On the finite space blow up of the solutions of the Swift-Hohenberg equation

The aim of this paper is to study the finite space blow up of the solutions for a class of fourth order differential equations. Our results answer a conjecture in [F. Gazzola and R. Pavani. Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations. Arch. Ration. Mech. Anal., 207(2):717--752, 2013] and they have implications on the nonexistence of beam oscillation given by traveling wave profile at low speed propagation.Comment: 24 pages, 2 figure

Periodic solutions for some fully nonlinear fourth order differential equations

In this paper we present sufficient conditions for the existence of periodic solutions to some nonlinear fourth order boundary value problems u(4)(x) = f(x; u(x); u′(x); u′′(x); u′′′(x)) u(i)(a) = u(i)(b); i = 0; 1; 2; 3; To the best of our knowledge it is the first time where this type of general nonlinearities is considered in fourth order equations with periodic boundary conditions. The difficulties in the odd derivatives are overcome due to the following arguments: the control on the third derivative is done by a Nagumo-type condition and the bounds on the first derivative are obtained by lower and upper solutions, not necessarily ordered. By this technique, not only it is proved the existence of a periodic solution, but also, some qualitative properties of the solution can be obtained

Application of Backward Differentiation Formula on Fourth-Order Differential Equations

Higher order ordinary differential equations are typically encountered in engineering, physical science, biological sciences, and numerous other fields. The analytical solution of the majority of engineering problems involving higher-order ordinary differential equations is not a simple task. Various numerical techniques have been proposed for higher-order initial value problems (IVP), but a higher degree of precision is still required. In this paper, we propose a novel two-step backward differentiation formula in the class of linear multistep schemes with a higher order of accuracy for solving ordinary differential equations of the fourth order. The proposed method was created by combining interpolation and collocation techniques with the use of power series as the basis function at some grid and off-grid locations to generate a hybrid continuous two-step technique. The method's fundamental properties, such as order, zero stability, error constant, consistency, and convergence, were explored, and the analysis showed that it is zero stable, consistent and convergent. The developed method is suitable for numerically integrating linear and nonlinear differential equations of the fourth order. Four Numerical tests are presented to demonstrate the efficiency and accuracy of the proposed scheme in comparison to some existing block methods. Based on what has been observed, the numerical results indicate that the proposed scheme is a superior method for estimating fourth-order problems than the method previously employed, confirming its convergence

Hamiltonian formulation of a class of constrained fourth-order differential equations in the Ostrogradsky framework

We consider a class of Lagrangians that depend not only on some configurational variables and their first time derivatives, but also on second time derivatives, thereby leading to fourth-order evolution equations. The proposed higher-order Lagrangians are obtained by expressing the variables of standard Lagrangians in terms of more basic variables and their time derivatives. The Hamiltonian formulation of the proposed class of models is obtained by means of the Ostrogradsky formalism. The structure of the Hamiltonians for this particular class of models is such that constraints can be introduced in a natural way, thus eliminating expected instabilities of the fourth-order evolution equations. Moreover, canonical quantization of the constrained equations can be achieved by means of Dirac's approach to generalized Hamiltonian dynamics.Comment: 8 page

Non-Resonant Oscillations for some Fourth-Order Differential Equations with Delay

We use coincidence degree arguments in order to derive the existence and uniqueness of periodic solution of equation (1.1}

Periodic Boundary-Value Problems for Fourth Order Differential Equations with Delay

We study the periodic boundary-value problem z(iul(t) + f (x )x{t) + OO:{t) + g(t , :i:(t - -r)) + dx = p(t) z {O) = z ( 2 . r ) , :i:(O) = :.i:(2.r), %(0) = %(2.r), : i i (O) = ·x(2.r) , Under some resonant conditions on t he asymptotic behaviour of the ratio g(t, y ) /(by ) for IYI -. oo. Uniqueness of periodic solutions is al5o examined