Second-order Nonlinear Equations Research Articles

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

On the Existence of Solutions for Fully Nonlinear Elliptic Equations Under Relaxed Convexity Assumptions

We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like H(v, Dv, D 2 v, x) = 0 in smooth domains without requiring H to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of H at points at which |D 2 v| ≤K, where K is any given constant. For large |D 2 v| some kind of relaxed convexity assumption with respect to D 2 v mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes.

Interval oscillation of a general class of second-order nonlinear impulsive differential equations with nonlinear damping

In this paper, a general class of second-order nonlinear impulsive differential equations with damping is studied. Several interval oscillation criteria which generalize and extend some known ones are established. Moreover, examples are given to illustrate the effectiveness and non-emptiness of our results.

Extended set-membership filter for power system dynamic state estimation

A new method for power system dynamic state estimation is presented. It is based on the application of the extended set-membership filter (ESMF). The ESMF provides an on-line nonlinear guaranteed estimation that assumes that noise sources are unknown but bounded, rather than stochastic as in probabilistic estimation algorithms such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). Thus, the ESMF provides 100% confidence to the estimated states for the safety and reliability of power systems. In this paper, the ESMF is derived and demonstrated by using two different single machine infinite bus (SMIB) systems. The performance of the ESMF is compared with the classical UKF by using one SMIB system modeled as a second-order nonlinear dynamic equation. The feasibility of the ESMF method with unknown inputs is also presented on the other SMIB system, in which the exciter output voltage may be unavailable but bounded. The ESMF method is also tested on a multi-machine system model to demonstrate its effectiveness.

A posteriori error estimates for mixed finite element approximation of nonlinear quadratic optimal control problems

In this paper, we obtain an a posteriori error analysis for mixed finite element approximation of convex optimal control problems governed by a nonlinear second-order elliptic equation. Our results are based on the approximation for both the coupled state variables and the control variable. We propose to improve the error estimates, which can be used to construct an adaptive finite element scheme. A numerical example demonstrating our theoretical results is also presented in this paper.

Poincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains

In [1], I. N. Vekua propose the Poincaré problem for some second order elliptic equations, but it can not be solved. In [2], the authors discussed the boundary value problem for nonlinear elliptic equations of second order in some bounded domains. In this article, the Poincaré boundary value problem for general nonlinear elliptic equations of second order in unbounded multiply connected domains have been completely investigated. We first provide the formulation of the above boundary value problem and corresponding modified well posed-ness. Next we obtain the representation theorem and a priori estimates of solutions for the modified problem. Finally by the above estimates of solutions and the Schauder fixed-point theorem, the solvability results of the above Poincaré problem for the nonlinear elliptic equations of second order can be obtained. The above problem possesses many applications in mechanics and physics and so on.

Boundary Value Problems for Nonlinear Elliptic Equations of Second Order in High Dimensional Domains

This paper mainly concerns oblique derivative problems for nonlinear nondivergent elliptic equations of second order with measurable coefficients in a multiply connected domain. Under certain condition, we derive a priori estimates of solutions. By using these estimates and the fixed-point theorem, we prove the existence of solutions.

AnH1-Galerkin Expanded Mixed Finite Element Approximation of Second-Order Nonlinear Hyperbolic Equations

We investigate anH1-Galerkin expanded mixed finite element approximation of nonlinear second-order hyperbolic equations, which model a wide variety of phenomena that involve wave motion or convective transport process. This method possesses some features such as approximating the unknown scalar, its gradient, and the flux function simultaneously, the finite element space being free of LBB condition, and avoiding the difficulties arising from calculating the inverse of coefficient tensor. The existence and uniqueness of the numerical solution are discussed. Optimal-order error estimates for this method are proved without introducing curl operator. A numerical example is also given to illustrate the theoretical findings.

Numerical methods of Lagrangian points and particles for one-dimensional wave gas-dynamics equations

In this paper, an approach is formulated to construct numerical methods of computational gas dynamics based on the approximation of the second-order nonlinear wave equations (NWE) in time and spatial variables. The NWE approach enables the construction of finite-difference and finite-element schemes with the balance (conservation) cells both in the form of finite volumes and Lagrangian points and particles. That is why the numerical methods based on the NWE approximation are of great interest for the solution of both one-dimensional (1D) and multidimensional problems of gas dynamics. In this work, we construct and study discrete NWE models for 1D problems of gas dynamics in Lagrangian variables and discuss the results of numerical experiments.

Interval Oscillation Criteria for Second-Order Nonlinear Forced Dynamic Equations with Damping on Time Scales

By using the Riccati transformation technique and constructing a class of Philos-type functions on time scales, we establish some new interval oscillation criteria for the second-order damped nonlinear dynamic equations with forced term of the form(r(t)xΔ(t))Δ+p(t)xΔσ(t)+q(t)(xσ(t))α=F(t,xσ(t))on a time scale𝕋which is unbounded, whereαis a quotient of odd positive integer. Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

Kamenev-Type Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales

Using functions from some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order nonlinear dynamic equations on time scales of the form(p(t)ψ(x(t))k∘xΔ(t))Δ+f(t,x(σ(t)))=0. Two examples are included to show the significance of the results.

Positive solutions for systems of second-order integral boundary value problems

We investigate the existence and nonexistence of positive solutions of a system of secondorder nonlinear ordinary dierential equations, subject to integral boundary conditions. 2010 AMS Subject Classication: 34B10, 34B18.

An Ersatz Existence Theorem for Fully Nonlinear Parabolic Equations without Convexity Assumptions

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable “coefficients” and bounded “free” term in the whole space or in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a continuous solution with the first and the second spatial derivatives under control: bounded in the case of the whole space and locally bounded in the case of equations in cylinders. The approximating equation is constructed in such a way that it modifies the original one only for large values of the second spatial derivatives of the unknown function. This is different from a previous work of Hongjie Dong and the author, where the modification was done for large values of the unknown function and its spatial derivatives.

Oscillatory properties of second-order nonlinear neutral dynamic equations with positive and negative coefficients

This paper is to investigate the oscillatory properties of second-order nonlinear neutral dynamic equation with positive and negative coefficients of the form on an arbitrary time scale $\mathbb{T}$ . We establish several oscillation criteria for the equation by using a generalized Riccati technique. The obtained results extend and improve certain existing results in the literature. Examples are shown to illustrate our main results.

Nonlinear run-ups of regular waves on sloping structures

Abstract. For coastal risk mapping, it is extremely important to accurately predict wave run-ups since they influence overtopping calculations; however, nonlinear run-ups of regular waves on sloping structures are still not accurately modeled. We report the development of a high-order numerical model for regular waves based on the second-order nonlinear Boussinesq equations (BEs) derived by Wei et al. (1995). We calculated 160 cases of wave run-ups of nonlinear regular waves over various slope structures. Laboratory experiments were conducted in a wave flume for regular waves propagating over three plane slopes: tan α =1/5, 1/4, and 1/3. The numerical results, laboratory observations, as well as previous datasets were in good agreement. We have also proposed an empirical formula of the relative run-up in terms of two parameters: the Iribarren number ξ and sloping structures tan α. The prediction capability of the proposed formula was tested using previous data covering the range ξ ≤ 3 and 1/5 ≤ tan α ≤ 1/2 and found to be acceptable. Our study serves as a stepping stone to investigate run-up predictions for irregular waves and more complex geometries of coastal structures.

Asymptotics of decreasing solutions of coupled $$p$$ -Laplacian systems in the framework of regular variation

Under the assumption that the coefficients are regularly varying functions, existence and asymptotic form of strongly decreasing solutions are here studied for a system of two coupled nonlinear second-order equations of Emden–Fowler type, satisfying a subhomogeneity condition. Several examples of application of the main result and a comparison with existing literature complete the paper.

Oscillation criteria for second-order nonlinear dynamic equations

This paper concerns the oscillation of solutions to the second-order dynamic equation on a time scale which is unbounded above. No sign conditions are imposed on , , and . The function is assumed to satisfy and for . In addition, there is no need to assume certain restrictive conditions and also the both cases are considered. Our results will improve and extend results in (Baoguo et al. in Can. Math. Bull. 54:580-592, 2011; Bohner et al. in J. Math. Anal. Appl. 301:491-507, 2005; Hassan et al. in Comput. Math. Anal. 59:550-558, 2010; Hassan et al. in J. Differ. Equ. Appl. 17:505-523, 2011) and many known results on nonlinear oscillation. These results have significant importance to the study of oscillation criteria on discrete time scales such as , , , or and the space of harmonic numbers . Some examples illustrating the importance of our results are also included. MSC:34K11, 39A10, 39A99.

Asymptotic behavior of increasing solutions to a system of [formula omitted] nonlinear differential equations

We consider the system xi′=ai(t)|xi+1|αisgnxi+1, i=1,…,n, n≥2, where ai, i=1,…,n, are positive continuous functions on [a,∞), αi∈(0,∞), i=1,…,n, with α1⋯αn<1, and xn+1 means x1. We analyze the asymptotic behavior of the solutions to this system whose components are eventually positive increasing. In particular, we derive exact asymptotic formulas for the extreme case, where all the solution components tend to infinity (the so-called strongly increasing solutions). We offer two approaches: one is based on the asymptotic equivalence theorem, and the other utilizes the theory of regular variation. The above-mentioned system includes, as special cases, two-term nonlinear scalar differential equations of arbitrary order n and systems of n/2 second-order nonlinear equations (provided n is even), which are related to elliptic partial differential systems. Applications to these objects are presented and a comparison with existing results is made. It turns out that some of our results yield new information even in the simplest case, a second-order Emden–Fowler differential equation.

Positive solutions of Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro-differential equation in Banach spaces

In this work, we investigate the existence of positive solutions of Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro differential equation in a real Banach space. Some new existence results of positive solutions are established by applying fixed-point index theory together with comparison theorem. Some discussions and an example are given to demonstrate the applications of our main results.

Hyers-Ulam Stability of Linear and Nonlinear Differential Equations of Second Order

In this paper we established the Hyers-Ulam stability of a nonlinear differential equation of second order with initial condition. We also proved the Hyers -Ulam stability of a linear differential equation of second order with initial condition.