Integrable Nonlinear Differential Equations Research Articles

Existence and attractivity of global solutions for a class of fractional quadratic integral equations in Banach space

In this paper, we present some results concerning the existence and attractivity of global solutions for a class of nonlinear fractional integral equations and fractional differential equations in a Banach space X, respectively. These results are new even in the case of $X=\mathbf{R}$ . Some examples are given to show the applications of the abstract results.

Henryk Hudzik – vita et opera

This article contains a short vita of Henryk Hudzik's as well as a non-exhaustive survey of his contribution to various areas of analysis. We focus on the theory of Orlicz−Sobolev spaces and the geometry of Banach spaces. We highlight criteria for some important geometric properties related to the metric fixed point theory in some classes of Banach lattices, including Orlicz and Orlicz−Lorentz spaces, but we do not forget Henryk Hudzik's contribution to nonlinear integral equations and partial differential equations.

Asymptotic integration of differential equations with singular $p$-Laplacian

In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p-$Laplacian, where $1 < p < 2$. We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t \rightarrow \infty $.

On Solvability of one Class of Nonlinear Integral-differential Equation with Hammerstein Non-compact Operator Arising in a Theory of Income Distribution

On Solvability of one Class of Nonlinear Integral-differential Equation with Hammerstein Non-compact Operator Arising in a Theory of Income Distribution

Milne quantization for non-Hermitian systems

We generalize the Milne quantization condition to non-Hermitian systems. In the general case the underlying nonlinear Ermakov–Milne–Pinney equation needs to be replaced by a nonlinear integral differential equation. However, when the system is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and one may employ the original energy integral to determine its quantization. We illustrate the working of the general framework with the Swanson model and two explicit examples for pairs of supersymmetric Hamiltonians. In one case both partner Hamiltonians are Hermitian and in the other a Hermitian Hamiltonian is paired by a Darboux transformation to a non-Hermitian one.

Dark-bright soliton solutions with nontrivial polarization interactions for the three-component defocusing nonlinear Schrödinger equation with nonzero boundary conditions

We derive novel dark-bright soliton solutions for the three-component defocusing nonlinear Schrödinger equation with nonzero boundary conditions. The solutions are obtained within the framework of a recently developed inverse scattering transform for the underlying nonlinear integrable partial differential equation, and unlike dark-bright solitons in the two component (Manakov) system in the same dispersion regime, their interactions display non-trivial polarization shift for the two bright components.

A simple and accurate numeric solution procedure for nonlinear buckling model of drill string with frictional effect

Abstract The nonlinear buckling analysis of drill string in a rigid well is presented in this paper. Considering effects of friction and boundary constraints, this problem could be taken as a model of a rod laterally constrained in a rigid cylinder (horizontal, oblique and vertical rigid cylinder could be included). After introducing a new variable, the resulting coupled nonlinear integral–differential equations are successfully solved by employing the extended system shooting method. Examples with various friction coefficients and combinations of boundary conditions are proposed. It is found that the axial frictional force plays a more significant part on buckling load for horizontal well than vertical one. Compared to experimental data or results obtained by using the discrete singular convolution algorithm (DSC) and the finite element method (FEM), the accuracy of the formulations and solution procedures, is verified. What’s more, the nonlinear buckling behaviors of two instances of vertical scientific wells are analyzed. The present results are useful for practical design applications related to calculation of buckling loads and selection of bottom hole assembly (BHA) elements.

Modelling of double reset spiking neuron and Fitzhugh–Nagumo equations with coupling kernel functions

Abstract Creation of a computationally efficient and biologically plausible neuron and neural network models is an exceptionally important occupation, the results of which are much needed for the neuroscience in general or for the neurophysiology in partial. Besides, the applying neural network technological areas are always looking for novel improvements of the methodological approaches, algorithms, computational facilities. In this paper, a new (in some aspects) hybrid integrate-and-fire (HIF) neuron model, based on N-shaped Na + and K + ionic current–voltage characteristics and a double discrete reset mechanism, is proposed. In the results of computational experiments, the spike series from a single spiking to the burst and chaotic burst trains are demonstrated. In the other part of the paper, analogous FitzHugh–Nagumo neuron model as a generalized neural networks system with an expansion to an inclusion of a coupling kernel function has been considered. The mean field approximation of neuronal potentials and recovery currents inside a neuron cluster was used. The biologically more realistic nonlinear N-shaped Na + characteristic and kernel functions (one - or two-dimensional) were applied. A possibility to present the nonlinear integral differential equations with kernel functions under the Fourier transformation by partial differential equations was used that allowed us to overcome the analytical and numerical modeling difficulties. The equivalence of two kinds of solutions was additionally confirmed based on the error evaluation. The approach of the equivalent partial differential equations was successfully employed to solve the system with the heterogeneous synaptic functions. The analytical studies are corroborated by many numerical modeling experiments.

A numerical method for reconstructing the coefficient in a wave equation

We present a numerical method for reconstructing the coefficient in a wave equation from a single measurement of partial Dirichlet boundary data. The original inverse problem is converted to a nonlinear integral differential equation, which is solved by an iterative method. At each iteration, one linear second‐order elliptic problem is solved to update the reconstruction of the coefficient, then the reconstructed coefficient is used to solve the forward problem to obtain the new data for the next iteration. The initial guess of the iterative method is provided by an approximate model. This model extends the approximate globally convergent method proposed by Beilina and Klibanov, which has been well developed for the determination of the coefficient in a special wave equation. Numerical experiments are presented to demonstrate the stability and robustness of the proposed method with noisy data.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 289–307, 2015

Regularity results for full nonlinear mixed type integraldifferential operators

We introduce a class of full nonlinear integral-differential equations with nonsymmetric positive kernels and study the regularity of their solutions. More precisely, we establish a nonlocal version of A-B-P estimate, a Harnack inequality, Holder regularity, and C 1, α of the solutions.

Integrable Background Geometries

This work has its origins in an attempt to describe systematically the integrable geometries and gauge theories in dimensions one to four related to twistor theory. In each such dimension, there is a nondegenerate integrable geometric structure, governed by a nonlinear integrable differential equation, and each solution of this equation determines a background geometry on which, for any Lie group $G$, an integrable gauge theory is defined. In four dimensions, the geometry is selfdual conformal geometry and the gauge theory is selfdual Yang-Mills theory, while the lower-dimensional structures are nondegenerate (i.e., non-null) reductions of this. Any solution of the gauge theory on a $k$-dimensional geometry, such that the gauge group $H$ acts transitively on an $\ell$-manifold, determines a $(k+\ell)$-dimensional geometry ($k+\ell\leqslant4$) fibering over the $k$-dimensional geometry with $H$ as a structure group. In the case of an $\ell$-dimensional group $H$ acting on itself by the regular representation, all $(k+\ell)$-dimensional geometries with symmetry group $H$ are locally obtained in this way. This framework unifies and extends known results about dimensional reductions of selfdual conformal geometry and the selfdual Yang-Mills equation, and provides a rich supply of constructive methods. In one dimension, generalized Nahm equations provide a uniform description of four pole isomonodromic deformation problems, and may be related to the ${\rm SU}(\infty)$ Toda and dKP equations via a hodograph transformation. In two dimensions, the ${\rm Diff}(S^1)$ Hitchin equation is shown to be equivalent to the hyperCR Einstein-Weyl equation, while the ${\rm SDiff}(\Sigma^2)$ Hitchin equation leads to a Euclidean analogue of Plebanski's heavenly equations.

TRAVELING WAVE SOLUTIONS OF NONLINEAR SCALAR INTEGRAL DIFFERENTIAL EQUATIONS ARISING FROM SYNAPTICALLY COUPLED NEURONAL NETWORKS

Consider the following nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks. These model equations generalize many important nonlinear scalar integral differential equations aris ing from synaptically coupled neuronal networks. The kernel functions K and W represent synaptic couplings between neurons in synaptically coupled neuronal networks. The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions), lateral inhibitions (modeled with Mexican hat kernel functions), lateral excitations (modeled with upside down Mexican hat kernel functions), but also synaptic couplings which may change sign for finitely many times or even infinitely many times. The function H = H(u − ) represents the Heaviside step function, which is defined by H(u − ) = 0 for all u . The functions and represent probability density functions defined on (0,1). The parameter c > 0 represents the speed of an action potential and the parameter > 0 represents a constant delay. In these equations, u = u(x, t) stands for the membrane potential of a neuron at position x and time t. The positive constants > 0 and > 0 represent synaptic rates. The positive constants > 0 and > 0 represent thresholds for excitation of neurons. The function f = f(u) represents the sodium currents in neuronal networks. The positive constant w 0 > 0 is to be given. The authors will establish the existence and stability of traveling wave solutions of these nonlinear scalar integral differential equations by coupling together speed index functions, stability index func tions (often called Evans functions, that is, complex analytic functions), implicit function theorem, intermediate value theorem, mean value theorem, global strong maximum principle for Evans func tions, linearized stability criterion and many other important techniques in dynamical systems. They will find sufficient conditions satisfied by the synaptic couplings, by the probability density functions, by the synaptic rate constants and by the thresholds so that the traveling wave solutions and their wave speeds exist, and the stability of the traveling wave solutions is true. The main results obtained in this paper greatly improve many previous results.

Generalized Bilinear Differential Operators, Binary Bell Polynomials, and Exact Periodic Wave Solution of Boiti-Leon-Manna-Pempinelli Equation

We introduce how to obtain the bilinear form and the exact periodic wave solutions of a class of (2+1)-dimensional nonlinear integrable differential equations directly and quickly with the help of the generalizedDp-operators, binary Bell polynomials, and a general Riemann theta function in terms of the Hirota method. As applications, we solve the periodic wave solution of BLMP equation and it can be reduced to soliton solution via asymptotic analysis when the value ofpis 5.

Existence of Wave Front Solutions of an Integral Differential Equation in Nonlinear Nonlocal Neuronal Network

An integral-differential model equation arising from neuronal networks with very general kernel functions is considered in this paper. The kernel functions we study here include pure excitations, lateral inhibition, lateral excitations, and more general synaptic couplings (e.g., oscillating kernel functions). The main goal of this paper is to prove the existence and uniqueness of the traveling wave front solutions. The main idea we apply here is to reduce the nonlinear integral-differential equation into a solvable differential equation and test whether the solution we get is really a wave front solution of the model equation.

EMPLOYMENT, PRODUCTION AND CONSUMPTION WITH RANDOM UPDATE: NON-EQUILIBRIUM STATIONARY STATE EQUATIONS

In this work, we investigate the Model of Employment, Production and Consumption, as introduced in a series of papers by I. Wright [1–3] from the perspective of statistical physics, and we focus on the presence of equilibrium. The model itself belongs to the class of multi-agent computational models, which aim to explain macro-economic behavior using explicit micro-economic interactions.&lt;br /&gt;Based on the mean-field approximation, we form the Fokker-Plank equation(s) and then formulate conditions forming the stationary solution, which results in a system of non-linear integral-differential equations. This approximation then allows the presence of non-equilibrium stationary states, where the model is a mixed additive-multiplicative model.

Thermal postbuckling behavior of 3D braided beams with initial geometric imperfection under different type temperature distributions

Thermal postbuckling analysis for 3D (three-dimension) braided beams to initial geometrical imperfection in general modes subjected to uniform, linear and non-linear temperature distribution through the thickness are presented. The cross-section of 3D braided composite beam may be treated as a cell system and the geometry of each cell is deeply dependent on its position. A generic imperfection function for one-dimensional composite beam is introduced to model various possible initial geometrical imperfection including sine type, local type, and global type imperfections. Based on first-order shear deformation beam theory incorporating von Kármán nonlinear strain displacement relations, the governing equation is nonlinear integral–differential equations. An analytical solution for thermal postbuckling of 3D braided beams with and without imperfection obtained as a function of the applied thermal load is employed to determine buckling temperatures and postbuckling equilibrium paths of 3D braided beams. The results reveal that the temperature dependent properties, temperature distribution, geometric parameter, fiber volume fraction, initial geometrical imperfections and braiding angle have a significant effect on thermal postbuckling behavior of braided composite beams.

Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks

Consider the following nonlinear scalar integral differential equation arising from delayed synaptically coupled neuronal networks\begin{eqnarray*}\frac{\partial u}{\partial t}+f(u)&=&\alpha\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y)H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\&+&\beta\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H(u(y,t-\tau)-\Theta){\rm d}y\right]{\rm d}\tau.\end{eqnarray*}This model equation generalizes many important nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks.The kernel functions $K$ and $W$ represent synaptic couplings between neurons in synaptically coupled neuronal networks.The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions),but also lateral inhibitions (modeled with Mexican hat kernel functions) and lateral excitations (modeled with upside down Mexican hat kernel functions).In this nonlinear scalar integral differential equation, $u=u(x,t)$ stands for the membrane potential of a neuron at position $x$ and time $t$.The integrals represent nonlocal spatio-temporal interactions between neurons. &nbsp; &nbsp We have accomplished the existence and stability of three traveling wave fronts $u(x,t)=U_k(x+\mu_kt)$ of the nonlinear scalar integral differential equation in an earlier work [42],where $\mu_k$ denotes the wave speed and $z=x+\mu_kt$ denotes the moving coordinate, $k=1,2,3$.In this paper, we will investigate how the neurobiological mechanisms represented by the synaptic couplings $(K,W)$,by the probability density functions $(\xi,\eta)$, by the synaptic rate constants $(\alpha,\beta)$ and by the firingthresholds $(\theta,\Theta)$ influence the wave speeds $\mu_k$ of the traveling wave fronts.We will define several speed index functions and use rigorous mathematical analysis to investigate the influence of the neurobiological mechanisms on the wave speeds.In particular, we will compare wave speeds of the traveling wave fronts of the nonlinear scalar integral differential equationwith different synaptic couplings and with different probability density functions;we will accomplish new asymptotic behaviors of the wave speeds; we will compare wave speeds of traveling wave fronts ofmany reduced forms of nonlinear scalar integral differential equations of the above model equation;we will establish new estimates of the wave speeds. All these will greatly improve results obtained in previous work [38], [40] and [41].

Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in its first kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers-Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second-order nonlinear equations

Vibration analysis of embedded curved carbon nanotube subjected to a moving harmonic load based on nonlocal theory

In the present study, forced vibration of a simply supported embedded curved single-walled carbon nanotube (ECSWCNT) subjected to a moving harmonic load is investigated based on nonlocal Euler-Bernoulli beam theory. By using a single-mode Galerkin approximation method, the nonlinear integral-differential equation governing the motion of the nanotube is converted into a second-order nonlinear ordinary differential equation. The differential equation of the model is solved using Magnus expansion method which is one of the geometric integration methods. In the numerical calculation, the effects of nonlocal parameter, aspect ratio of ECSWCNT, velocity and the elastic medium constant is discussed. The results show that the above mentioned effects play an important role on the dynamic behavior of ECSWCNT.

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.