The Bernstein theorem for a class of fourth order equations

In this paper, we prove the 2-dimensional Bernstein theorem for a class of fourth order equations including Abreu’s equation. The main ingredients of the paper are the a priori estimates and the proof of the strict convexity.     

Nonlinear partial differential equations of fourth order under mixed boundary conditions

Solutions to nonlinear partial differential equations of fourth order are studied. Boundary regularity is proved for solutions that satisfy mixed boundary conditions. Various geometric situations including so called triple points are considered. Regularity is measured in Sobolev-Slobodeckii spaces and the results are sharp in this scale. The approach is based on the use of a first order difference quotient method.     

Oscillation of fourth order sub-linear differential equations

In this paper, we deal with oscillatory and asymptotic properties of solutions of a fourth order sub-linear differential equation with the oscillatory operator. We establish conditions for the nonexistence of positive and bounded solutions and an oscillation criterion.     

On a class of nonlinear fourth order differential equations

Summary In this paper we study the nonlinear fourth order differentiai equation u^iv±F(u, v)u=0, where F is a pssitive monotone function of u. Asymptotic behaviour of certain solutions are treated in sections 2 and 4 while a two point boundary value problem is studied in section 3. Entrata in Redazione il 26 agosto 1968.     

Oscillation properties of fourth order differential equations

We define various ways in which linear-quadratic control problems can be robust and show that there are conditions which are simultaneously necessary and sufficient for robust problems to have a solution when such conditions do not exist for nonrobust problems.     

Existence of non-trivial solutions for systems of n fourth order partial differential equations

In this paper, employing a very recent local minimum theorem for differentiable functionals due to Bonanno, the existence of at least one nontrivial solution for a class of systems of n fourth order partial differential equations coupled with Navier boundary conditions is established.     

A maximum principle for fourth order ordinary differential equations

We present a maximum principle for fourth order ordinary differential equations, based on a new approach involving counting of inflection points. We use our results to compute solutions of nonlinear equations describing static displacements of a uniform beam     

A maximum principle for fourth order ordinary differential equations

A maximum principle is obtained for solutions of fourth order ordinary differential equations. As an application, the existence of a nonnegative (positive) solution of a nonlinear boundary problem is established. Extension of the maximum principle for solutions of higher order equations is also indicated.     

Bernstein series solution of linear second‐order partial differential equations with mixed conditions

The purpose of this study is to present a new collocation method for numerical solution of linear PDEs under the most general conditions. The method is given with a priori error estimate. By using the residual correction procedure, the absolute error can be estimated. Also, one can specify the optimal truncation limit n, which gives better result in any norm ∥ ∥ . Finally, the effectiveness of the method is illustrated in some numerical experiments. Numerical results are consistent with the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Oscillation criteria of a class of fourth order differential equations

In the present paper, we consider the oscillation of a class of self‐adjoint fourth order differential equation. Some oscillation and non‐oscillation criteria on a perturbation equation of Euler differential equation are given. In particular, we find the exact value of a constant in some oscillation criteria, which has answered the open problem presented by Do?lý in 2004. Copyright © 2011 John Wiley & Sons, Ltd.     

Nonlinear oscillation of fourth order functional differential equations

Summary In the oscillation theory of nonlinear differential equations one of the important problems is to find necessary and sufficient conditions for the equations under consideration to be oscillatory. Beginning with the pionearing work of F. V. Atkinson, there have been a number of papers. Recently, Kusano and Naito proved the interesting results to the jourth order nonlinear ordinary differential equations of the from [r(t)y″(t)]″+y(t)F(y(t) ₂ ,t)=0. In the present paper, we will extend them to the more general functional differential equations and improve the not clear parts of them. Also, we will propose a new simple definition of nonlinearity of the functional differential equations. Entrata in Redazione il 5 settembre 1977.     

B-splines methods with redefined basis functions for solving fourth order parabolic partial differential equations

In this work, we discuss two methods for solving a fourth order parabolic partial differential equation. In Method-I, we decompose the given equation into a system of second order equations and solve them by using cubic B-spline method with redefined basis functions. In Method-II, the equation is solved directly by applying quintic B-spline method with redefined basis functions. Stability of these methods have been discussed. Both methods are unconditionally stable. These methods are tested on four examples. The computed results are compared wherever possible with those already available in literature. We have developed Method-I for fourth order non homogeneous parabolic partial differential equation from which we can obtain displacement and bending moment both simultaneously, while Method-II gives only displacement. The results show that the derived methods are easily implemented and approximate the exact solution very well.     

Boundedness theorems for some fourth order differential equations

Summary In this paper a new approach involving the use of two signum functions together with a suitably chosen Lyapunov function is employed to investigate the boundedness property of solutions of two special cases of(1.3). This approach makes for considerable reduction in the conditions imposed on f, g in an earlier paper[1]. Entrata in Redazione il 25 ottobre 1970.     

Oscillation theorems for fourth order functional differential equations

The authors investigate the oscillatory behavior of all solutions of the fourth order functional differential equations $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})+q(t)f(x[g(t)])=0$ and $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})=q(t)f(x[g(t)])+p(t)h(x[\sigma(t)])$ in the case where ∫ ^∞ a ^?1/α (s)ds<∞. The results are illustrated with examples.     

Nonlocal boundary-value problems for differential equations of the fourth order

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 12, pp.1653–1659, December, 1989.     

一类非线性偏微分方程的四阶格子Boltzmann模型

通过Chapman-Enskog展开技术和多尺度分析,建立了一种新的D1Q4带修正项的四阶格子Boltzmann模型,一类非线性偏微分方程从连续的Boltzmann方程得到正确恢复．统一了KdV和Burgers等已知方程类型的格子BGK模型,还首次给出了组合KdV-Burgers,广义Burgers—Huxley等方程...