Fourth-order Beam Equation Research Articles

Existence of Solutions to Nonlinear Fourth-Order Beam Equation

This paper studies the boundary value problem for a fourth-order difference equation with three quasidifferences. The new existence criterion of at least one solution to the issues considered is obtained using the theory of variational methods. The main result is illustrated in some examples.

On the nonlinear system of fourth-order beam equations with integral boundary conditions

<abstract><p>The purpose of this paper is to establish an existence theorem for a system of nonlinear fourth-order differential equations with two parameters</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{rcl} u^{(4)}+A(x)u& = &\lambda f(x, u, v, u'', v''), \ 0<x<1, \\ v^{(4)}+B(x)v& = &\mu g(x, u, v, u'', v''), \ 0<x<1 \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>subject to the coupled integral boundary conditions:</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{rcl} u(0) = u'(1) = u'''(1) = 0, \ u''(0)& = & \int_{0}^{1}p(x)v''(x)dx, \\ v(0) = v'(1) = v'''(1) = 0, \ v''(0)& = & \int_{0}^{1}q(x)u''(x)dx, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ A, \ B \in C[0, 1], $ $ p, q\in L^{1}[0, 1], $ $ \lambda > 0, \mu > 0 $ are two parameters and $ f, g: [0, 1]\times[0, \infty)\times[0, \infty)\times(-\infty, 0)\times(-\infty, 0) \rightarrow \mathbb{R} $ are two continuous functions satisfy the growth conditions.</p></abstract>

Noether symmetries and exact solutions of an Euler–Bernoulli beam model

In this paper, a Noether symmetry analysis is carried out for an Euler–Bernoulli beam equation via the standard Lagrangian of its reduced scalar second-order equation which arises from the standard Lagrangian of the fourth-order beam equation via its Noether integrals. The Noether symmetries corresponding to the reduced equation is shown to be the inherited Noether symmetries of the standard Lagrangian of the beam equation. The corresponding Noether integrals of the reduced Euler–Lagrange equations are deduced which remarkably allows for three families of new exact solutions of the static beam equation. These are shown to contain all the previous solutions obtained from the standard Lie analysis and more.

Strong continuity of the Lidstone eigenvalues of the Beam equation in potentials

In this paper we study the dependence of the Lidstone eigenvalues λm(q), m ∈ N, of the fourth-order beam equation on potentials q ∈ L p (0,1) ,1 p ∞ .T hefirst result is that λm(q) have a strongly continuous dependence on potentials, i.e., as nonlinear functionals, λm(q) are continuous in q ∈ L p (0,1) when the weak topology is considered. The second result is that λm(q) are continuously Frechet differentiable in potentials q ∈ L p (0,1) when the L p norm is considered. These results will be used in studying the optimal estimations for these eigenvalues in later works.

Infinitely Many Sign-Changing Solutions for Some Nonlinear Fourth-Order Beam Equations

Several new existence theorems on positive, negative, and sign-changing solutions for the following fourth-order beam equation are obtained:u(4)=f(t,u(t)), t∈[0,1]; u(0)=u(1)=u′′(0)=u′′(1)=0, wheref∈C([0,1]×ℝ1,ℝ1). In particular, an infinitely many sign changing solution theorem is established. The method of the invariant set of decreasing flow is employed to discuss this problem.

Linear stability analysis for travelling waves of second order in time PDE's

We study travelling waves φc of second order in time PDE's . The linear stability analysis for these models is reduced to the question of the stability of quadratic pencils in the form , where .If is a self-adjoint operator, with a simple negative eigenvalue and a simple eigenvalue at zero, then we completely characterize the linear stability of φc. More precisely, we introduce an explicitly computable index , so that the wave φc is stable if and only if . The results are applicable both in the periodic case and in the whole line case.The method of proof involves a delicate analysis of a function , associated with , whose positive zeros are exactly the positive (unstable) eigenvalues of the pencil . We would like to emphasize that the function is not the Evans function for the problem, but rather a new object that we define herein, which fits the situation rather well.As an application, we consider three classical models—the ‘good’ Boussinesq equation, the Klein–Gordon–Zakharov (KGZ) system and the fourth order beam equation. In the whole line case, for the Boussinesq case and the KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable travelling waves (and for all powers of p in the case of Boussinesq). This result is new for the KGZ system, while it generalizes the results of Alexander et al (2012, personal communication) and Alexander and Sachs (1995 Nonlinear World 2 471–507), which apply to the case p = 2. For the beam equation, we provide an implicit formula (depending only on the function , which works for all p and for both the periodic and the whole line cases.Our results complement (and exactly match, whenever they exist) the results of a long line of investigation regarding the related notion of orbital stability of the same waves. Informally, we have found that in all the examples that we have looked at, our theory applies, whenever the Grillakis–Shatah–Strauss (GSS) theory applies. We believe that the results in this paper (or a variation thereof) will enable the linear stability analysis as well as asymptotic stability analysis for most models in the form .

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω ⊂ ℝn be a bounded domain, H = L2(Ω), L: D(L) ⊂ H → H be an unbounded linear operator, f ∈ C(\(\bar \Omega\) × ℝ, ℝ) and λ ∈ ℝ. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem $$Lu = \lambda f\left( {x,u} \right), u \in D\left( L \right),$$ which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a secondorder ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.

Existence and Uniqueness of Positive Solutions for Discrete Fourth-Order Lidstone Problem with a Parameter

This work presents sufficient conditions for the existence and uniqueness of positive solutions for a discrete fourth-order beam equation under Lidstone boundary conditions with a parameter; the iterative sequences yielding approximate solutions are also given. The main tool used is monotone iterative technique.

Positive fixed points and fourth-order equations

This work presents sufficient conditions for the existence of at least one positive solution for a nonlinear fourth-order beam equation under Lidstone boundary conditions. The main tool used is a fixed point theorem that essentially combines the monotone iterative technique with fixed point theorems of cone expansion or compression type. The last section contains two examples that complement some related results existent in the literature.

Positive Solutions for Some Beam Equation Boundary Value Problems

A new fixed point theorem in a cone is applied to obtain the existence of positive solutions of some fourth-order beam equation boundary value problems with dependence on the first-order derivative where is continuous.

Multiple solutions of some nonlinear fourth-order beam equations

Several new existence theorems on three solutions and infinitely many solutions for the following fourth-order beam equation are obtained: u ( 4 ) = f ( t , u ( t ) ) , t ∈ [ 0 , 1 ] ; u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f ∈ C 1 ( [ 0 , 1 ] × R 1 , R 1 ) . The Morse theory is employed to discuss this problem.

Existence of Positive Solutions to Fourth Order Quasilinear Boundary Value Problems

In this paper, we generalize the fixed point theorem of cone expansion and compression of norm type to the theorem of functional type. As an application, the existence of positive solutions for some fourth–order beam equation boundary value problems is obtained. The emphasis is put on that the nonlinear term is dependent on all lower order derivatives.

Positive solutions of the nonlinear fourth-order beam equation with three parameters

This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation u ( 4 ) ( t ) + η u ″ ( t ) − ζ u ( t ) = λ f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f ( t , u ) : [ 0 , 1 ] × [ 0 , + ∞ ) → [ 0 , + ∞ ) is continuous and ζ, η and λ are parameters. We show that there exists a λ * > 0 ˙ such that the above boundary value problem (BVP) has at least two, one and no positive solutions for 0 < λ < λ * , λ = λ * and λ > λ * , respectively. Furthermore, by using the semiorder method on cones of Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution u λ ( t ) of the BVP depends continuously on the parameter λ, i.e., u λ ( t ) is nondecreasing in λ, lim λ → 0 + ‖ u λ ( t ) ‖ = 0 and lim λ → + ∞ ‖ u λ ( t ) ‖ = + ∞ for any t ∈ [ 0 , 1 ] .

On positive solutions of some nonlinear fourth-order beam equations

The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered: u (4)(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0, where λ>0 is a constant, f :[0,1]×[0,+∞)→[0,+∞) is continuous.