Problem For Nonlinear Equation Research Articles

Unexpected Solutions of the Nehari Problem

The Nehari characteristic numbers λn(a,b) are the minimal values of an integral functional associated with a boundary value problem (BVP) for nonlinear ordinary differential equation. In case of multiple solutions of the BVP, the problem of identifying of minimizers arises. It was observed earlier that for nonoscillatory (positive) solutions of BVP those with asymmetric shape can provide the minimal value to a functional. At the same time, an even solution with regular shape is not a minimizer. We show by constructing the example that the same phenomenon can be observed in the Nehari problem for the fifth characteristic number λn(a,b) which is associated with oscillatory solutions of BVP (namely, with those having exactly four zeros in (a,b)).

Quasilinearization for nonlinear boundary value problems for delay-type difference equations with maxima

The paper deals with an approximate method for solving a mixed boundary value problem for nonlinear difference equations containing a maximum of the unknown function over a past time interval. Every successive approximation to the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with maxima, and an algorithm for its explicit obtaining is suggested. Also, each approximation is a lower/upper solution of the given mixed problem. The rapid convergence of the successive approximations is proved. The suggested algorithm is realized as a computer program and it is applied to an example.MSC: 39A23, 39A99, 65Q10.

Approximation methods in the theory of hybrid differential equations with linear perturbations of second type

In this paper, some existence theorems for the extremal solutions are proved for an initial value problem of nonlinear hybrid differential equations via constructive methods. The monotone iterative techniques for initial value problems of first order hybrid differential equations are developed and it is shown that the sequences of successive iterations defined in a certain way converge to the minimal and maximal solutions of the hybrid differential equations.

Nonlinear interaction of the concentrated waves

The nonlinear problem of the wave propagation is considered. In addition to Kerr nonlinearity the question of the existence of concentrated solutions is analyzed for the threshold and saturable nonlinearity. It is shown that both in the case of threshold nonlinearity, and in the case of saturable nonlinearity solitary waves – concentrated solutions of the corresponding wave equations exist. For the nonlocal nonlinearity, it is taken into account that the diffusion process transforms the interaction of the electromagnetic field with the environment. This phenomenon is described by the system of differential equations including the equation for the perturbation of the dielectric permittivity. The mathematical problem is reduced to the eigenvalue problem for nonlinear integro-differential equation of Hartree type. The computational procedure is constructed.

Direct and inverse problems for nonlinear one-dimensional poroelasticity equations

Direct and inverse problems for nonlinear one-dimensional poroelasticity equations

Turing Computability of the Solution Operator of the Cauchy Problem for Nonlinear Davey-Stewartson Equation

The computability of the solution operator of the Cauchy problem for the nonlinear Davey-Stewartson equation is studied in this paper. Firstly, a nonlinear map KRH5(R2)→C(R;H5(R2)), where H5(R2) is inhomogeneous Sobolev space on R2, defined from the initial value Φ to the solutionu. Then we used the relevant knowledge of type-2 theory of effectivity, functional analysis and Sobolev space to prove that when s>4/7 the solution operator of the Cauchy problem for the nonlinear Davey-Stewartson equation is computable. The conclusion enriches the theories of computability.

Three-point boundary value problems for nonlinear second-order impulsive q-difference equations

The quantum calculus on finite intervals was studied recently by the authors in Adv. Differ. Equ. 2013:282, 2013, where the concepts of q k -derivative and q k -integral of a function f: J k :=[ t k , t k + 1 ]→R have been introduced. In this paper, we prove existence and uniqueness results for nonlinear second-order impulsive q k -difference three-point boundary value problems, by using Banach’s contraction mapping principle and Krasnoselskii’s fixed-point theorem.MSC:26A33, 39A13, 34A37.

Eigenvalue of boundary value problem for nonlinear singular third-order q-difference equations

In this paper, we establish the existence of positive solutions of a boundary value problem for nonlinear singular third-order q-difference equations D q 3 u(t)+λa(t)f(u(t))=0, t∈ I q , u(0)=0, D q u(0)=0, α D q u(1)+β D q 2 u(1)=0, by using Krasnoselskii’s fixed-point theorem on a cone, where λ is a positive parameter. Finally, we give an example to demonstrate the use of the main result of this paper. The conclusions in this paper essentially extend and improve known results.

Application of the symmetry classification to the boundary value problem of nonlinear partial differential equations

In this paper, we study the application of the symmetry classification to the boundary value problem of nonli-near partial differential equations. Firstly, by using differential characteristic set algorithm for the complete symmetry classification of partial differential equations, the complete symmetry classification of a given boundary value problem of nonlinear partial differential equations is proposed. Secondly, by using an extended symmetry, the boundary value problem of nonlinear partial differential equations is reduced to an initial value problem of the original differential equations. Finally, we numerically solve the initial value problem of the original differential equations by using Runge-Kutta method.

The Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equations

We consider the existence of positive solutions for the nonlinear fractional differential equations boundary value problem-D0+αu(t)=f(t,u(t)), 0<t<1, u(0)=u'(0)=u'(1)=0,where2<α≤3is a real number,D0+αis the Riemann-Liouville fractional derivative of orderα, andfis a given continuous function. Our analysis relies on the fixed point index theory in cones.

Impulsive Antiperiodic Boundary Value Problems for Nonlinearqk-Difference Equations

We show the existence and uniqueness of solutions for an antiperiodic boundary value problem of nonlinear impulsiveqk-difference equations by applying some well-known fixed point theorems. An example is presented to illustrate the main results.

On Approximate Controllability of Generalized KdV Solitons

We consider the approximate control of solitons in generalized Korteweg--de Vries equations. By introducing a suitable internal bilinear control on the equation, we prove that any soliton is approximate null controllable, and, moreover, any soliton can be accelerated to any particular positive velocity, after a suitable large amount of time. Precise estimates on the error terms and the rate of decay in the approximate null controllability result are also given. Our method introduces a new insight into the control of nonlinear objects from the point of view of interaction and collision problems for nonlinear dispersive equations, recently developed by Martel and Merle [Ann. Math. (2), 174 (2011), pp. 757--857], [Invent. Math., 183 (2011), pp. 563--648]. It can be applied in principle to several other models with soliton solutions.

Existence of solutions with exponential growth for nonlinear differential-functional parabolic equations

We consider the Cauchy problem for nonlinear parabolic equations with functional dependence. We prove Schauder-type existence results for unbounded solutions. We also prove existence of maximal solutions for a wide class of differential functional equatio

Solutions of Second-Orderm-Point Boundary Value Problems for Impulsive Dynamic Equations on Time Scales

We study a general second-orderm-point boundary value problems for nonlinear singular impulsive dynamic equations on time scalesuΔ∇(t)+a(t)uΔ(t)+b(t)u(t)+q(t)f(t,u(t))=0,t∈(0,1),t≠tk,uΔ(tk+)=uΔ(tk)-Ik(u(tk)), andk=1,2,…,n,u(ρ(0))=0,u(σ(1))=∑i=1m-2αiu(ηi)‍.The existence and uniqueness of positive solutions are established by using the mixed monotone fixed point theorem on cone and Krasnosel’skii fixed point theorem. In this paper, the function items may be singular in its dependent variable. We present examples to illustrate our results.

Optimal control for a sixth order nonlinear parabolic equation

In this paper, we consider the optimal control problem for a sixth order nonlinear parabolic equation, which arising in oil‐water‐surfactant mixtures. Based on Lions' theory, we prove the existence of optimal solution. The optimality system is also established. Copyright © 2013 John Wiley & Sons, Ltd.

平面V形切口塑性应力奇异性分析

A higher-order analysis of the stress singularity of the plane V-notches and cracks in power law hardening materials is studied. First the asymptotic displacement field in terms of radial coordinates at the notch tip is adopted. When the material near the notch tips arises in plastic deformation, the Von Mises yield criterion and the plastic‘total strain’theory are used. By introducing the displacement expressions into the governing differential equations of the plastic theory, it results in a set of the eigenvalue problem of nonlinear ordinary differential equations with the stress singularity orders and the associated eigenfunctions. Then the interpolating matrix method established by the first author is used to solve the eigenvalue problem by an iteration process. The several leading plastic stress singularity orders of plane V-notches and cracks have been obtained. Simultaneously the associated eigenvectors of the displacement and stress fields in the notch tip region have been determined with the same degree of accuracy. Finally two numerical examples have been given to illustrate the accuracy and the effectiveness of the present method to determine the singularity orders of the plastic V-notch and crack. Based on the computed results, some characteristics related to the stress singularity of the plastic notches and cracks are discussed.

The global Cauchy problems for the nonlinear dispersive equations on modulation spaces

In this paper, we discuss decay estimates and Strichartz estimates for dispersive equations with non-homogeneous symbols on modulation spaces Mp,qs to obtain the global well-posedness of the Cauchy problems for nonlinear dispersive equations. As a result, we have a generalization of the result in [19] which treated the Schrödinger equations with a nonlinearity of wider class.

Solutions of the boundary value problem for a 2nth-order nonlinear difference equation containing both advance and retardation

In this paper, we consider the boundary value problem for a 2n th-order nonlinear difference equation containing both advance and retardation. By using the critical point theory, some sufficient conditions of the existence of solutions of the boundary value problem are obtained. The proof is based on the linking theorem. An example is given to illustrate our results.

Existence Results for Singular Boundary Value Problems on Unbounded Domains in Banach Spaces

In this paper, the Krasnosels’kii fixed point theorem in cones for strict set-contractions is used to investigate the existence of single and twin positive solutions for a class of a two-point boundary value problem of second-order nonlinear differential equations posed on an infinite interval. The nonlinearity, which may have a time-singularity, takes values in a general Banach space and has at most polynomial growth with respect to the unknown.

On the new critical exponent for the nonlinear Schrödinger equations

We consider the Cauchy problem for the pth order nonlinear Schrodinger equation in one space dimension $$\left\{\begin{array}{ll}iu_{t} + \frac{1}{2} u_{xx} = |u|^{p}, x \in {\bf R}, \, t > 0, \qquad u(0, x) = u_{0} (x), \; x \in {\bf R},\end{array}\right.$$ where \({p > p_{s} = \frac{3 + \sqrt{17}}{2}}\). We reveal that p = 4 is a new critical exponent with respect to the large time asymptotic behavior of solutions. We prove that if ps 4.