Second-order Nonlinear Equations Research Articles

Analysis of the dependence of surfatron acceleration of electrons by an electromagnetic wave in space plasma on the particle momentum along the wave front

Based on the numerical solution of the nonlinear nonstationary second-order equation for the wave phase on the particle trajectory, the dynamics of surfatron acceleration of electrons by an electromagnetic wave propagating across the external magnetic field in space plasma is analyzed as a function of the electron momentum along the wave front. Numerical calculations show that, for strongly relativistic initial values of the electron momentum component along the wave front gy(0) (the other parameters of the problem being the same), electrons are trapped into the regime of ultrarelativistic surfatron acceleration within a certain interval of the initial wave phase Ψ(0) on the particle trajectory. It is assumed in the calculations that |Ψ(0)| ≤ π. For strongly relativistic values of gy(0), electrons are immediately trapped by the wave for 19% of the initial values of the phase Ψ(0) (favorable phases). For the rest of the values of Ψ(0), trapping does not occur even at long times. This circumstance substantially simplifies estimations of the wave damping due to particle acceleration in subsequent calculations. The dynamics of the relativistic factor and the components of the electron velocity and momentum under surfatron acceleration is also analyzed. The obtained results are of interest for the development of modern concepts of possible mechanisms of generation of ultrarelativistic particle fluxes in relatively calm space plasma, as well as for correct interpretation of observational data on the fluxes of such particles and explanation of possible reasons for the deviation of ultrarelativistic particle spectra detected in the heliosphere from the standard power-law scalings and the relation of these variations to space weather and large-scale atmospheric processes similar to tropical cyclones.

On the Oscillation of Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

Based on Riccati transformation and the inequality technique, we establish some new sufficient conditions for oscillation of the second-order neutral delay dynamic equations on time scales. Our results not only extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales. At the end of this paper, we give an example to illustrate the main results.

A note on "oscillation criteria for second-order nonlinear neutral dynamic equations on time scales

In this work, we give a counter example for the main result of the paper \small[Tamkang J. of Math., 43 (1)(2012), 109--122.] and we give the correct formula for some theorems given in their work.

Nonlinear parabolic equations with nonstandard growth

In this paper, we study the solvability of the initial-boundary value problem for second-order nonlinear parabolic equations with nonstandard growth conditions and -source terms. In the model case, these equations include the p-Laplacian with a variable exponent p(x, t). We prove that if the variable exponent p is bounded away from both 1 and and is log-Hölder continuous, then the problem has a weak solution which satisfies the energy equality.

Symmetries and reductions of order for certain nonlinear third- and second-order differential equations with arbitrary nonlinearity

We examine the reductions of the order of certain third- and second-order nonlinear equations with arbitrary nonlinearity through their symmetries and some appropriate transformations. We use the folding transformation which enables one to change from a nonlinearity with an arbitrary exponent to a nonlinearity with a specific numerical exponent.

On the asymptotic behaviors of solutions to nonlinear differential equations of second order

We study the asymptotic behavior of solutions to a nonlinear differential equation of the second order whose coefficient of nonlinearity is a bounded function for arbitrarily large values of \(x\) in \(R\). We obtain certain sufficient conditions which guarantee boundedness of solutions, their convergence to zero as \(x\rightarrow \infty\) and their unboundedness.

Oscillation results for nonlinear second-order damped dynamic equations

Oscillation results for nonlinear second-order damped dynamic equations

Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients

Purpose – The purpose of this paper is to present the computational modeling of second-order two-dimensional nonlinear hyperbolic equations by using cosine expansion-based differential quadrature method (CDQM). Design/methodology/approach – The CDQM reduced the equations into a system of second-order differential equations. The obtained system is solved by RK4 method by converting into a system of first ordinary differential equations. Findings – The computed numerical results are compared with the results presented by other workers (Mohanty et al., 1996; Mohanty, 2004) and it is found that the present numerical technique gives better results than the others. Second, the proposed algorithm gives good accuracy by using very less grid point and less computation cost as comparison to other numerical methods such as finite difference methods, finite elements methods, etc. Originality/value – The author extends CDQM proposed in (Korkmaz and Dağ, 2009b) for two-dimensional nonlinear hyperbolic partial differential equations. This work is new for two-dimensional nonlinear hyperbolic partial differential equations.

Maximum Principles for Second-Order Impulsive Integro-Differential Equations with Integral Jump Conditions

We are concerned with two new maximum principles for second-order impulsive integro-differential equations with integral jump conditions. The impulse effects or jump conditions of this paper are involved in terms of integral of past states. As an application, we introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear second-order impulsive functional integro-differential equation with integral jump conditions.

A sharp ultimate bound of the solutions to some second-order evolution equations with nonlinear damping

We establish a sharp ultimate bound of the solution to a second-order dissipative nonlinear equation with bounded forcing term.

New criteria for oscillation of certain second-order nonlinear dynamic equations on time scales

New criteria for oscillation of certain second-order nonlinear dynamic equations on time scales

Critical edge behavior in the modified Jacobi ensemble and Painlevé equations

We study the Jacobi unitary ensemble perturbed by an algebraic singularity at t > 1. For fixed t, this is the modified Jacobi ensemble studied by Kuijlaars et al. The main focus here, however, is the case when the algebraic singularity approaches the hard edge, namely t → 1+.In the double scaling limit case when t − 1 is of the order of magnitude of 1/n2, n being the size of the matrix, the eigenvalue correlation kernel is shown to have a new limiting kernel at the hard edge 1, described by the ψ-functions for a certain second-order nonlinear equation. The equation is related to the Painlevé III equation by a Möbius transformation. It also furnishes a generalization of the Painlevé V equation, and can be reduced to a particular Painlevé V equation via the Bäcklund transformations in special cases. The transitions of the limiting kernel to Bessel kernels are also investigated, with n2(t − 1) being large or small.In the present paper, the approach is based on the Deift–Zhou nonlinear steepest descent analysis for Riemann–Hilbert problems.

Построение приближенного решения одного нелинейного дифференциального уравнения второго порядка в окрестности подвижной особой точки

Построение приближенного решения одного нелинейного дифференциального уравнения второго порядка в окрестности подвижной особой точки

Oscillation Criteria for Functional Nonlinear Dynamic Equations with $${\gamma}$$ γ -Laplacian, Damping and Nonlinearities Given by Riemann–Stieltjes Integrals

In this paper, we present oscillation criteria for the second-order nonlinear dynamic equation \({[a(t)\phi_{\gamma} (x^{\Delta}(t))]^{\Delta} + p(t)\phi_{\gamma}(x^{\Delta^{\sigma}}(t)) + q_{0}(t) \phi_{\gamma}(x(g_{0}(t)))+\sum_{i=1}^{2}\int_{a_{i}}^{b_{i}}q_{i}(t,s)\phi_{\alpha_{i}(s)}(x(g_{i}(t,s))) \Delta \zeta_{i}(s)=0}\) on a time scale \({\mathbb{T}}\) which is unbounded above. Our results generalize and improve some known results for oscillation of second-order nonlinear dynamic equation. Some examples are given to illustrate the main results.

Online fault diagnosis for nonlinear power systems

This paper considers the problem of automatic fault diagnosis for transmission lines in large scale power networks. Since faults in transmission lines threatens stability of the entire power network, fast and reliable fault diagnosis is an important problem in transmission line protection. This work is the first paper exploiting sparse signal recovery for the fault-diagnosis problem in power networks with nonlinear swing-type dynamics. It presents a novel and scalable technique to detect, isolate and identify transmission faults using a relatively small number of observations by exploiting the sparse nature of the faults. Buses in power networks are typically described by second-order nonlinear swing equations. Based on this description, the problem of fault diagnosis for transmission lines is formulated as a compressive sensing or sparse signal recovery problem, which is then solved using a sparse Bayesian formulation. An iterative reweighted ℓ1-minimisation algorithm based on the sparse Bayesian learning update is then derived to solve the fault diagnosis problem efficiently. With the proposed framework, a real-time fault monitoring scheme can be built using only measurements of phase angles at the buses.

Stabilisation of second-order nonlinear equations with variable delay

For a wide class of second-order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilisation by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.

Kamenev-type criteria for nonlinear damped dynamic equations

We establish a new Kamenev-type theorem for a class of second-order nonlinear damped delay dynamic equations on a time scale by using the generalized Riccati transformation technique. The criterion obtained improves related contributions to the subject. An example is provided to illustrate assumptions in our theorem are less restrictive.

Group classification of nonlinear evolution equations: Semi-simple groups of contact transformations

We generalize and modify the group classification approach of Zhdanov and Lahno (1999) to make it applicable beyond Lie point symmetries. This approach enables obtaining exhaustive classification of second-order nonlinear evolution equations in one spatial dimension invariant under semi-simple groups of contact transformations. What is more, all inequivalent second-order nonlinear evolution equations which admit semi-simple groups or groups having nontrivial Levi decompositions are constructed in explicit forms.

Nonlocal problem for a general second-order elliptic operator

The paper deals with a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the strong problem essentially increases its complexity and the necessary total computational work. The existence and uniqueness of the weak solution is established. The nonlinear weak formulation is reduced to the minimization of a nonlinear functional. Finite element discretizations by Lagrangian finite elements are applied to obtain an approximate minimization problem. A two-point step size gradient method with an original steplength is used for finding approximate solutions of the problem under consideration. No line search is necessary in the new approach. The present method is computer implemented and tested on different triangulations. The test examples indicate that the method slightly depends on the initial guesses.

Oscillation of second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales

This paper concerns second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales of the form $$\bigl(r(t) \bigl(\bigl(y(t)+p(t)y\bigl(\tau(t)\bigr)\bigr)^{\Delta}\bigr)^{\gamma}\bigr)^{\Delta}+\int_{a}^{b}f \bigl(t,y\bigl(\delta (t,\xi)\bigr)\bigr)\Delta\xi=0, $$ where $\gamma>0$ is a quotient of odd positive integers. By using the generalized Riccati technique and integral averaging techniques, we derive new oscillation criteria for the above equations, which generalize and improve some existing results in the literature.