Integrable Nonlinear Differential Equations Research Articles

Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is investigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.

Mean Field Modeling of the FitzHugh–Nagumo Neuronal Network Model with Kernel Functions and Time-Delayed Couplings

In this paper, the nonlinear neural network FitzHugh–Nagumo model with an expansion by the excited neuronal kernel function has been investigated. The mean field approximation of neuronal potentials and recovery currents inside neuron ensembles was used. The biologically more realistic nonlinear sodium ionic current–voltage characteristic and kernel functions were applied. A possibility to present the nonlinear integral differential equations with kernel functions under the Fourier transformation by partial differential equations allows us to overcome the analytical and numerical modeling difficulties. An equivalence of two kinds solutions was confirmed basing on the errors analysis. The approach of the equivalent partial differential equations was successfully employed to solve the system with the heterogeneous synaptic functions as well as the FitzHugh–Nagumo nonlinear time-delayed differential equations in the case of the Hopf bifurcation and stability of stationary states. The analytical studies are corroborated by many numerical modeling experiments.The digital simulation at the transient and steady-state conditions was carried out by using finite difference technique. The comparison of the simulation results revealed that some of the calculated parameters, i.e. response and sensitivity is the same, while the others, i.e. half-time of the steady-state is significantly different for distinct models.

High Accuracy of Finite Element Method for a Class of Nonlinear Hyperbolic Integral Differential Equation

The hyperbolic integral differential equations are widely used in practical problems of heat conduction,nuclear reactions dynamics, viscous elastic mechanics, biomechanics, the pressure in loose medium with memory materials. We shall study a class of nonlinear hyperbolic integral differential equation in this paper. Using finite element method, we obtain some superconvergence results.

Stability of the μ-Camassa–Holm Peakons

The μ-Camassa–Holm (μCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa–Holm equation. We prove that the periodic peaked traveling wave solutions (peakons) of the μCH equation are orbitally stable.

A class of retarded nonlinear integral inequalities and its application in nonlinear differential-integral equation

In this paper, we discuss a class of retarded nonlinear integral inequalities and give an upper bound estimation of an unknown function by the integral inequality technique. This estimation can be used as a tool in the study of differential-integral equations with the initial conditions.MSC:26D10, 26D15, 26D20, 34A12, 34A40.

Systems of singularly perturbed fractional integral equations

The solution of a singularly perturbed nonlinear system fractional integral (differential) equations of order ς, 0 < ς < 1 is investigated. The leading order formal asymptotic solution is derived and proved to have the required properties.

Investigation of frictional effects on the nonlinear buckling behavior of a circular rod laterally constrained in a horizontal rigid cylinder

Although buckling of circular rods laterally constrained by a cylinder received considerable attention in the past, relatively fewer attempts have been made to study effects of friction and boundary constraints on the nonlinear buckling behavior, especially for relatively short rods. Thus a model of rods laterally constrained in a horizontal rigid cylinder considering effects of friction and boundary constraints is built. The resulting coupled nonlinear integral–differential equations are successfully solved for the first time by using the discrete singular convolution (DSC), a relatively new numerical approach, together with the Newton–Raphson method. Detailed formulations and solution procedures are given. Examples with various friction coefficients and combinations of boundary conditions are analyzed. To verify the formulations and solution procedures, some DSC results are compared to experimental data or results obtained by using the finite element method. Results reveal that both lateral and helical buckling loads usually increase with the increase of the friction coefficients. However, the helical buckling loads of the rod with both ends fixed, defined as the load at which the rod loses wall contact, slightly decrease with the increase of the friction coefficients. Boundary conditions have obvious effects on the buckling behavior for relatively short rods.

A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods

AbstractThe Boltzmann equation (BE) for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinate method (DOM) using a finite set of velocity quadrature points. Such simplification reduces the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main difficulty of the currently used (conventional) numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite quadrature. The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position. By this dynamic nature in the physical space, this new quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM) and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM and DLBM have been tested and validated with several testing problems. Of the same accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.

ON THE NONLOCAL SYMMETRIES OF THE μ-CAMASSA–HOLM EQUATION

The μ-Camassa–Holm (μCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa–Holm and the Hunter–Saxton equations. This equation admits quadratic pseudo-potentials which allow us to compute some first-order nonlocal symmetries. The found symmetries preserve the mean of solutions. Finally, we discuss also the associated μCH equation.

Postbuckling Investigation on the Micromachined Double-Clamped Beams

In this paper, thermal postbuckling of a beam-like structure subjected to a uniform thermal loading is studied both theoretically and experimentally. In theory aspect, the equations governing the axial and transverse deformations of beams are derived. The two equations are reduced to a single nonlinear fourth-order integral-differential equation governing the transverse deformations. The analytical method presented here offers a simple yet efficient solution approach to analyze the bistable behavior of microbeam or beam-like structures under thermal loading. Then, microfabricated double-clamped beams are chosen as experimental object due to its precise dimensions and easy loading method. The lab-made HF vapor system was used to release the suspended structures with satisfactory results. After the fabrication and measurement, by comparing the theoretical results with experimental results, an excellent agreement is gotten. It proves not only the validity of the solution obtained here, on the other hand, importantly, it also demonstrates a new field for MEMS as another easy and accurate tool to investigate mechanics problems. With the accurate solutions for posbuckling beams, we apply it into new electrothermal postbuckling actuators, a big deflection promising the possibility of a new kind of actuator.

Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces

A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM). Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the 'squared solutions' (generalized exponentials) are derived. Next, expansions of the potential and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Finally, the Hamiltonian structures of these generalized multi-component Heisenberg ferromagnetic (MHF) type integrable models on A.III-type symmetric spaces are briefly analyzed.

Rational bundles and recursion operators for integrable equations on A.III-type symmetric spaces

We analyze and compare methods for constructing the recursion operators for a special class of integrable nonlinear differential equations related to symmetric spaces of the type A.III in Cartan’s classification and having additional reductions.

A New Perturbative Approach in Nonlinear Singularity Analysis

Problem statement: The study is devoted to the “mirror” method which enables one to study the integrability of nonlinear differential equations. Approach: A perturbative extension of the mirror method is introduced. Results: The mirror system and its first perturbation are then utilized to gain insights into certain nonlinear equations possessing negative Fuchs indices, which were poorly understood in the literatures. Conclusion/Recommendations: In particular, for a nonprincipal but maximal Painleve family the first-order perturbed series solution is already a local representation of the general solution, whose convergence can also be proved.

Reductions of integrable equations on A.III-type symmetric spaces

We study a class of integrable nonlinear differential equations related to the A.III-type symmetric spaces. These spaces are realized as factor groups of the form SU(N)/S(U(N − k) × U(k)). We use the Cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic Lax operators to this symmetric space. The symmetries of the Lax operator are inherited by the fundamental analytic solutions and give a characterization of the corresponding Riemann–Hilbert data.

On certain new integrable second order nonlinear differential equations and their connection with two dimensional Lotka–Volterra system

In this paper, we consider a second order nonlinear ordinary differential equation of the form ẍ+k1(ẋ2/x)+(k2+k3x)ẋ+k4x3+k5x2+k6x=0, where ki’s, i=1,2,…,6, are arbitrary parameters. By using the modified Prelle–Singer procedure, we identify five new integrable cases in this equation besides two known integrable cases, namely (i) k2=0, k3=0 and (ii) k1=0, k2=0, k5=0. Among these five, four equations admit time-dependent first integrals and the remaining one admits time-independent first integral. From the time-independent first integral, nonstandard Hamiltonian structure is deduced, thereby proving the Liouville sense of integrability. In the case of time-dependent integrals, we either explicitly integrate the system or transform to a time-independent case and deduce the underlying Hamiltonian structure. We also demonstrate that the above second order ordinary differential equation is intimately related to the two dimensional Lotka–Volterra system. From the integrable parametric choices of the above nonlinear equation all the known integrable cases of the LV system can be deduced.

Analysis of dynamical response to large deformation of piles with initial displacements

In this paper, a nonlinear mathematical model for analyzing dynamical response to the large deformation of piles with initial displacements is firstly established with the arc-coordinate, and it is a set of nonlinear integral-differential equations, in which, the Winkeler model is used to simulate the resistance of the soil to the pile. Secondly, a set of new auxiliary functions are introduced. The differential-integral equations are transformed into a set of nonlinear differential equations, and the differential quadrature method (DQM) and the finite difference method (FDM) are applied to discretize the set of nonlinear equations in the spatial and time domains, respectively. Then, the Newton-Raphson method is used to solve the set of discretization algebraic equations at each time step. Finally, numerical examples are presented, and the dynamical responses to the deformation of piles, including configuration, bending moment and shear force, are graphically illuminated. In calculation, two types of initial displacements and dynamical loads are applied, and the effects of parameters on the dynamical responses of piles are analyzed in detail.

Stochastic processes in Q p associated with systems of nonlinear PIDEs

We construct a Markov process and its multiplicative operator functional associated with a system of nonlinear partial integral-differential equations (PIDE). As a result we derive a probabilistic representation of the solution to the Cauchy problem for a system of nonlinear PIDEs in Qp.

A new technique for solving nonlinear integral–differential equations

In this paper the variational iteration method is used to solve a system of nonlinear integral–differential equations. This method is based on optimal identification of Lagrange multipliers in the correction functionals. Numerical examples are given to show effectiveness, convenience and accuracy of the method.

An Interactive Desktop Computer Program for Simulating Nanometer Scale Surface Pattern Formation

AbstractNanometer-scale patterns may form as one or more chemical components deposit on a solid substrate. This self-assembly process can be described by a set of nonlinear integral-differential diffusion equations accounting for two opposing factors: phase separation to minimizing Gibb's free energy in individual surface phases and reduction in phase boundaries to minimize surface energy created by phase separation. I here present a desktop computer program that allows us to interactively simulate self-assembly of nanometer-scale surface patterns. In particular, this program provides a convenient tool for studying the effects of temperature variations and preexisting patterns on the self-assembly process. Computer simulations show that an increase in temperature may enlarge pattern sizes and can eventually lead to the disappearance of the patterns.

Method of Green’s function of nonlinear vibration of corrugated shallow shells

Based on the dynamic equations of nonlinear large deflection of axisymmetric shallow shells of revolution, the nonlinear free vibration and forced vibration of a corrugated shallow shell under concentrated load acting at the center have been investigated. The nonlinear partial differential equations of shallow shell were reduced to the nonlinear integral-differential equations by using the method of Green’s function. To solve the integral-differential equations, the expansion method was used to obtain Green’s function. Then the integral-differential equations were reduced to the form with a degenerate core by expanding Green’s function as a series of characteristic function. Therefore, the integral-differential equations became nonlinear ordinary differential equations with regard to time. The amplitude-frequency relation, with respect to the natural frequency of the lowest order and the amplitude-frequency response under harmonic force, were obtained by considering single mode vibration. As a numerical example, nonlinear free and forced vibration phenomena of shallow spherical shells with sinusoidal corrugation were studied. The obtained solutions are available for reference to the design of corrugated shells.