Fourth-order Boundary Value Problem Research Articles

Three solutions for a discrete fourth-order boundary value problem with four parameters

This paper presents several sufficient conditions for the existence of at least three classical solutions of a boundary value problem for a fourth-order difference equation. Fourth-order boundary value problems act as models for the bending or deforming of elastic beams. In different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and many others,the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. Our technical approach is based on variational methods. An example is included in the paper.Numerical computations of the example confirm our theoretical results.

Derivative block methods for the solution for fourth-order boundary value problems of ordinary differential equations

The application of block methods in the study of dynamical system is one area which needs more research. Therefore, this work presents block methods and its application to solve some problems in solid and fluid mechanics. The method was developed directly using the approaches of collocation and interpolation, and using power series polynomial as a trial solution. First, the system of linear equations is solved to obtain the unknown coefficients. The coefficients gotten are then substituted into the approximate solution to obtain continuous scheme. The continuous scheme, its first, second and third derivatives are evaluated at all the grid points to generate the block methods. The derived methods were applied to solve fourth-order boundary value problems of ordinary differential equations arising from beams and chemical problems. The results demonstrate the reliability and efficiency of the proposed method.

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.

A posteriori error analysis of an ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

A posteriori error analysis of an ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity

Abstract The impact of variable thermal conductivity plays a key role in the analysis of fluid mechanics. The applications of liquid are significant in nuclear reactors, automobiles, technical and manufacturing industry, electronic appliances, and so on. The novelty of this analysis is to exhibited the variable thermal conductivity in motion of Casson liquid via porous stretching sheet. In energy equation is consider thermal radiation and heat generation. Rosseland approximation plays a key role in the current work. Under the liquid motion assumptions, BL approximation is applied on numerical model and developed partial differential equations (PDE). The similarity transportation variable is taken by transporting PDE’s to ordinary differential equations. Numerical model is explored with the help of fourth-order boundary value problem with R–K–F procedure via shooting technique. The main key points noticed are: the heat transfer rate is more effective in the presence of heat generation than that in the absence of heat generation parameter. The temperature is enhanced in presence of variable thermal radiation while comparing absence of variable thermal radiation for large numerical values of Biot Number.

Shear-induced wrinkling in accelerating thin elastic discs

The wrinkling instabilities produced by in-plane angular accelerations in a rotating disc are discussed here in a particular limit of relevance to very thin plates. By coupling the classical linear elasticity solution for this configuration with the Föppl–von Kármán plate buckling equation, a fourth-order boundary-value problem with variable coefficients is obtained. The singular-perturbation character of the resulting problem arises from a combination of factors encompassing both the pre-stress (due to the spinning motion) and the geometry of the annular domain. With the help of a simplified multiple-scale perturbation method in conjunction with matched asymptotics, we succeed in capturing the dependence of the critical (wrinkling) acceleration on the instantaneous speed of the disc as well as other physical parameters. We show that the asymptotic predictions compare well with the results of direct numerical simulations of the original bifurcation problem. The limitations of the formulae obtained are also considered, and some practical suggestions for improving their accuracy are suggested.

Global structure of positive solutions for a fourth-order boundary value problem with singular data

Global structure of positive solutions for a fourth-order boundary value problem with singular data

Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter

Abstract A kind of fourth-order boundary value problem with eigenparameter-dependent boundary and transmission conditions is investigated. By constructing the characteristic function, we prove that the problems consist of a finite number of eigenvalues. We obtain that the number of eigenvalues of the problems not only depend on the order of the equation but also depend on the partition of the domain interval, the boundary conditions, and the eigenparameter-dependent transmission conditions.

Multiple solutions of a nonlinear elastic beam equation supported at two ends

In this paper, several new existence theorems on pairs of solutions are obtained for the fourth order boundary value problem ${u^{(4)}}(t) = h(t,u(t))$ subject to $u(0) = u(1) = u''(0) = u''(1) = 0$. Further, if $h(t,u) = \mu f(t,u)$, we prove that there exists $ \mu ^ * > 0$ such that there is no solution to the above boundary value problem for $\mu > {\mu ^ * }$, and that there are multiple solutions of the above boundary value problem for $0 < \mu < {\mu^ * }$.

Superconvergent postprocessing of the C1-conforming finite element method for fourth-order boundary value problems

We develop a very simple but efficient postprocessing technique for enhancing the accuracy of the C1-conforming finite element method for fourth-order boundary value problems. The key idea of the postprocessing technique is to add certain generalized Jacobi polynomials of degree larger than k to the Galerkin approximation of degree k. We prove that the postprocess improves the order of convergence of the Galerkin approximation under L2- and H2-norms. Numerical experiments are provided to illustrate the theoretical results. We also present computational results for a steady-state problem with to demonstrate the effectiveness of the postprocessing.

On the solvability of fourth-order boundary value problems with accretive operators

On the solvability of fourth-order boundary value problems with accretive operators

Positive solutions for a fourth-order p-Laplacian boundary value problem

Abstract The authors consider a fourth-order boundary value problem with a p-Laplacian appearing in both the equation and the boundary conditions. Upper and lower estimates for positive solutions to the problem are proved. Sufficient conditions for the existence and nonexistence of positive solutions to the problem are obtained and illustrated with an example.

Extraction and application of super-smooth cubic B-splines over triangulations

The space of C1 cubic Clough–Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a preferred choice in computer aided geometric design and isogeometric analysis. A B-spline basis is a locally supported basis that forms a convex partition of unity. In this paper, we explore several alternative C1 cubic spline spaces over triangulations equipped with a B-spline basis. They are defined over a Powell–Sabin refined triangulation and present different types of C2 super-smoothness. The super-smooth B-splines are obtained through an extraction process, i.e., they are expressed in terms of less smooth basis functions. These alternative spline spaces maintain the same optimal approximation power as Clough–Tocher splines. This is illustrated with a selection of numerical examples in the context of least squares approximation and finite element approximation for second and fourth order boundary value problems.

Stability of a flexible missile and asymptotics of the eigenvalues of fourth order boundary value problems

Fourth order problems with the differential equation y^{(4)}-(gy')'=lambda ^2y, where gin C^1[0,a] and a>0, occur in engineering on stability of elastic rods. They occur as well in aeronautics to describe the stability of a flexible missile. Fourth order Birkhoff regular problems with the differential equation y^{(4)}-(gy')'=lambda ^2y and eigenvalue dependent boundary conditions are considered. These problems have quadratic operator representations with non-self-adjoint operators. The first four terms of the asymptotics of the eigenvalues of the problems as well as those of the eigenvalues of the problem describing the stability of a flexible missile are evaluated explicitly.

A polynomial-augmented RBF collocation method for fourth-order boundary value problems

A polynomial-augmented RBF collocation method for fourth-order boundary value problems

An Existence Result of Positive Solutions for the Bending Elastic Beam Equations

This paper is concerned with the existence of positive solutions to the fourth-order boundary value problem u(4)(x)=f(x,u(x),u″(x)) on the interval [0, 1] with the boundary condition u(0)=u(1)=u″(0)=u″(1)=0, which models a statically bending elastic beam whose two ends are simply supported. Without assuming that the nonlinearity f(x, u, v) is nonnegative, an existence result of positive solutions is obtained under the inequality conditions that |(u, v)| is small or large enough. The discussion is based on the method of lower and upper solutions.

The reproducing kernel method for nonlinear fourth-order BVPs

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

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

&lt;abstract&gt;&lt;p&gt;In this paper, we are concerned with the existence of positive solutions for boundary value problems of nonlinear fourth-order differential equations&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{eqnarray*} &amp;amp;&amp;amp;u^{(4)}+c(x)u = \lambda a(x)f(u), \; \; x\in (0,1),\\ &amp;amp;&amp;amp;u(0) = u(1) = u''(0) = u''(1) = 0, \end{eqnarray*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;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 $.&lt;/p&gt;&lt;/abstract&gt;

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.

Multiple Nonnegative Solutions for a Class of Fourth-Order BVPs Via a New Topological Approach

In this paper, we study a class of fourth-order boundary value problems with integral boundary conditions. The nonlinearity may have time-singularity and change sign. Moreover, it satisfies general polynomial growth conditions. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions. An example of application illustrates the existence result.