. For this purpose, the dynamics of three different rotor systems are analysed in this study. First, the dynamics of a system of an unbalanced flexible rotor shaft is examined. Second, vibration analysis and a parametric study are carried out on a system of an unbalanced rigid disc located at the mid-point of a rotor flexible shaft. The third system is composed of a flexible rotor shaft carrying a longitudinally dispositioned unbalanced rigid disc. The flexible shaft is modelled using the Euler-Bernoulli beam theory. Geometrical and inertial nonlinearities of the flexible shaft are considered in the mathematical modelling of the structure. The extended Hamilton principle is utilized to derive the nonlinear equations of motion of the described system. The obtained nonlinear governing equations are discretized using the assumed mode method and applying the Galerkin method to the equations of motion. In this study, a two-mode reduced order model is used to investigate the nonlinear dynamics of the rotor systems. The resultant discrete nonlinear equations of motion are solved using direct numerical integration. The results of the first case study are used to verify the accuracy of the theory and methods used in this study. Then, the results of the parametric study are presented. The effects of various parameters of the system are investigated on the dynamic behaviour of the structure. Finally, the chaotic response of the third case is investigated using the numerical simulation of the dynamics of the system.

Fracture analysis of plastically graded material with thermo-mechanical J-integral

In this paper, the influence of plasticity graded property and thermal boundary conditions have been investigated on the fracture parameter, i.e. J-integral using the extended finite element method. A complete computational methodology has been presented to model elasto-plastic fracture problems with geometrical and material nonlinearities. For crack discontinuity modeling, a partition of unity enrichment concept was employed with additional mathematical functions like Heaviside and branch enrichment for crack discontinuity and stress field gradient, respectively. The modeling of the stress–strain relationship of the material is implemented using the Ramberg–Osgood material model and geometric nonlinearity is modeled using an updated Lagrangian approach. The isotropic hardening and von-Mises yield criteria are considered to check the plasticity condition. The elastic predictor–plastic corrector algorithm is employed to capture elasto-plastic stress in a cracked domain. The variation in plasticity properties for plastically graded material is modeled by exponential law. Furthermore, the nonlinear discrete equations are numerically solved using a Newton–Raphson iterative scheme. Various cracked problem geometries subjected to thermal (adiabatic and isothermal conditions) and thermo-mechanical loads are simulated for stress contours and J-integrals using the elasto-plastic fracture mechanics approach. A comparison of the results obtained using extended finite element method with literature and the finite element analysis (FEA) package shows the accuracy and effectiveness of the presented computational approach. A component-based problem, i.e. a Brazilian disc subjected to thermo-mechanical loading, has been solved to show the adaptability of this work.

A robust and efficient iterative strategy for nonlinear analysis of structures subjected to buckling

A robust and efficient iterative strategy for nonlinear analysis of structures subjected to buckling

On oscillatory second order nonlinear impulsive systems of neutral type

In this work, the necessary and sufficient conditions for oscillation of a class of second order neutral impulsive systems are established and our impulse satisfies a discrete neutral nonlinear equation of similar type. Further, one illustrative example showing applicability of the new result is included.

Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type* *Project supported by the National Natural Science Foundation of China (Grant Nos. 12071042 and 61471406), the Beijing Natural Science Foundation, China (Grant No. 1202006), and Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCP-B201704).

We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction, which may have potential applications in electric circuits. Nonlocal infinitely many conservation laws are constructed based on its Lax pair. Nonlocal discrete generalized (m, N – m)-fold Darboux transformation is extended and applied to solve this system. As an application of the method, we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized (1, N – 1)-fold Darboux transformation, respectively. By using the asymptotic and graphic analysis, structures of one-, two-, three- and four-soliton solutions are shown and discussed graphically. We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures. It is shown that the soliton structures are quite different between discrete local and nonlocal systems. Results given in this paper may be helpful for understanding the electrical signals propagation.

Nonlinear vibration and chaos of a moving flexible graphene smart electronic web

Flexible graphene smart electronics are a kind of electronic device made of silver or other inorganic material and a flexible graphene web substrate. Because of their flexibility and excellent electronic properties, flexible graphene smart electronics have become a research topic of interest. However, during the mass production process, the high-speed transmission of the continuous tension state is susceptible to external disturbances, which seriously restricts the improvement of the flexible graphene smart electronic printing speed. Therefore, studying the dynamic characteristics of moving flexible graphene electronic materials is of great significance for improving their quality and production efficiency. This paper considers a roll-to-roll smart RFID label graphene web as the research object. Based on the energy functional variational theory, the Bubnov-Galerkin method is used to eliminate the residual value of the dynamic Von Karman partial differential equation, and a discrete forced nonlinear vibration control equation for a moving flexible graphene smart electronic web is established. The Melnikov method is used to obtain the Melnikov function of the orbit parameter for the vibration control equation and solve the conditions of the chaos criterion. Using the fourth-order Runge-Kutta method, the nonlinear vibration phase-plane portraits, Poincare maps, time history diagram, and power spectra of the moving flexible graphene smart electronic web are obtained. The effects of the web moving speed, external damping, and initial system conditions on nonlinear dynamic behaviour are calculated, and the stable and unstable working regions of the moving flexible graphene smart electronic web are obtained. This research provides a theoretical basis for the dynamic stability and vibration control of a moving graphene smart electronic web.

The Asymptotic Behavior of Solutions of Discrete Nonlinear Fractional Equations

In this study, we consider a class of nonlinear fractional difference equations following form $$\Delta \left( {a\left( t \right)g\left( {{\Delta ^\alpha }x\left( t \right)} \right)} \right) + F\left( {t,\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{\left( {t - s - 1} \right)}^{\left( { - \alpha } \right)}}x\left( s \right)} } \right) = 0,$$ where t ∈ Nt0+1−α and Δα denotes the Riemann-Liouville fractional difference operator of order α. Using the generalized Riccati function, we obtain some oscillation criteria. Finally, we give some illustrative examples. MSC 2010: Primary 26A33; Secondary 34A08, 39A21, 39A10