Second-order Nonlinear Equations Research Articles

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.

On the global existence and boundedness of solutions to a nonlinear integro-differential equations of second order

The purpose of this paper is to study the global existence and boundedness of solutions to a kind of nonlinear integro-differential equations of second order by integral inequalities. Some illustrative examples are also provided. Our results complement and improve some ones in the literature.

Mathematical modeling vibration protection system for the motor of the boat

The nonlinear mathematical model of the system to mount the motor in a niche of the outboard motor boat, which protects against external influences by means nonlinear shock absorbers and dampers, has been developed. The mathematical model of the system is represented by three nonlinear differential equations of second order. The nonlinear shock absorbers and dampers, as a function of the generalized coordinates are represented by polynomials up to the fifth degree inclusive. For approximate analytical solution of the nonlinear mathematical model is applied the modified method of polynomial transformations and the numerical method of Runge-Kutta.

On the Evolutionary Fractionalp-Laplacian

In this work, existence results on nonlinear first order as well as doubly nonlinear second-order evolution equations involving the fractional |$p$|-Laplacian are presented. The proofs do not exploit any monotonicity assumption but rely on a compactness argument in combination with regularity of the Galerkin scheme and the nonlocal character of the operator.

Existence results for nonlinear second-order q-difference equations with q-integral boundary conditions

Existence results for nonlinear second-order q-difference equations with q-integral boundary conditions

On a nonlinear hyperbolic equation with a bistable reaction term

Nonlinear second-order hyperbolic equations are gaining ground as models in many areas of application, as extensions of parabolic reaction–diffusion equations that might otherwise be used. The theory of travelling-wave solutions of such reaction–diffusion equations is well established. The present paper is concerned with its counterpart for the wider class of equations in the particular case that the reaction term is bistable. Conditions that are necessary and sufficient for the existence and uniqueness of these solutions are determined. A combination of traditional ordinary differential equation techniques and an innovatory integral equation approach is employed.

Boundary estimates for non-negative solutions to non-linear parabolic equations

This paper is mainly devoted to the boundary behavior of non-negative solutions to the equation $$\begin{aligned} \mathrm{H }u =\partial _tu-\nabla \cdot \mathrm{A }(x,t,\nabla u) = 0 \end{aligned}$$in domains of the form \(\Omega _T=\Omega \times (0,T)\) where \(\Omega \subset \mathbb R^n\) is a bounded non-tangentially accessible (NTA) domain and \(T>0\). The assumptions we impose on \(A\) imply that \(H\) is a non-linear parabolic operator with linear growth. Our main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary. Furthermore, to each such solution one can associate a natural Riesz measure supported on the lateral boundary and one of our main result is a proof of the doubling property for this measure. Our results generalize, to the setting of non-linear equations with linear growth, previous results concerning the boundary behaviour, in Lipschitz cylinders and time-independent NTA-cylinders, established for non-negative solutions to equations of the type \(\partial _tu-\nabla \cdot (\mathrm{A }(x,t)\nabla u)=0\), where \(\mathrm{A }\) is a measurable, bounded and uniformly positive definite matrix-valued function. In the latter case the measure referred to above is essentially the caloric or parabolic measure associated to the operator and related to Green’s function. At the end of the paper we also remark that our arguments are general enough to allow us to generalize parts of our results to general fully non-linear parabolic partial differential equations of second order.

The unified group classification method for the generalized nonlinear wave equation and its partial difference schemes

This paper is concerned with the generalized nonlinear second-order equation and its discrete counterpart furnished with different lattices, respectively. By the unified symmetry analysis method, the complete group classifications of the equations are performed. In the sense of point symmetry, all of the vector fields of the continuous nonlinear equation are obtained. As its special cases, the vector fields of some other important nonlinear equations are provided. Then, we develop the unified group classification method for dealing with partial difference schemes (PΔS), all of the point symmetries of the difference schemes are presented with respect to the given arbitrary analytic function f(u).

Oscillation of Second-Order Nonlinear Neutral Dynamic Equations with Noncanonical Operators

The study of half-linear differential equations has become an important area of research due to the fact that such equations occur in a variety of real-world problems such as in the study of \(p\)-Laplace equations, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium. On the basis of these background details, we study oscillatory behavior of a class of second-order neutral functional dynamic equations on a time scale. New criteria improve and complement related results reported in the literature. Some examples are included to illustrate the results obtained. In particular, an example regarding the second-order neutral differential equation is also provided to show that these theorems improve those in the continuous case.

FINITE ELEMENT MODELING OF NONLINEAR WAVE TRANSFORMATION USING ELLIPTIC MILD SLOPE EQUATION

Accurate modeling of nonlinear wave transformation is important for studies related to harbor and nearshore design. While sophisticated finite-element models based on the elliptic mild-slope equation are often used for wave prediction, they do not include wave-wave interactions. These interactions, in general, involve transfer of energy and wave phase coupling among spectral components and are known to be quite significant especially in shoaling regions and inside harbors. To overcome this limitation, and to provide a basis for the eective modeling of nonlinear wave transformation in complex coastal and harbor environments, the development of a finite-element model based on the second-order nonlinear mild-slope equation of Kaihatu and Kirby (1995) is considered. The model uses an iterative procedure for solving the second-order boundary value problem. To ensure eective boundary treatment, a combination of frequently-used boundary conditions and the method of internal wave generation is used. Two cases involving nonlinear shoaling and harbor resonance are considered for model validation. Modeled results are compared with experimental data, and good agreement is observed in most cases. The methodology described in this paper can improve the applicability of existing finite-element based mild-slope models.

Effect of the longitudinal momentum of electrons on their surfatron acceleration by an electromagnetic wave in space plasma

By numerically calculating the second-order nonlinear time-dependent equation for the wave phase on a particle trajectory, the effect of the longitudinal (with respect to the external magnetic field) momentum of electrons on the dynamics of their surfatron acceleration by an electromagnetic wave propagating across the external magnetic field in space plasma is analyzed. It is shown that, for strongly relativistic initial values of the longitudinal component of the electron momentum (the other parameters of the problem being fixed), the electrons are trapped into the ultrarelativistic regime of surfatron acceleration within a definite interval of the initial wave phase Ψ(0) on the particle trajectory. It was assumed in the calculations that Ψ(0) ≤ π. For the initial wave phases lying within the interval of 0 < Ψ(0) ≤ π, the electrons are immediately trapped by the wave, whereas at π ≤ Ψ(0) ≤ 0, no electron trapping is observed even at long computation times. This result substantially simplifies estimates of the wave damping caused by particle acceleration. The dynamics of the velocity components, momentum, and relativistic factor of electrons in the course of their ultrarelativistic acceleration are considered. The obtained results present interest for the development of modern concepts of the mechanisms for the generation of ultrarelativistic particles in space plasma, correct interpretation of experimental data on the flows of such particles, explanation of possible reasons for the deviation of the fast particle spectra observed in the heliosphere from the standard power-law scaling, and analysis of the relation between such deviations and the space weather.

On the solutions of a nonlinear ‘pseudo’-oscillator equation

The second-order nonlinear equation has been proposed as a simple model to describe the dynamics of electrons in plasma physics. This equation is assumed to have periodic solutions by many authors who argue physical reasons. A great variety of approximate methods have been used in the recent literature in order to detect these periodic solutions. It is the objective of this paper to show that this equation has no periodic solutions whatsoever. In addition, the general solution can be obtained by showing that the equation is equivalent to a planar solvable Hamiltonian system.

Dynamics of a second-order nonlinear equation with a large coefficient of delay control

Dynamics of a second-order nonlinear equation with a large coefficient of delay control

NEW OSCILLATION AND NON-OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS

In this paper we have present new oscillation and non-oscillation criteria for nonlinear differential equations of second order with damping term (r(t)x ' (t)) ' + p(t)x ' (t) + q(t)f(g(x(t))) = 0, t ≥ t0 > 0, (1)

Microlocal approach towards construction of nonreflecting boundary conditions

This paper addresses the problem of construction of non-reflecting boundary condition for certain second-order nonlinear dispersive equations. It is shown that using the concept of microlocality it is possible to relax the requirement of compact support of the initial data. The method is demonstrated for a class of initial data such that outside the computational domain it behaves like a continuous-wave. The generalization is detailed for two existing schemes in the framework of pseudo-differential calculus, namely, Szeftel's method (Szeftel (2006) [1]) and gauge transformation strategy (Antoine et al. (2006) [2]). Efficient numerical implementation is discussed and a comparative performance analysis is presented. The paper also briefly surveys the possibility of extension of the method to higher-dimensional PDEs.

Nonlinear second-order impulsive q-difference Langevin equation with boundary conditions

In this paper, we discuss the existence and uniqueness of solutions for Langevinimpulsive q-difference equations with boundary conditions. Our studyrelies on Banach’s and Schaefer’s fixed point theorems. Illustrativeexamples are also presented.MSC: 26A33, 39A13, 34A37.

Periodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations

We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of Hamilton–Jacobi, “viscous” Hamilton–Jacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate “periodizations” of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization.

Paul Painlevé and his contribution to science

The life and career of the great French mathematician and politician Paul Painleve is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painleve and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painleve transcendents. The contribution of Paul Painleve to the study of algebraic nonintegrability of the N-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.

Oscillation criteria for second-order quasi-linear delay dynamic equations on time scales

This paper is concerned with oscillations of the second-order delay nonlinear dynamic equation on a time scale , where a and q are real-valued rd-continuous positive functions on , α and β are ratios of odd positive integers, , , , and . We establish some new sufficient conditions for this equation. MSC:34K11, 39A10, 39A99.