Fourth-order Two-point Boundary Value Problem Research Articles

Computation and convergence of fixed points in graphical spaces with an application to elastic beam deformations

This paper aims to discuss the convergence behavior of fixed points for graphical Bc-Kannan-contractions using numerical iterations within the structure of graphical extended b-metric spaces. We provide innovative fixed point results using Bc-Kannan-contraction and demonstrate that every Kannan contraction is a graphical Bc-Kannan, but the converse is not valid in general. The findings, which explain the convergence of fixed points for graph-Kannan mappings, are the first of their kind in state of the art. We give examples using the framework of graphical analysis to show that our important findings are more general than the research that backs them up. Lastly, utilizing our results, we show that a solution to a fourth-order two-point boundary value problem expressing elastic beam deformations exists.

NEW FIXED POINT RESULTS FOR ASYMPTOTIC CONTRACTIONS AND ITS APPLICATION TO CANTILEVER BEAM PROBLEMS

In this article, we deal with some interesting variants of asymptotic contractions, namely Reich type and Chatterjea type weak asymptotic contractions defined on the usual metric spaces. We derive a couple of fixed point results concerning such contractions. Moreover, we look over the existence of solutions to a fourth-order two-point boundary value problem which is a particular type of cantilever beam problems. Furthermore, we construct numerical examples to justify our obtained results.

A STUDY ON THE SOLUTIONS OF NOTABLE ENGINEERING MODELS

In the commenced work, we establish some novel results concerning graph contractions in a more generalized setting. Furthermore, we deliver some examples to elaborate and explain the usability of the attained results. By virtue of nontrivial examples, we show our results improve, extend, generalize, and unify several noteworthy results in the existing state-of-art. We adopt computer simulation validating our results. To arouse further interest in the subject and to show its efficacy, we devote this work to recent applications which emphasize primarily the applications for the existence of the solution of various models related to engineering problems viz. fourth-order two-point boundary value problems describing deformations of an elastic beam, ascending motion of a rocket, and a class of integral equations. This approach is entirely new and will open up some new directions in the underlying graph structure.

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.

Positive solutions to a semipositone superlinear elastic beam equation

A semipositone fourth-order two-point boundary value problem is considered. In mechanics, the problem describes the deflection of an elastic beam rigidly fastened on the left and simply supported on the right. Under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, the existence of nontrivial solutions and positive solutions to this boundary value problem is obtained. The main results are obtained by using the topological method and the fixed point theory of superlinear operators.

A Note on the Eigenvalue Criteria for Positive Solutions of a Cantilever Beam Equation with Free End

In this paper, we study the existence of at least one positive solution to the fourth-order two-point boundary value problem (BVP) which models a cantilever beam equation, where one end is kept free. Here $$f \in {\mathcal {C}}\left( [0,1] \times {\mathbb {R}}_{+}, {\mathbb {R}}_{+}\right) $$ , $$g \in {\mathcal {C}}\left( [0,1] , {\mathbb {R}}_{+}\right) $$ and $$\lambda $$ is a positive parameter. The sufficient conditions are interesting, new and easy to verify. We have used some inequalities on the nonlinear function f and eigenvalues of a linear integral operator as bounds for the parameter $$\lambda $$ to obtain our results. Our approach is based on a revised version of a fixed point theorem due to Gustafson and Schmitt.

A Computationally Efficient Iterative Scheme for Solving Fourth-Order Boundary Value Problems

A new rapid-converging analytical scheme is introduced to obtain the approximate analytical solutions of nonlinear fourth-order two-point boundary value problems, which appear in various physical phenomena. The idea of the method to obtain the solution of such problems is essentially based on reducing the solution of the main problem to the solution of an integral problem. The introduced technique consists of two steps. First, construct an integral operator by introducing Green’s function, and then, the Normal-S iterative scheme is applied to this integral operator to construct the iterative approach for such problems, which yields a simple way to improve the convergence of the iterative solutions to the problem. We also discuss the convergence of the introduced iterative method. To exhibit the performance of the method, we consider some numerical test examples. The obtained results are compared with the existing analytical and numerical approaches to reveal the superiority and computational efficiency of the proposed approach. In fact, it is a direct recursive and computationally cost-effective method for dealing with strong nonlinearity. The numerical simulations signify the applicability and effectiveness of the present work.

On the Solvability of Fourth-Order Two-Point Boundary Value Problems

In this paper, we study the solvability of various two-point boundary value problems for x ( 4 ) = f ( t , x , x ′ , x ″ , x ‴ ) , t ∈ ( 0 , 1 ) , where f may be defined and continuous on a suitable bounded subset of its domain. Imposing barrier strips type conditions, we give results guaranteeing not only positive solutions, but also monotonic ones and such with suitable curvature.

Solvability for some fourth order two-point boundary value problems

Some fourth-order two-point boundary value problems are considered in this paper. Firstly, the Green's function is obtained by the use of the Laplace transform. Secondly, the first eigenvalue is given by using Ritz method. Then, by the use of the properties of self-adjoint operators and the fixed point index theory, the existence of positive solutions is obtained. Finally, an example is given to illustrate the main results.

Existence of Symmetric Positive Solutions for the FourthOrder Boundary Value Problem

In this paper, we study the existence of positive solutions for a class of fourthorder two-point boundary value problems:

Solution of fourth-order two-point BVPs with cubic non-polynomial spline and SOR iterative method

No Abstract Keywords: successive over-relaxation; cubic non-polynomial spline; fourth-order two-point boundary value problem

Some fixed point results for (β-ψ1-ψ2)-contractive conditions in ordered b-metric-like spaces

This paper extends and generalizes results of Mukheimer [(?,?,?)-contractive mappings in ordered partial b-metric spaces, J. Nonlinear Sci. Appl. 7(2014), 168-179]. A new concept of (?-?1-?2)-contractive mapping using two altering distance functions in ordered b-metric-like space is introduced and basic fixed point results have been studied. Useful examples are illustrated to justify the applicability and effectiveness of the results presented herein. As an application, the existence of solution of fourth-order two-point boundary value problems is discussed and rationalized by a numerical example.

Boundary integral solution for a class of fourth-order two-point boundary value problems

Boundary integral solution for a class of fourth-order two-point boundary value problems

Positive Solutions of a Nonlinear Fourth-order Integral Boundary Value Problem

Abstract In this paper, the existence of positive solutions for a nonlinear fourth-order two-point boundary value problem with integral condition is investigated. By using Krasnoselskii’s fixed point theorem on cones, sufficient conditions for the existence of at least one positive solutions are obtained.

Approximate series solution of fourth-order boundary value problems using decomposition method with Green’s function

This paper proposes a new efficient approach for obtaining approximate series solutions to fourth-order two-point boundary value problems. The proposed approach depends on constructing Green’s function and Adomian decomposition method (ADM). Unlike existing methods like ADM or modified ADM, the proposed approach avoids solving a sequence of nonlinear equations for the undetermined coefficients. In fact, the proposed method gives a direct recursive scheme for obtaining approximations of the solution with easily computable components. We also discuss the convergence and error analysis of the proposed scheme. Moreover, several numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed approach. The numerical results reveal that the proposed method is very effective and simple.

FAST INTEGRATION OF ONE-DIMENSIONAL BOUNDARY VALUE PROBLEMS

Two-point nonlinear boundary value problems (BVPs) in both unbounded and bounded domains are solved in this paper using fast numerical antiderivatives and derivatives of functions of L2(-∞, ∞). This differintegral scheme uses a new algorithm to compute the Fourier transform. As examples we solve a fourth-order two-point boundary value problem (BVP) and compute the shape of the soliton solutions of a one-dimensional generalized Korteweg–de Vries (KdV) equation.

Bifurcation from interval and positive solutions for a class of fourth-order two-point boundary value problem

We consider the fourth-order two-point boundary value problem x ���� + kx �� + lx = f (t, x), 0 < t <1 ,x(0) = x(1) = x � (0) = x � (1) = 0, which is not necessarily linearizable. We give conditions on the parameters k, l and f (t, x) that guarantee the existence of positive solutions. The proof of our main result is based upon topological degree theory and global bifurcation techniques. MSC: 34B15

Successive Iteration of Positive Solutions for Fourth-Order Two-Point Boundary Value Problems

We are concerned with a fourth-order two-point boundary value problem. We prove the existence of positive solutions and establish iterative schemes for approximating the solutions. The interesting point of our method is that the nonlinear term is involved with all lower-order derivatives of unknown function, and the iterative scheme starts off with a known cubic function or the zero function. Finally we give two examples to verify the effectiveness of the main results.

一类四阶两点边值问题正解的存在性与多解性

一类四阶两点边值问题正解的存在性与多解性

Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations

In this paper, we establish certain fixed point theorems in metric spaces with a partial ordering. Presented theorems extend and generalize several existing results in the literature. As application, we use the fixed point theorems obtained in this paper to study existence and uniqueness of solutions for fourth-order two-point boundary value problems for elastic beam equations.