Second-order Nonlinear Differential Equations Research Articles

MODIFIED OSCILLATION CRITERIA OF SECOND-ORDER NEUTRAL DIFFERENCE EQUATIONS

In this paper we shall consider a class of second-order nonlinear difference equations with a negative neutral term. Some new oscillation criteria are obtained via Riccati transformation technique. These criteria improve and modify the existing results mentioned in the literature. Some examples are given to show the applicability and significance of the main results.

Plasma Oscillations at Cathode Spot Region of Vacuum Arc

The model of ionization processes in high-temperature plasma of the near-cathode region of vacuum arc is given. The model is mathematically represented as a system of second-order nonlinear differential equations of drift diffusion for the electron density and mean energy of electrons. The solution of the presented system of equations was derived using the finite element method according to two coordinates in numerical form using the COMSOL Multiphysics program. It is shown that Langmuir waves, which result in a multiple-step ionization of metallic vapor, are generated in plasma owing to nonstationary processes.

Steady-state heat transfer analysis in a spherical domain revisited

The paper discusses the numerical solution of the one-dimensional radially axi-symmetric non-linear second-order differential equation to model the conduction and radiation transfer through a spherical domain as a result of an exothermic heat source. The equation is transformed to a non-dimensional form. The dimensionless numbers emanating from the transformation represent the effect of the reaction rate, reaction type, activation energy, radiation and the convection on the temperature. The non-dimensional differential equation for the temperature distribution was previously solved using the Runge-Kutta-Fehlberg method coupled with a Shooting technique. In this paper the solution of the non-dimensional differential equation using an iterative Galerkin finite element method approach employing the Picard method is described. The commercial finite element code Comsol is also employed to solve the non-dimensional differential equation. The current study was motivated by inconsistencies that were observed in the previous results that were presented. Although the assumed underlying physics is used to evaluate the results, the study focuses purely on the numerical solution of the non-dimensional differential equation. The results obtained by the Galerkin finite element code and Comsol were found to be in exact agreement and also exhibit no inconsistencies.

Diffusion Model for Forecasting Events in News Feeds

On the basis of the diffusion theory, we suggested a model for forecasting event in news feeds, which is based on the use of stochastic dynamics of changes in the structure of non-stationary time series in news text clusters (states of the information space). Forecasting events in a news feed is based on their text description, vectorization, and finding the cosine value of the angle between the given vector and the centroids of various information space semantic clusters. Changes over time in the cosine value of such angle between the above vector and centroids can be represented as a point wandering on [0,1] segment. This segment contains a trap at the event occurrence threshold point. The wandering point can fall into this trap over time. We have considered probability patterns of transitions between states in the information space. We have derived a nonlinear second-order differential equation; formulated and solved the boundary value problem of forecasting news events. We have obtained theoretical time dependence for the probability density function of the parameter distribution of non-stationary time series that describe the information space evolution. The results of simulating the time dependence of the event probability (with sets of parameter values of the developed model, which have been experimentally determined for already occurred events) show that the model is consistent and adequate. Experimental verification of the proposed model was carried out using a corpus of texts written in Russian.

A Class of Optimal Liquidation Problem with a Nonlinear Temporary Market Impact

We extend the self-exciting model by assuming that the temporary market impact is nonlinear and the coefficient of the temporary market impact is an exponential function. Through optimal control method, the optimal strategy satisfies the second-order nonlinear ordinary differential equation. The specific form of the optimal strategy is given, and the decreasing property of the optimal strategy is proved. A numerical example is given to illustrate the financial implications of the model parameter changes. We find that the optimal strategy of a risk-neutral investor changes with time and investment environment.

The Existence of Positive Solutions of Singular Initial-Value Problem for Second Order Differential Equations-2

We consider the singular initial value problem for the second order differential equation. We interested in the existence of positive solutions and proved an easily applicable theorem on the existence of positive solutions to initial-value problems for second-order nonlinear singular differential equations. The previously established result on the existence of positive solution by Agarwal and O'Regan is not easy for the applications due to very complex definition of a new function and their properties in the statement of their theorem. We used the Lebesgue dominated convergence theorem and the Schauder--Tychonoff theorem in the proof of the main result. The main result can be easily applied for the singular and regular type of problems.

A nonlinear solver based on an adaptive neural network, introduction and application to porous media flow

Sensitivity to derivatives and a need for proper initial guesses are the main disadvantages of classic nonlinear solvers like Newton's method. To overcome the obstacles, a numerical solver for second-order nonlinear Partial Differential Equations (PDEs) based on an Adaptive Neural Network (AdNN) is introduced. While using Newton's method needs to form the Jacobian matrix and its inversion, which both of them are time-consuming and dependent on the nature of the problem, the proposed solver tries to find the roots of the algebraic equations by changing the weights of AdNN by the help adaptive laws. The proposed approach has been applied to solve the governing PDEs of the gas flow in shale resources and the immiscible two-phase flow of water and oil in hydrocarbon reservoirs as two highly nonlinear phenomena. The generated profiles of pressures and saturations show a satisfying match with the outputs of Newton's method. However, using the presented algorithm not only removes the former necessities but also helps the community to solve the relevant PDEs governing the critical elements of the energy market in the future with a higher level of confidence.

Multiple positive solutions of second-order nonlinear difference equations with discrete singular \u03d5-Laplacian

We consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation {−∇[ϕ(△u(t))]=λa(t,u(t))+μb(t,u(t)),t∈T,u(1)=u(N)=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}$$ \\textstyle\\begin{cases} -\\nabla [\\phi (\\triangle u(t))]=\\lambda a(t,u(t))+\\mu b(t,u(t)), \\quad t\\in \\mathbb{T}, \\\\ u(1)=u(N)=0, \\end{cases} $$\\end{document} where lambda ,mu geq 0, mathbb{T}={2,ldots ,N-1} with N>3, phi (s)=s/sqrt{1-s^{2}}. The function f:=lambda a(t,s)+mu b(t,s) is either sublinear, or superlinear, or sub-superlinear near s=0. Applying the topological method, we prove the existence of either one or two, or three positive solutions.

Analysis of the start-up process of the tower crane slewing mechanism with a steady state motion mode of its load trolley

provided that the load trolley moves at a constant velocity. On the basis of the second-order Lagrange equations, a corresponding mathematical model is obtained, which also describes the oscillations of the load on a flexible suspension in tangential and radial directions. The driving torque applied to the system is modeled using the Kloss equation. The obtained mathematical model is represented by a system of four second-order nonlinear differential equations, so numerical methods are used to integrate it. To assess the level of dynamic and energy loads in the system elements, we propose a set of indicators that reflect the maximum and root-mean-square values. It is suggested to consider the evolution of the system in two cases: the position of the trolley near the tower (the trolley moves from it) and the position of the trolley near the end of the boom (the trolley moves towards the tower). For both cases, the values of the estimation parameters were calculated, which together with the corresponding graphical dependencies allowed to identify the most significant factors that have an impact on the energetic, dynamic and kinematic processes of the system. In particular, these include: centrifugal force, Coriolis force, damping ability of the asynchronous electric drive of the crane slewing mechanism. The analysis of the load oscillation on a flexible suspension, which was carried out on the basis of phase portraits in the plane of the trolley movement and perpendicular to it, revealed their dependence on the initial position of the trolley on the boom. Apart from this, the drive power consumption, a part of which is spent on overcoming the centrifugal force which acts on the trolley and the load, significantly depends on this factor.

Asymptotic Problem for Second-Order Ordinary Differential Equation with Nonlinearity Corresponding to a Butterfly Catastrophe

For the second-order nonlinear ordinary differential equation $$ {u}_{xx}^{\hbox{'}\hbox{'}}={u}^5-{tu}^3-x, $$ we prove the existence and uniqueness of a strictly increasing solution, which satisfies an initial condition and a limit condition at infinity and whose graph lies between the zero equation and the continuous graph of the root of the nondifferential equation u5 − tu3 − x = 0. For this solution, we find an asymptotics, which is uniform on the ray t ∈ (−∞,−Mt) as x → +∞; separately, we construct asymptotics on the ray s > Ms and on the segment 0 ≤ s ≤ Ms, where s = |t|−5/2x is the variable compressed with respect to x. Using the method of matching of asymptotic expansions, we construct a composite asymptotic expansion of the solution to the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem. Finally, we construct a uniform asymptotic expansion under the restriction t ≤ 0 as x2 + t2 →∞.

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integro-differential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton’s principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with Galerkin’s method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the anti-symmetric modes being excited. Bifurcations including the saddle-node, Hopf, period-doubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.

On the Asymptotic Behavior of a Class of Second-Order Non-Linear Neutral Differential Equations with Multiple Delays

In this work, we present some new sufficient conditions for the oscillation of a class of second-order neutral delay differential equation. Our oscillation results, complement, simplify and improve recent results on oscillation theory of this type of non-linear neutral differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

Approximations for the inverse cumulative distribution function of the gamma distribution used in wireless communication

The use of quantile functions of probability distributions whose cumulative distribution is intractable is often limited in Monte Carlo simulation, modeling, and random number generation. Gamma distribution is one of such distributions, and that has placed limitations on the use of gamma distribution in modeling fading channels and systems described by the gamma distribution. This is due to the inability to find a suitable closed-form expression for the inverse cumulative distribution function, commonly known as the quantile function (QF). This paper adopted the Quantile mechanics approach to transform the probability density function of the gamma distribution to second-order nonlinear ordinary differential equations (ODEs) whose solution leads to quantile approximation. Closed-form expressions, although complex of the QF, were obtained from the solution of the ODEs for degrees of freedom from one to five. The cases where the degree of freedom is not an integer were obtained, which yielded values closed to the R software values via Monte Carlo simulation. This paper provides an alternative for simulating gamma random variables when the degree of freedom is not an integer. The results obtained are fast, computationally efficient and compare favorably with the machine (R software) values using absolute error and Kullback–Leibler divergence as performance metrics.

Two-grid IPDG discretization scheme for nonlinear elliptic PDEs

Two-grid interior penalty discontinuous Galerkin (IPDG) method for the mildly nonlinear second-order elliptic partial differential equations is studied in this paper. The IPDG finite element discretizations are developed and the corresponding well-posedness is established by introducing the equivalent weak formulation of IPDG method and combining Brouwer’s fixed point theorem. Some priori error estimates for discrete solution in the broken H1-norm, L2-norm and L∞-norm are derived, respectively. Two-grid method is designed for solving IPDG discretization scheme and the corresponding error estimate is provided. Numerical experiments are also shown to confirm the efficiency of the proposed approach.

On the Impact of Fault Ride-Through on Transient Stability of Autonomous Microgrids: Nonlinear Analysis and Solution

Fault ride-through (FRT) is essential for inverter-interfaced distributed generation (IIDG) units to protect semiconductor switches from being imposed to overcurrent conditions while the transients are securely passed. To this end, a current limiting strategy is adopted for IIDG units, mostly embedded in control loops, to limit the current within the withstand-able band and to make the IIDG units stay connected to the (micro) grid during the transient. However, the FRT/current limiting of grid-forming inverters affects the transient stability of the autonomous droop-based microgrids and may make them unstable, which yet has not been well-explored in the literature. This issue is considered in this work through a scrupulous observation of the second-order nonlinear differential equation describing the frequency-phase angle dynamics and investigating the problem through the Lyapunov theory. The Chetaev's instability theorem, which is developed based on the Lyapunov direct method, is used to explore the instability conditions due to the FRT. It is revealed that the phase angle variation, as a consequence of the current limiting, and the arbitrary/resistive transient impedance of grid-forming inverters make the system unstable. Numerical and time-domain results through MATLAB/Simulink platform prove the validity of the models.

Painlevé V and the Hankel determinant for a singularly perturbed Jacobi weight

We study the Hankel determinant generated by a singularly perturbed Jacobi weightw(x,t):=(1−x2)αe−tx2,x∈[−1,1],α>0,t≥0. If t=0, it is reduced to the classical symmetric Jacobi weight. For t>0, the factor e−tx2 induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities.In the finite n dimensional case, we obtain two auxiliary quantities Rn(t) and rn(t) by using the ladder operator approach. We show that the Hankel determinant has an integral representation in terms of Rn(t), where Rn(t) is closely related to a particular Painlevé V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant. This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto σ-function of a particular Painlevé V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. n→∞ and t→0 such that s=2n2t is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large s and small s asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.

Harmonic Linearization Method for Computing the Motions of a Specific Type of Three-Joint Arms

The paper presents a three-joint arm that has two translational fifth-class kinematic pairs and a rotational one. The first two kinematic pairs are axially orthogonal, while the second and the third one are axially parallel. Such robots find use in machinery handling for arc welding. Systems of three nonlinear second-order differential equations were obtained to describe the motions of this arm. As it was necessary to assess the moment of load emerging on the control-drive shaft, the external generalized forces were reduced to a corresponding generalized coordinate. Harmonic linearization of nonlinearities was applied to analyze the arm system. This method can be used to fine the core properties of the system (frequency, amplitude, and phase of the oscillations; dependence of the form of nonlinearities or on the parameters of the linear part; etc.) without studying the transient.Ratios were obtained to find the parameters of natural oscillational motion occurring in such arms. These rations can be used to design and set up such an arm. The finding is that where the system of equations describing the motion of such an arm (including the drive motion equations) contains no prominent nonlinearities, these rations are a system of two algebraic frequency equations: a sixth-degree one and a fifth-degree one.

Solving second-order nonlinear evolution partial differential equations using deep learning**The project is supported by the National Natural Science Foundation of China (No. 11675054), Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

Solving nonlinear evolution partial differential equations has been a longstanding computational challenge. In this paper, we present a universal paradigm of learning the system and extracting patterns from data generated from experiments. Specifically, this framework approximates the latent solution with a deep neural network, which is trained with the constraint of underlying physical laws usually expressed by some equations. In particular, we test the effectiveness of the approach for the Burgers’ equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions. The results also indicate that for soliton solutions, the model training costs significantly less time than other initial conditions.

Instanton solutions in a model of AdS4/CFT3 correspondence

From the wrapping of the (anti)membranes of 11-dimensional supergravity over , on the internal directions along with an ansatz for its 4-form flux, by solving the original equations and identities, we arrive at scalar differential equations in the Euclidean space; note that the associated bulk solutions and setups break all supersymmeties, parity and scale invariance; the resulting (pseudo) scalar potential, which is Higgs-like with two nearly homogeneous vacua, provides the first-order phase transition and tunneling from the false- to true- vacuum. Here, concentrating on the three (pseudo) scalar modes m2=-2, 4, 10 , which are, in turn, realizable in Wick-rotated and skew-whiffed M2-branes backgrounds, we employ approximate methods and, particularly, Adomian decomposition method to solve the nonlinear second-order partial differential equations, valid in the probe approximation, with the Dirichlet boundary condition or the initial data from a basic exact solution, to get solutions in series expansions near the boundary in different orders of perturbation. Next, making use of the AdS4/CFT3 correspondence rules, after swapping the three fundamental representations of for gravitino, we build the dual singlet operators from the (scalar, fermion and gauge) fields in a 3-dimensional Chern-Simons-matter gauge field theory living on the resultant anti-M2-brnaes; after that, by deforming the corresponding boundary actions with the operators, we get invariant solutions with nonzero finite actions, which ,in turn, are small instantons sitting at the origin of a 3-sphere at infinity, causing instability and mediating false vacuum decay. In other words, the boundary potentials unbounded from below are duals for the collapse of the bulk (thin-wall) vacuum bubbles and big crunch singularities.

О численных методах построения эталонных решений для нелинейного уравнения теплопроводности с особенностью

The paper deals with the construction of exact solutions to a nonlinear heat equation with degeneration in the case of the zero value of the required function. Generically self-similar solutions and traveling wave solutions are considered, the construction of which reduces to solving Cauchy problems for a nonlinear second-order ordinary differential equation with a singularity before the higher derivative. Two approaches are proposed to solve the Cauchy problems: the analytical solution by the power series method and the numerical solution by the boundary element method on a specified segment. A complex computational experiment is carried out to compare the above two methods with each other and with the finite difference methods, namely the Euler method and the fourth-order Runge-Kutta method. Power series segments are used on the first step of the finite difference solutions in order to resolve the singularity. The comparison of the application domains, the accuracy of the solutions and their dependence on the parameters of a certain problem shows that the boundary element method is the most universal, although not the most accurate for some particular examples. The conclusions drawn allow us to construct benchmark solutions to verify the approximate solutions of the nonlinear heat equation by various methods in a wide range of parameter values.