Nonlinear Fourth-order Boundary Value Problem Research Articles

Positive solution for a class of nonlinear fourth-order boundary value problem

<abstract><p>In this paper, we are concerned with the existence of positive solutions for boundary value problems of nonlinear fourth-order differential equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} &&u^{(4)}+c(x)u = \lambda a(x)f(u), \; \; x\in (0,1),\\ &&u(0) = u(1) = u''(0) = u''(1) = 0, \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ a(x) $ may change signs. The proof of main results is based on Leray-Schauder's fixed point theorem and the properties of Green's function of the fourth-order differential operator $ L_cu = u^{(4)}+c(x)u $.</p></abstract>

Solving the fourth-order nonlinear boundary value problem by a boundary shape function method

A method to construct the boundary shape function (BSF) and then two novel methods are developed to obtain the solutions of fourth-order singularly perturbed beam equation and nonlinear boundary value problem (BVP). In the first-type algorithm, the free function is a series of complete basis functions, while the corresponding BSFs are new bases. The trial functions with fractional powers exponential are suitable for the singularly perturbed beam equation under fixed-end and simply supported boundary conditions. With the aid of the BSF, we can improve the asymptotic and uniform approximations to exactly satisfy the prescribed boundary conditions. In the second-type algorithm, the solution of a nonlinear BVP is viewed as a boundary shape function, while the free function is regarded as a new variable. With this means, the fourth-order nonlinear BVP is exactly converted to an initial value problem with a new variable, the terminal value of which is unknown, when the initial conditions are given. The computed order of convergence and an error estimation are given. Numerical illustrations, including the singularly perturbed examples, show that the present methods, based on the new idea of the BSF, are highly effective, accurate, and fast convergent.

A study on solvability of the fourth-order nonlinear boundary value problems

Abstract The purpose of the paper is devoted to proving the solvability of the fourth order boundary value problem. Firstly, we build a maximum principle for the corresponding linear equation, by the use of this maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to solve the nonlinear equation, secondly, the existence and uniqueness results for the problem is obtained. In addition, an example is presented to show the application of our main results.

Global structure of positive solutions for semipositone nonlinear Euler-Bernoulli beam equation with Neumann boundary conditions

In this paper, we focus on the existence of positive solutions for nonlinear fourth-order Neumann boundary value problem where k 1 and k 2 are constants, λ > 0 is the bifurcation parameter, f ∈ C([0, 1] × ℝ+, ℝ), ℝ+ := [0, ∞). We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we show that there exists a global branch of solutions emanating from infinity under some different growth conditions. In addition, we prove that for λ near the bifurcation points, solutions of large norm are indeed positive. The technique for dealing with this paper relies on the global bifurcation theory.

Generalized Iteration Method via Green's Function for the Solution of the Fourth-order Boundary Value Problems

The aim of this paper is to extend and generalize Picard-Green’s fixed point iteration method for the solution of fourth-order Boundary Value Problems. Several numerical applications to linear and nonlinear fourth-order Boundary Value Problems are discussed to illustrate the main results.

Existence of solutions for fourth-order nonlinear boundary value problems

In this paper, we discuss the existence and approximation of solutions for a fourth-order nonlinear boundary value problem by using a quasilinearization technique. In the presence of a lower solution α and an upper solution β in the reverse order alpha geq beta , we show the existence of (extreme) solution.

Existence-uniqueness and fixed-point iterative method for general nonlinear fourth-order boundary value problems

Existence and uniqueness results are established for the general nonlinear fourth-order two-point boundary value problem with general linear homogeneous boundary conditions. The proofs are based on the Banach fixed-point theorem under the conditions that the nonlinear term is bounded and Lipschitz in a specific region. A fixed-point iterative method for finding the solution of the problem is also proposed with error estimates. The resulting iterative solutions automatically satisfy the boundary conditions provided that the initial iterate does so. Two examples are given to illustrate the applicability of the proposed approach and the efficiency of the iterative method.

Multiple Solutions for a Class of Nonlinear Fourth-Order Boundary Value Problems

This paper is concerned with multiple solutions for a class of nonlinear fourth-order boundary value problems with parameters. By constructing a special cone and applying fixed point index theory, the multiple solutions for the considered systems are obtained under some suitable assumptions. The main feature of obtained solutions (u(t),v(t)) is that the solution u(t) is positive, and the other solution v(t) may change sign. Finally, two examples with continuous function f1 being positive and f2 being semipositone are worked out to illustrate the main results.

Analysis of optimal superconvergence of the local discontinuous Galerkin method for nonlinear fourth-order boundary value problems

This paper is concerned with the convergence and superconvergence of the local discontinuous Galerkin (LDG) finite element method for nonlinear fourth-order boundary value problems of the type $u^{(4)}=f(x,u,u^{\prime },u^{\prime \prime },u^{\prime \prime \prime })$ , x ∈ [a,b] with classical boundary conditions at the endpoints. Convergence properties for the solution and for all three auxiliary variables are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th-order convergence, when polynomials of degree p are used. We also prove that the derivatives of the errors between the LDG solutions and Gauss-Radau projections of the exact solutions in the L2 norm are superconvergent with order p + 1. Furthermore, a (2p + 1)th-order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages is also obtained under quasi-uniform meshes. Finally, we prove that the LDG solutions are superconvergent with an order of p + 2 toward particular projections of the exact solutions. The error analysis presented in this paper is valid for p ≥ 1. Numerical experiments indicate that our theoretical findings are optimal.

New Fixed Point Theorem for Discontinuous Operators in Cones and Applications

We provide new fixed point theorems for a class of discontinuous operators by combining a new fixed point theorem of compression-expansion type for these discontinuous operators with monotone iterative methods. As an application we study the existence of positive solutions for a nonlinear fourth-order discontinuous boundary value problem.

Exponential Spline Solution of Boundary Value Problems Occurring in the Plate Deflection Theory

In this paper, we develop a class of methods based on exponential spline at off-step points for the solution of the nonlinear and linear fourth-order boundary value problems (BVPs). Spline relations and boundary equations are developed, and the methods of order two, four and six have been obtained. Convergence analysis of the method is discussed. Numerical examples for linear and nonlinear BVPs are given to illustrate the efficiency of our method, and results are compared with other known methods.

A short review on analytical methods for a fully fourth-order nonlinear integral boundary value problem with fractal derivatives

Purpose This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives. Design/methodology/approach Boundary value problems arise everywhere in engineering, hence two-scale thermodynamics and fractal calculus have been introduced. Some analytical methods are reviewed, mainly including the variational iteration method, the Ritz method, the homotopy perturbation method, the variational principle and the Taylor series method. An example is given to show the simple solution process and the high accuracy of the solution. Findings An elemental and heuristic explanation of fractal calculus is given, and the main solution process and merits of each reviewed method are elucidated. The fractal boundary value problem in a fractal space can be approximately converted into a classical one by the two-scale transform. Originality/value This paper can be served as a paradigm for various practical applications.

Analytical estimates of the pull-in voltage for carbon nanotubes considering tip-charge concentration and intermolecular forces

Two-side accurate analytical estimates of the pull-in parameters of a carbon nanotube switch clamped at one end under electrostatic actuation are provided by considering the proper expressions of the electrostatic force and van der Waals interactions for a carbon nanotube, as well as the contribution of the charge concentration at the free end. According to the Euler–Bernoulli beam theory, the problem is governed by a fourth-order nonlinear boundary value problem. Two-side estimates on the centreline deflection are derived. Then, very accurate lower and upper bounds to the pull-in voltage and deflection are obtained as function of the geometrical and material parameters. The analytical predictions are found to agree remarkably well with the numerical results provided by the shooting method, thus validating the proposed approach. Finally, a simple closed-form relation is proposed for the minimum feasible gap and maximum realizable length for a freestanding CNT cantilever.

一类非线性四阶边值问题正解的存在性

一类非线性四阶边值问题正解的存在性

Existence results and iterative method for a fully fourth-order nonlinear integral boundary value problem

We consider the boundary value problem $$ \begin{array}{@{}rcl@{}} u^{\prime\prime\prime\prime}(t)&=&f(t,u(t),u^{\prime}(t),u^{\prime\prime}(t),u^{\prime\prime\prime}(t)), \quad 0<t<1,\\ u^{\prime}(0)&=&u^{\prime\prime}(0)= u^{\prime}(1)=0, u(0)={{\int}_{0}^{1}} g(s)u(s) ds, \end{array} $$ where $f: [0, 1] \times \mathbb {R}^{4} \rightarrow \mathbb {R}^{+},\ a: [0, 1] \rightarrow \mathbb {R}^{+}$ are continuous functions. For f = f(u(t)), very recently in Benaicha and Haddouchi (An. Univ. Vest Timis. Ser. Mat.-Inform. 1(54): 73–86, 2016) the existence of positive solutions was studied by employing the fixed point theory on cones. In this paper, by the method of reducing the boundary value problem to an operator equation for the right-hand sides we establish the existence, uniqueness, and positivity of solution and propose an iterative method on both continuous and discrete levels for finding the solution. We also give error analysis of the discrete approximate solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.

A fourth-order orthogonal spline collocation method to fourth-order boundary value problems

In this article, we use orthogonal spline collocation methods (OSCM) for the fourth-order linear and nonlinear boundary value problems (BVPs). Cubic monomial basis functions and piecewise Hermite cubic basis functions are used to approximate the solution. We establish the existence and uniqueness solution to the discrete problem. We discuss dynamics of the stationary Swift–Hohenberg equation for different values of α. Finally, we perform some numerical experiments and using grid refinement analysis, we compute the order of convergence of the numerical method. Comparative to existing methods, we show that the OSCM gives optimal order of convergence for norms and superconvergent result for -norm at the knots with minimal computational cost.

A fast, spectrally accurate homotopy based numerical method for solving nonlinear differential equations

We present an algorithm for constructing numerical solutions to one-dimensional nonlinear, variable coefficient boundary value problems. This scheme is based upon applying the Homotopy Analysis Method (HAM) to decompose a nonlinear differential equation into a series of linear differential equations that can be solved using a sparse, spectrally accurate Gegenbauer discretisation. Uniquely for nonlinear methods, our scheme involves constructing a single, sparse matrix operator that is repeatedly solved in order to solve the full nonlinear problem. As such, the resulting scheme scales quasi-linearly with respect to the grid resolution. We demonstrate the accuracy, and computational scaling of this method by examining a fourth-order nonlinear variable coefficient boundary value problem by comparing the scheme to Newton-Iteration and the Spectral Homotopy Analysis Method, which is the most commonly used implementation of the HAM.

Existence and Iteration of Monotone Positive Solution for a Fourth-Order Nonlinear Boundary Value Problem

This paper is concerned with the following fourth-order three-point boundary value problem BVP \[ u^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right)\right),\quad t\in\left[0,1\right], \] \[ u'\left(0\right)=u''\left(0\right)=u\left(1\right)=0,\;u'''\left(\eta\right)+\alpha u\left(0\right)=0, \] where $f\in C\left(\left[0,1\right]\times\left[0,+\infty\right),\left[0,+\infty\right)\right)$ , $\alpha\in\left[0,6\right)$ and $\eta\in\left[\frac{2}{3},1\right)$. Although corresponding Green\textquoteright s function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions on $f$ by applying iterative method. An example is also given to illustrate the main results.

Existence and Nonexistence of Solutions for Fourth-Order Nonlinear Difference Boundary Value Problems via Variational Methods

This paper is concerned with boundary value problems for a fourth-order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence ofndistinct pairs of nontrivial solutions, and nonexistence of solutions. Some examples are provided to show the effectiveness of the main results.

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 &lt; t &lt; 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.