Solutions and Green’s functions for some linear second-order three-point boundary value problems

Zhen, Zhao

doi:10.1016/j.camwa.2007.11.037

Cited by 19 publications

(12 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The existence of some three-point BVPs determined by non-linear DEs of order three is assessed in [26] by means of Green functions. In [27], for a class of ordinary differential equations of order two, a method is presented in order to find the Green functions for three-point BVPs. Paper [28] is dedicated to a specific class of third-order three-point BVPs.…”

Section: Introductionmentioning

confidence: 99%

Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations

2023

View full text Add to dashboard Cite

It is our main objective to find the critical load for three beams with cross sectional heterogeneity. Each beam has three supports, of which the intermediate one is a spring support. Determination of the critical load for these beams leads to three point boundary value problems (BVPs) associated with homogeneous boundary conditions—the mentioned BVPs constitute three eigenvalue problems. They are solved by using a novel solution strategy based on the Green functions that belong to these BVPs: the eigenvalue problems established for the critical load are transformed into eigenvalue problems governed by homogeneous Fredholm integral equations with kernels that can be given in closed forms provided that the Green function of each BVP is known. Then the eigenvalue problems governed by the Fredholm integral equations can be manipulated into algebraic eigenvalue problems solved numerically by using effective algorithms. It is an advantage of the way we attack these problems that the formalism established and the results obtained remain valid for homogeneous beams as well. The numerical results for the critical forces can be applied to solve some stability problems in the engineering practice.

show abstract

Section: Introductionmentioning

confidence: 99%

Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations

2023

View full text Add to dashboard Cite

show abstract

“…When it is about degenerated ordinary differential equation systems, the definition is provided in [21]. For some second-order ordinary differential equations, a technique is given in [25] for the construction of the Green functions for three-point boundary value problems. Furthermore, constructing the Green function is shown [19] for a special class of third-order three-point boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

Vibration of an axially loaded heterogeneous fixed–fixed beam with an intermediate roller support

Kiss

Szeidl

Messaoudi

2022

J Braz. Soc. Mech. Sci. Eng.

View full text Add to dashboard Cite

The present paper investigates how an axial load can change the natural frequencies of heterogeneous fixed–fixed beams with an intermediate roller support. The problem is treated as a three-point boundary value problem (eigenvalue problem) that is paired with homogeneous boundary conditions. The Green functions are determined for the unloaded and pre-loaded beams as well—in the later case, both for compression and tension. With these, the eigenvalue problems can be transformed into eigenvalue problems governed by a homogeneous Fredholm integral equations. It is then replaced by an algebraic eigenvalue problem, that is solved numerically with an effective solution algorithm which is based on the boundary element method.

show abstract

“…A large amount of theoretical work have been done on TBVPs 3–5 and the references given there. In addition, some numerical methods are applied to approximate the solutions of TBVPs due to the lack of methods of finding analytical solutions of TBVPs, and many efficient numerical methods have been adopted, such as the optimal homotopy asymptotic method, 6 shifted Jacobi spectral method, 7 Newton interpolating polynomials, 8 difference scheme, 9 the differential quadrature method, 10 Adomian decomposition method, 11 semi‐analytic numerical method, 12 and He's variational iteration method 13 …”

Section: Introductionmentioning

confidence: 99%

Reproducing kernel representation of the solution of second order linear three‐point boundary value problem

Bai

Leong

Dang

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper investigates the analytical solutions and numerical solutions of second order linear boundary value problems with three-point boundary conditions by means of the weak pre-orthogonal adaptive Fourier decomposition (W-POAFD) method introduced by Qian et al. Using the reproducing kernel functions of the Sobolev spaces on the unit interval [0, 1], we construct a dictionary to implement the W-POAFD. Based on orthonormalizataion, we can select the crucial parameters successively by adopting the weak maximal selection strategy of the W-POAFD method. Then the numerical solutions have "sparse" representations, which converge to the analytic solution with fast convergence rate. Some numerical experiments demonstrate the high accuracy and efficiency of the W-POAFD method in this three-point boundary value problem.

show abstract

Solutions and Green’s functions for some linear second-order three-point boundary value problems

Cited by 19 publications

References 18 publications

Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations

Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations

Vibration of an axially loaded heterogeneous fixed–fixed beam with an intermediate roller support

Reproducing kernel representation of the solution of second order linear three‐point boundary value problem

Contact Info

Product

Resources

About