Nonlinear Fourth-order Boundary Value Problem Research Articles

Existence of positive solutions of a nonlinear fourth-order boundary value problem

In this paper, we study the existence of positive solutions of fourth-order boundary value problem u ( 4 ) ( t ) = f ( t , u ( t ) , u ″ ( t ) ) , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f : [ 0 , 1 ] × [ 0 , ∞ ) × ( − ∞ , 0 ] → [ 0 , ∞ ) is continuous. The proof of our main result is based upon the Krein–Rutman theorem and the global bifurcation techniques.

Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems

In this paper, we study the existence of positive solutions of a two-point boundary value problem for a system of fourth-order nonlinear singular semipositone differential equations by the fixed point index theorem. Some new existence results are established, and an example is given to demonstrate the application of our main results.

Searching the least value method for solving fourth-order nonlinear boundary value problems

This paper obtains a searching least value (SLV) method for a class of fourth-order nonlinear boundary value problems is investigated. The argument is based on the reproducing kernel space W 5 [ 0 , 1 ] . The approximate solutions u n ( x ) and u n ( k ) ( x ) are uniformly convergent to the exact solution u ( x ) and u ( k ) ( x ) ( k = 1 , 2 , 3 , 4 ) respectively. Numerical results are verified that the method is quite accurate and efficient for this kind of problem.

On discrete fourth-order boundary value problems with three parameters

In this paper, the existence, multiplicity, and nonexistence results of nontrivial solutions are obtained for discrete nonlinear fourth-order boundary value problems with three parameters. The methods used here are based on the critical point theory and monotone operator theory.

Constructive proof of existence for a class of fourth-order nonlinear BVPs

A new existence proof of solutions for a class of fourth-order nonlinear boundary value problems is proposed. The proof of the main results is based on the reproducing kernel theorem. It is worthwhile to point out that the method presented in this paper can be applied for the existence proof of diverse kinds of boundary conditions.

Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem

We study the existence of positive solutions of the nonlinear boundary value problem u ( 4 ) ( t ) = f ( t , u ( t ) , u ″ ( t ) ) , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , which is not necessarily linearizable. Our approaches are based on Krein–Rutman theorem, topological degree theory and global bifurcation techniques.

A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems

This paper presents a new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between our method and other methods for solving an open fourth-order boundary value problem presented by Scott and Watts. The method is also applied to a nonlinear fourth-order boundary value problem. The numerical results demonstrate that the new method is quite accurate and efficient for fourth-order boundary value problems.

ConstantT-curvature conformal metrics on 4-manifolds with boundary

In this paper we prove that, given a compact four-dimensional smooth Riemannian manifold (M,g) with smooth boundary, there exists a metric conformal to g with constant T-curvature, zero Q-curvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth-order nonlinear elliptic boundary value problem (BVP) with boundary conditions given by a third-order pseudodifferential operator and homogeneous Neumann operator. It has a variational structure, but since the corresponding Euler�Lagrange functional is in general unbounded from below, we look for saddle points. We do this by using topological arguments and min-max methods combined with a compactness result for the corresponding BVP.

Application of the homotopy perturbation method to linear and nonlinear fourth-order boundary value problems

In this study, we applied the homotopy perturbation (HP) method for solving linear and nonlinear fourth-order boundary value problems. The analytical results of the boundary value problems have been obtained in terms of a convergent series with easily computable components. Comparisons between the results of the HP method and the analytical solution showed that this method gives very precise results with a few terms. In the implied HP method, some unknown parameters in the initial guess are introduced, which are identified after applying boundary conditions. This improvement results in higher accuracy.

Existence and uniqueness of positive solutions for fourth-order nonlinear singular continuous and discrete boundary value problems

Using the mixed monotone method we establish the existence and uniqueness of positive solutions for some fourth-order nonlinear singular continuous and discrete boundary value problem. The theorems obtained are very general and complement previously known results.

Fifth-degree B-spline solution for nonlinear fourth-order problems with separated boundary conditions

In this paper, we discussed a fifth-degree B-spline solution for the numerical solution to nonlinear fourth-order boundary value problems (BVPs) with separated boundary conditions. Two numerical examples are given to illustrate the efficiency and performance of the method. The method gives accurate results for both the linear and nonlinear cases.

A boundary value problem for fourth-order elastic beam equations

Multiplicity results for a fourth-order nonlinear boundary value problem are presented. The proof of the main result is based on the critical point theory.

Numerical methods for nonlinear fourth-order boundary value problems with applications

In this paper, we present efficient numerical algorithms for the approximate solution of nonlinear fourth-order boundary value problems. The first algorithm deals with the sinc–Galerkin method (SGM). The sinc basis functions prove to handle well singularities in the problem. The resulting SGM discrete system is carefully developed. The second method, the Adomian decomposition method (ADM), gives the solution in the form of a series solution. A modified form of the ADM based on the use of the Laplace transform is also presented. We refer to this method as the Laplace Adomian decomposition technique (LADT). The proposed LADT can make the Adomian series solution convergent in the Laplace domain, when the ADM series solution diverges in the space domain. A number of examples are considered to investigate the reliability and efficiency of each method. Numerical results show that the sinc–Galerkin method is more reliable and more accurate.

Fourth-order compact finite difference method for fourth-order nonlinear elliptic boundary value problems

A compact finite difference method with non-isotropic mesh is proposed for a two-dimensional fourth-order nonlinear elliptic boundary value problem. The existence and uniqueness of its solutions are investigated by the method of upper and lower solutions, without any requirement of the monotonicity of the nonlinear term. Three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently. The convergence and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.

Existence of positive solutions for fourth-order m-point boundary value problem with variable parameters

In this paper, we investigate the existence of positive solutions for a class of nonlinear fourth-order m-point boundary value problem with two variable parameters. By using the fixed point index theory, we give some new existence results.

Positive Solutions of Boundary Value Problems for System of Nonlinear Fourth-Order Differential Equations

Some existence theorems of the positive solutions and the multiple positive solutions for singular and nonsingular systems of nonlinear fourth-order boundary value problems are proved by using topological degree theory and cone theory.

Positive solutions of fourth-order nonlinear singular boundary value problems

We consider the existence of positive solutions for the following fourth-order singular Sturm–Liouville boundary value problem: { 1 p ( t ) ( p ( t ) u ‴ ( t ) ) ′ − g ( t ) F ( t , u ) = 0 , 0 < t < 1 , α 1 u ( 0 ) − β 1 u ′ ( 0 ) = 0 , γ 1 u ( 1 ) + δ 1 u ′ ( 1 ) = 0 , α 2 u ″ ( 0 ) − β 2 lim t → 0 + p ( t ) u ‴ ( t ) = 0 , γ 2 u ″ ( 1 ) + δ 2 lim t → 1 − p ( t ) u ‴ ( t ) = 0 , where g , p may be singular at t = 0 and/or 1. Moreover F ( t , x ) may also have singularity at x = 0 . The existence and multiplicity theorems of positive solutions for the fourth-order singular Sturm–Liouville boundary value problem are obtained by using the first eigenvalue of the corresponding linear problems. Our results significantly extend and improve many known results including singular and nonsingular cases.

Monotone iterative technique for numerical solutions of fourth-order nonlinear elliptic boundary value problems

This paper is concerned with finite difference solutions of a class of fourth-order nonlinear elliptic boundary value problems. The nonlinear function is not necessarily monotone. A new monotone iterative technique is developed, and three basic monotone iterative processes for the finite difference system are constructed. Several theoretical comparison results among the various monotone sequences are given. A simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. Numerical results for a model problem with known analytical solution are given.

Multiply sign-changing solutions for fourth-order nonlinear boundary value problems

This paper deals with the existence of sign-changing solutions for the fourth-order nonlinear boundary value problem u ( 4 ) = f ( u , u ″ ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f : R 2 → R is a continuously odd function. Some conditions on f guaranteeing the existence and multiplicity of sign-changing solutions are presented. The existence and multiplicity conditions concern the eigenline of the associated two-parameter linear eigenvalue problem. The discussion is based on the fixed point index theory in cones and an anti-symmetrical extension method of solution.

Elastic beam problem with higher order derivatives

We apply a fixed point theorem to obtain sufficient conditions for existence of triple positive solutions of the symmetric nonlinear fourth-order boundary value problem u ( 4 ) ( t ) = a ( t ) f ( t , u ( t ) , | u ′ ( t ) | , u ″ ( t ) , | u ‴ ( t ) | ) , t ∈ ( - 1 , 1 ) , u ( - 1 ) = u ′ ( - 1 ) = 0 , u ( - t ) = u ( t ) , t ∈ [ - 1 , 1 ] , where f satisfies certain growth conditions. In the process, we obtain nontrivial a priori bounds on the derivatives of the solution.