Problem For Nonlinear Equation Research Articles

On Global Dynamics in Duffing Equation with Quasiperiodic Perturbation

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

On an iterative method for solution of direct problem for nonlinear hyperbolic differential equations

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

$H_{\infty }$ Sampled-Data Fuzzy Observer Design for Nonlinear Parabolic PDE Systems

This article considers the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> sampled-data fuzzy observer (SDFO) design problem for nonlinear parabolic partial differential equation (PDE) systems under spatially local averaged measurements (SLAMs). Initially, the nonlinear PDE system is accurately represented by the Takagi-Sugeno (T-S) fuzzy PDE model. Then, based on the T-S fuzzy PDE model, an SDFO under SLAMs is constructed for the state estimation. To attenuate the effect of the exogenous disturbance and the design disturbance, an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> SDFO design under SLAMs is developed in terms of linear matrix inequalities by utilizing Lyapunov functional and inequality techniques, which can guarantee the exponential stability and satisfy an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance for the estimation error fuzzy PDE system. Finally, simulation results on the state estimation of the FitzHugh-Nagumo equation are given to support the presented H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> SDFO design method.

Almost-sure scattering for the radial energy-critical nonlinear wave equation in three dimensions

We study the Cauchy problem for the radial energy critical nonlinear wave equation in three dimensions. Our main result proves almost sure scattering for radial initial data below the energy space. In order to preserve the spherical symmetry of the initial data, we construct a radial randomization that is based on annular Fourier multipliers. We then use a refined radial Strichartz estimate to prove probabilistic Strichartz estimates for the random linear evolution. The main new ingredient in the analysis of the nonlinear evolution is an interaction flux estimate between the linear and nonlinear components of the solution. We then control the energy of the nonlinear component by a triple bootstrap argument involving the energy, the Morawetz term, and the interaction flux estimate.

Constructive Existence of Solutions of Multi-point Boundary Value Problem for Hilfer Fractional Differential Equation at Resonance

In this paper, we study the constructive existence of solutions of multi-point boundary value problem for Hilfer fractional differential equation at resonance. The constructive existence of solutions is very important in practice, because it gives approximation method to solve equation as well as the existence. First, we obtain reversibility of some fractional integral operator which is useful for description and get analytic expressions of the inverse operator with Mittag–Leffler function. Next, we have the solution expressions of initial value problem and boundary-value problem for linear Hilfer fractional differential equations. And we get the sufficient conditions for the existence and nonexistence of solutions of multi-point value problem for nonlinear Hilfer fractional differential equation at resonance and establish the approximation method to solve it by the method of upper and lower solutions. Finally, we illustrate our results with an example.

Existence of a Positive Solution for a Boundary Value Problem of some Nonlinear Fractional Differential Equation

In this paper we will study the existence, uniqueness and positivity of solution of a boundary value problem for nonlinear fractional differential equation, by using Leray-Schauder nonlinear alternative, Banach contraction and Guo-Krasnosel’skii fixed point theorems.

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

We study optimal control problems for stochastic nonlinear Schrodinger equations in both the mass subcritical and critical case. For general initial data of the minimal $$L^2$$ regularity, we prove the existence and first order Lagrange condition of an open loop control. In particular, these results apply to the stochastic nonlinear Schrodinger equations with the critical quintic and cubic nonlinearities in dimensions one and two, respectively. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type $$U^2$$ and $$V^2$$ , adapted to evolution operators related to linear Schrodinger equations with lower order perturbations. These estimates yield a new temporal regularity of (backward) stochastic solutions, which is crucial for the tightness of approximating controls induced by Ekeland’s variational principle.

Integrability Properties of Functions with a Given Behavior of Distribution Functions and Some Applications

We establish that if the distribution function of a measurable function v defined on a bounded domain Ω in ℝn (n ≥ 2) satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ k−αϕ(k)/ψ(k), where α > 0, ϕ: [1,+∞) → ℝ is a nonnegative nonincreasing measurable function such that the integral of the function s → ϕ(s)/s over [1,+∞) is finite, and ψ: [0,+∞) → ℝ is a positive continuous function with some additional properties, then |v|αψ(|v|) ∈ L1(Ω). In so doing, the function ψ can be either bounded or unbounded. We give corollaries of the corresponding theorems for some specific ratios of the functions ϕ and ψ. In particular, we consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ Ck−α(ln k)−β with C, α > 0 and β ≥ 0. In this case, we strengthen our previous result for β > 1 and, on the whole, we show how the integrability properties of the function v differ depending on which interval, [0, 1] or (1,+∞), contains β. We also consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ Ck−α(ln ln k)−β with C, α > 0 and β ≥ 0. We give examples showing the accuracy of the obtained results in the corresponding scales of classes close to Lα(Ω). Finally, we give applications of these results to entropy and weak solutions of the Dirichlet problem for second-order nonlinear elliptic equations with right-hand side in some classes close to L1(Ω) and defined by the logarithmic function or its double composition.

Lower Bounds for the Blow up Time to a Coupled Nonlinear Hyperbolic Type Equations

The initial and Dirichlet boundary value problem of nonlinear hyperbolic type equations in a bounded domain is studied. We established a lower bounds for the blow up time.

Existence and multiplicity of positive radial solutions to the Dirichlet problem for nonlinear elliptic equations on annular domains

In this paper, we study the existence and nonexistence of mono- tone positive radial solutions of elliptic boundary value problems on bounded annular domains subject to local boundary condition. By using Krasnoselskii's xed point theorem of cone expansion-compression type we show that there exists > 0 such that the elliptic equation has at least two, one and no radial positive solutions for 0 respectively. We include an example to illustrate our results.

Global existence of solutions of initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

We prove the global existence and blow‐up of solutions of an initial boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.

Existence and Ulam stability results for a class of boundary value problem of neutral pantograph equations with complex order

The research reported in this paper deals with the existence of solutions of boundary value problem (BVP) of nonlinear neutral pantograph equations by means of complex fractional derivative in weighted spaces. The results were interpreted with the aid of classical fixed point theorems. The Ulam–Hyers–Rassias stability and Ulam–Hyers stability of differential equations are studied by utilizing the complex fractional derivative through the fixed point method.

Subsonic solutions to a shock diffraction problem by a convex cornered wedge for the pressure gradient system

We establish the global existence of subsonic solutions to a two dimensional Riemann problem governed by a self-similar pressure gradient system for shock diffraction by a convex cornered wedge. Since the boundary of the subsonic region consists of a transonic shock and a part of a sonic circle, the governing equation becomes a free boundary problem for nonlinear degenerate elliptic equation of second order with a degenerate oblique derivative boundary condition. We also obtain the optimal \begin{document}$ C^{0,1} $\end{document} -regularity of the solutions across the degenerate sonic boundary.

Существование решений квазилинейных эллиптических уравнений в пространствах Музилака-Орлича-Соболева для неограниченных областей

Рассматривается вопрос существования решений задачи Дирихле для нелинейных эллиптических уравнений второго порядка в неограниченных областях. Ограничения на структуру квазилинейных уравнений формулируются в терминах специального класса выпуклых функций - обобщенных $N$-функций. А именно, нелинейности определяются функциями Музилака-Орлича такими, что дополнительные к ним функции подчиняются $\Delta_2$-условию. Соответствующее пространство Музилака-Орлича-Соболева не обязано быть рефлексивным. Именно этот факт является существенной проблемой, поскольку теорема для псевдомонотонных операторов здесь не применима. Для рассматриваемого класса уравнений доказательство теоремы существования проводится на основе абстрактной теоремы для дополнительных систем. Важным инструментом, который позволил обобщить имеющиеся результаты существования решений рассматриваемых уравнений для ограниченных областей на неограниченные области, является теорема вложения пространств Музилака-Орлича-Соболева. Таким образом, в работе найдены условия на структуру квазилинейных уравнений в терминах функций Музилака-Орлича, достаточные для разрешимости задачи Дирихле в неограниченных областях. Кроме того, приведены примеры уравнений, показывающие, что класс нелинейностей, рассматриваемый в работе, шире, чем нестепенные и степенные нелинейности.

Начальная задача для нелинейного интегро-дифференциального уравнения с гиперболическим оператором высшего порядка и с отражением аргумента

In this paper it is studied the questions of one value solvability of initial value problem for nonlinear integro-differential equation with hyperbolic operator of the higher order, with degenerate kernel and reflective argument for regular values of spectral parameter. It is expressed the partial differential operator on the left-hand side of equation of higher order by the superposition of first-order partial differential operators. This is allowed us to present the considering integro-differential equation as an integral equation, describing the change of the unknown function along the characteristic. Further is applied the method of degenerate kernel. In proof of the theorem on one-value solvability of initial value problem is applied the method of successive approximations. Also is proved the stability of this solution with respect to the initial functions.

Local Existence and Uniqueness Theorem for a Nonlinear Schr&amp;amp;#246;dinger Equation with Robin Inhomogeneous Boundary Condition

In recent years, a vast amount of work has been done on initial value problems for important nonlinear evolution equations like the nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV). No comparable attention has been given to mixed initial-boundary value problems for these equations, i.e. forced nonlinear systems. But in many cases of physical interest, the mathematical model leads precisely to the forced problems. For example, the launching of solitary waves in a shallow water channel, the excitation of ion-acoustic solitons in a double plasma machine, etc. In this article, we present the PDE (Partial Differential Equation) method to study the following iut = uxx - g|u|pu, g ∈ R, p > 3, x ∈ Ω = [0,L], 0 ≤ t u (x,0) = u0 (x) ∈ H2 (Ω) and Robin inhomogeneous boundary condition ux (0,t) + αu (0,t) = R1(t), t ≥ 0 and ux (L,t) + αu (L,t) = R2 (t), t ≥ 0 (here α is a real number). The equation is posed in a semi-infinite strip on a finite domain Ω. Such problems are called forced problems and have many applications in other fields like physics and chemistry. The main tool of PDE method is semi-group theory. We are able to prove local existence and uniqueness theorem for the nonlinear Schrödinger equation under initial condition and Robin inhomogeneous boundary condition.

Remarks on nonlinear elastic waves with null conditions

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math.142(2000) 225–250) and T. C. Sideris (Ann. Math.151(2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

Evolutionary Problems of Nonlinear Magnetoelasticity

We consider mixed problems for nonlinear equations of magnetoelasticity. Our main result in the three-dimensional case is the proof of an existence and uniqueness theorem; uniqueness is established under some extra restrictions on the smoothness of solutions. We also manage to prove the existence and uniqueness of a weak solution to the problem in the two-dimensional case; uniqueness is established without any additional a priori assumptions on the smoothness of solutions.

An Enhanced Distance Vector-Hop Algorithm using New Weighted Location Method for Wireless Sensor Networks

Location is an indispensable segment for Wireless Sensor Network (WSN), since when events happened, we need to know location. The distance vector-hop (DV-Hop) technique is a popular range-free localization algorithm due to its cost efficiency and non-intricate process. Nevertheless, it suffers from poor accuracy, and it is highly influenced by network topology; Especially, more hop counts lead to more errors. In the final phase, least squares are employed to address nonlinear equation, which will gain greater location errors. Aimed at addressing problems mentioned above, an enhanced DV-Hop algorithm based on weighted factor, along with new weighted least squares location technique, is proposed in this paper, and it is called WND-DV-Hop. First, the one hop count of unknown node was corrected by employed received signal strength indication (RSSI) technology. Next, in order to reduce average hop distance error, a weighted coefficient based on beacon node hop count was constructed. A new weighted least squares method was embedded to solve nonlinear equation problem. Finally, considerable experiments were carried out to estimate the performance of WND-DV-Hop, compared the outcomes with state-of-the-art DV-Hop, IDV-Hop, Checkout-DV-Hop, and New-DV-Hop depicted in literature. The empirical findings demonstrated that WND-DV-Hop significantly outperformed other localization algorithms.

Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type

In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Caratheodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.