Nonlinear Fourth-order Boundary Value Research Articles

Eleventh Degree Polynomial Series Solution Approach of Special Non-linear Fourth Order Boundary Value Problems

In this paper a new eleventh degree polynomial series solution approach is developed for the solution of special non-linear fourth order boundary value problems. The method consist first of obtaining a linear differentials system of twelve equations from the boundary conditions, governing equation and its three successive derivatives which were evaluated at boundary points. Secondly we assume the approximate solution in the form of a polynomial of degree eleventh with twelve unknown coefficients. To determine the unknown coefficients we incorporate the assumed solution into linear differentials systems of twelve equations which results into a linear systems of twelve equations with twelve unknown and which can be solve uniquely. It is clear from the tables and figures that the method is in good agreement with the exact and with some existing results in the literatures. Also it can be seen from example 3.3 that the exact solution is reproduced which is an added advantage of the method.

Numerical method of sixth order convergence for solving a fourth order nonlinear boundary value problem

In this paper, we construct an iterative method of sixth order convergence for solving a fourth order nonlinear boundary value problem. The idea of the method is to construct difference schemes of sixth order accuracy for solving Dirichlet problems for second order equation at each iteration of the iterative method on continuous level constructed before by ourselves. Some numerical examples demonstrate the validity of obtained theoretical results.

The reproducing kernel method for nonlinear fourth-order BVPs

<abstract><p>Based on the reproducing kernel theory, we solve the nonlinear fourth order boundary value problem in the reproducing kernel space $ W_{2}^{5}[0, 1] $. Its approximate solution is obtained by truncating the n-term of the exact solution and using the $ \varepsilon $-best approximate method. Meanwhile, the approximate solution $ u^{(i)}_{n}(x) $ converges uniformly to the exact solution $ u^{(i)}(x), (i, 0, 1, 2, 3, 4) $. The validity and accuracy of this method are verified by some examples.</p></abstract>

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.

Distributional Henstock-Kurzweil integral for fourth order nonlinear boundary value problems

In this paper existence of solution of fourth order nonlinear boundary value problems involving distributional Henstock-Kurzweil integral is obtained using fixed point theorem and distributional derivative.

Solution of nonlinear boundary value problem by S-iteration

In this paper we import a novel approach to solve the non-linear fourth order boundary value problem. The basic strategy depends on integral operator equation which includes Green’s function and fixed point iteration. To ensure the method, three illustrations were conferred. From the residual or absolute error calculations, it is shown that the S-iteration for contraction operator gives fast convergence than Krasnoselskii–Mann’s iteration.

Distributional Henstock-Kurzweil integral for fourth order nonlinear boundary value problems

In this paper existence of solution of fourth order nonlinear boundary value problems involving distributional Henstock-Kurzweil integral is obtained using fixed point theorem and distributional derivative.

A class of fourth order nonlinear boundary value problem with singular perturbation

A class of singularly perturbed fourth order nonlinear differential equation is considered in this article. The existence of solution to boundary value problem (BVP) without singularly perturbed is obtained via nonlinear analysis method and upper and lower solution theory firstly. In addition, based upon some differential inequality techniques, as well as appropriate upper–lower solutions, the existence, uniqueness and corresponding asymptotic estimation of the solution to singularly perturbed boundary value problem (SPBVP) is considered.

On Approximate Stationary Radial Solutions for a Class of Boundary Value Problems Arising in Epitaxial Growth Theory

In this paper, we consider a non-self-adjoint, singular, nonlinear fourth order boundary value problem which arises in the theory of epitaxial growth. It is possible to reduce the fourth order equation to a singular boundary value problem of second order given by w''-1/r w'=w^2/(2r^2 )+1/2 λ r^2. The problem depends on the parameter λ and admits multiple solutions. Therefore, it is difficult to pick multiple solutions together by any discrete method like finite difference method, finite element method etc. Here, we propose a new technique based on homotopy perturbation method and variational iteration method. We compare numerically the approximate solutions computed by Adomian decomposition method. We study the convergence analysis of both iterative schemes in C^2 ([0,1]). For small values of λ, solutions exist whereas for large values of λ solutions do not exist.

Kansa radial basis function method with fictitious centres for solving nonlinear boundary value problems

A Kansa–radial basis function (RBF) collocation method is applied to two–dimensional second and fourth order nonlinear boundary value problems. The solution is approximated by a linear combination of RBFs, each of which is associated with a centre and a different shape parameter. As well as the RBF coefficients in the approximation, these shape parameter values are taken to be among the unknowns. In addition, the centres are distributed within a larger domain containing the physical domain of the problem. The size of this larger domain is controlled by a dilation parameter which is also included in the unknowns. In fourth order problems where two boundary conditions are imposed, two sets of (different) boundary centres are selected. The Kansa–RBF discretization yields a system of nonlinear equations which is solved by standard software. The proposed technique is applied to four problems and the numerical results are analyzed and discussed.

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.

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.

Existence and Iterative Method for Some Riemann Fractional Nonlinear Boundary Value Problems

In this paper, we prove the existence and uniqueness of solution for some Riemann–Liouville fractional nonlinear boundary value problems. The positivity of the solution and the monotony of iterations are also considered. Some examples are presented to illustrate the main results. Our results generalize those obtained by Wei et al (Existence and iterative method for some fourth order nonlinear boundary value problems. Appl. Math. Lett. 2019, 87, 101–107.) to the fractional setting.

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.

Variable shape parameter Kansa RBF method for the solution of nonlinear boundary value problems

We apply a variable shape parameter Kansa–radial basis function (RBF) collocation method for the numerical solution of second and fourth order nonlinear boundary value problems in two dimensions. In the current approach, each RBF in the solution approximation is associated with a different shape parameter. These shape parameters are considered to be part of the unknowns along with the values of the coefficients of the RBFs in the solution approximation. The system of nonlinear equations resulting from the Kansa–RBF discretization is solved by directly applying a standard nonlinear solver. The proposed method is applied to several numerical examples.

Existence Results and Numerical Method for a Fourth Order Nonlinear Problem

In this paper we consider a fully fourth order nonlinear boundary value problem which models the bending equilibrium of an extensible beam. Firstly we establish the existence and uniqueness of solution under some easily verified conditions. Next, for finding the approximate solution we propose an iterative method at continuous level and prove its convergence. To numerically realize the iterative method we carry out its discretization with the use of difference schemes of the fourth order of accuracy. Finally, some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method.

Nonlinear elastic beam problems with the parameter near resonance

AbstractIn this paper, we consider the nonlinear fourth order boundary value problem of the form$$ \begin{array}\text \left\{ \begin{aligned} &amp;u^{(4)}(x)-\lambda u(x)=f(x, u(x))-h(x), \ \ x\in (0,1),\\ &amp;u(0)=u(1)=u'(0)=u'(1)=0,\\ \end{aligned}\right. \end{array} $$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.

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.