Fourth-order Boundary Value Problem Research Articles

Eigenvalues of Fourth-Order Boundary Value Problems with Self-Adjoint Canonical Boundary Conditions

Fourth-order boundary value problems with general real self-adjoint boundary conditions are investigated. It is obtained that the eigenvalues of the problem depend not only continuously on all parameters of the problem but also smoothly on the boundary conditions. Furthermore, the derivatives of the eigenvalues with respect to boundary conditions are given in the sense of the fundamental canonical forms of fourth-order self-adjoint boundary conditions including each type of separated, mixed and coupled cases.

Positive Periodic Solutions for a Class of Fourth-Order Nonlinear Differential Equations

In this paper, we obtain conditions of existence and uniqueness of periodic solutions for a nonlinear fourth-order differential equation utilizing explicit Green’s function and fixed point index theorem combining with operator spectral theorem. We discuss an iteration method for constant coefficient nonlinear differential equations and establish the theorem on the existence of positive solutions for a fourth-order boundary value problem with a variable parameter. Finally, we give an example to illustrate our results.

Convergence analysis of the Adini element on a Shishkin mesh for a singularly perturbed fourth-order problem in two dimensions

We consider the singularly perturbed fourth-order boundary value problem e2Δ2u −Δu = f on the unit square ${\Omega }\subset \mathbb {R}^{2}$ , with boundary conditions u = ∂u/∂n = 0 on ∂Ω. Here, e ∈ (0,1) is a small parameter. The problem is solved numerically by means of Adini finite elements—a simple nonconforming finite element method for this problem. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with N2 elements is constructed and convergence of the method is proved in a ‘broken’ version of the Sobolev norm $v\mapsto \left (\varepsilon ^{2}|v|_{2}^{2} + |v|_{1}^{2} \right )^{1/2}$ . For a particular choice of the mesh, the error in the computed solution is at most C [e1/2(N− 1 lnN)2 + min {e1/2,e− 3/2N− 2} + N− 3], where the constant C is independent of e and N. Numerical results support our theoretical convergence rates, even for an example where not all the hypotheses of our theory are satisfied.

Optimal error estimates and superconvergence of an ultra weak discontinuous Galerkin method for fourth-order boundary-value problems

In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.

Multiplicity results for a boundary value problem with integral boundary conditions

The main purpose of this paper is to establish existence and multiplicity of positive solutions for a system of fourth-order boundary value problem with multi-point and integral conditions. To prove our results, we used Leggett–Williams fixed point theorem. An example is presented to illustrate our main results.

Existence and multiple solutions to a discrete fourth order boundary value problem

In this article, we study a discrete fourth order boundary value problem. By making use of variational methods and critical point theory, we obtain some criteria for the existence and multiple solutions. Moreover, two examples are included to illustrate the applicability of the main results.

Discrete fourth-order boundary value problems with four parameters

The theory of nonlinear difference equations and discrete boundary value problems has been widely used to study discrete models in many fields such as computer science, economics, mechanical engineering, control systems, artificial or biological neural networks, ecology, cybernetics, and so on. Fourth-order difference equations derived from various discrete elastic beam problems. In this paper, we seek further study of the multiplicity results for discrete fourth-order boundary value problems with four parameters. In fact, using a consequence of the local minimum theorem due Bonanno we look for the existence one solution under algebraic conditions on the nonlinear term and two solutions for the problem under algebraic conditions with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear term. Furthermore, by employing two critical point theorems, one due Averna and Bonanno, and another one due Bonanno we guarantee the existence of two and three solutions for our problem in a special case.

Nonlinear elastic beam problems with the parameter near resonance

AbstractIn this paper, we consider the nonlinear fourth order boundary value problem of the form$$ \begin{array}\text \left\{ \begin{aligned} &u^{(4)}(x)-\lambda u(x)=f(x, u(x))-h(x), \ \ x\in (0,1),\\ &u(0)=u(1)=u'(0)=u'(1)=0,\\ \end{aligned}\right. \end{array} $$which models a statically elastic beam with both end-points cantilevered or fixed. We show the existence of at least one or two solutions depending on the sign of λ−λ1, where λ1 is the first eigenvalue of the corresponding linear eigenvalue problem and λ is a parameter. The proof of the main result is based upon the method of lower and upper solutions and global bifurcation techniques.

New fixed point approach for a fully nonlinear fourth order boundary value problem

In this paper we propose a method for investigating the solvability and iterative solution of a nonlinear fully fourth order boundary value problem. Namely, by the reduction of the problem to an operator equation for the right-hand side function we establish the existence and uniqueness of a solution and the convergence of an iterative process. Our method completely differs from the methods of other authors and does not require the condition of boundedness or linear growth of the right-hand side function on infinity. Many examples, where exact solutions of the problems are known or not, demonstrate the effectiveness of the obtained theoretical results.

A spectral method for fourth-order boundary value problems

A spectral method for fourth-order boundary value problems

Existence and iterative method for some fourth order nonlinear boundary value problems

This work is devoted to proving the existence of the iterative solution to a fourth order boundary value problem. By the use of a novel efficient method for nonlinear fourth order boundary value problem, the existence and uniqueness of solution for the problem is obtained. The monotony of iterations is also considered. Some examples are presented to illustrate the main results.

Chebyshev Wavelet collocation method for solving a class of linear and nonlinear nonlocal boundary value problems

This study proposes the Chebyshev Wavelet Colocation method for solving a class of rth-order Boundary-Value Problems (BVPs) with nonlocal boundary conditions. This method is an extension of the Chebyshev wavelet method to the linear and nonlinear BVPs with a class of nonlocal boundary conditions. In this study, the method is tested on second and fourth-order BVPs and approximate solutions are compared with the existing methods in the literature and analytical solutions. The proposed method has promising results in terms of the accuracy.

Boundary value technique based on a fifth derivative method of order 10 for fourth order initial and boundary value problems

In this paper, the fourth order ordinary differential equation (ODE) of the form $$y^{(iv)}=f(x,y,y',y'',y''')$$ subject to initial or boundary conditions is solved directly without reducing the higher ODE into an equivalent first order system. This is accomplished by constructing a continuous fifth derivative method (CFIDM) which is used to produce the main and additional discrete fifth derivative methods (FIDMs). The FIDMs are then assembled and applied as a single block matrix equation to simultaneously provide all approximations on the entire interval of interest for the initial or boundary value problem. The convergence analysis of the method is discussed. The accuracy of the method is illustrated via numerical experiments.

Existence of three solutions to the discrete fourth-order boundary value problem with four parameters

In this work, we willproving the existence of three solutionsf or the discrete nonlinear fourth order boundary value problems with four parameters. The methods used here are based on the critical point theory.

Existence of solution for nonlinear fourth-order three-point boundary value problem

In this paper, we study the existence of nontrivial solution for the fourth-order three- point boundary value problem having the following formu(4) (t) + f (t, u(t)) = 0, 0 < t < 1,u(0) = α(η), u'(0) = u''(0) = 0, u(1) = βu(η),where η ∈ (0, 1), α, β ∈ R, f ∈ C ([0, 1] × R, R). We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.

Existence of solutions to a discrete fourth order boundary value problem

Criteria are established for the existence of at least two nontrivial solutions to the discrete fourth order boundary value problem where is an integer, , and is continuous in the second argument. Applications of the results to a related eigenvalue problem are also presented. The proofs are mainly based on the variational method and the classic mountain pass lemma of Ambrosetti and Rabinowitz. Examples are included to illustrate the applicability of the results.

Lyapunov-type inequalities for fourth-order boundary value problems

This paper is concerned with some two-point boundary value problems for fourth-order linear differential equations. Our study is based on the absolute maximums of Green’s functions corresponding to two-point boundary value problems. Thus, the best constants of Lyapunov-type inequalities for the problems are proved. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for two-point boundary value problems in this direction. In addition, some applications of the obtained inequalities are given.

Block Nystrom type method and its block extension for fourth order initial and boundary value problems

Block Nystrom type method and its block extension for fourth order initial and boundary value problems

Exponential spline approach for the solution of nonlinear fourth-order boundary value problems

Exponential spline approach for the solution of nonlinear fourth-order boundary value problems

An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition

This article discusses an iterative reproducing kernel method with respect to its effectiveness and capability of solving a fourth-order boundary value problem with nonlinear boundary conditions modeling beams on elastic foundations. Since there is no method of obtaining reproducing kernel which satisfies nonlinear boundary conditions, the standard reproducing kernel methods cannot be used directly to solve boundary value problems with nonlinear boundary conditions as there is no knowledge about the existence and uniqueness of the solution. The aim of this paper is, therefore, to construct an iterative method by the use of a combination of reproducing kernel Hilbert space method and a shooting-like technique to solve the mentioned problems. Error estimation for reproducing kernel Hilbert space methods for nonlinear boundary value problems have yet to be discussed in the literature. In this paper, we present error estimation for the reproducing kernel method to solve nonlinear boundary value problems probably for the first time. Some numerical results are given out to demonstrate the applicability of the method.