Existence of Multiple Positive Solutions for Third-Order Three-Point Boundary Value Problem

In this paper, we study the existence of positive solutions for a class of third-order three-point boundary value problem. By employing the fixed point theorem on cone, some new criteria to ensure the three-point boundary value problem has at least three positive solutions are obtained. An example illustrating our main result is given. Moreover, some previous results will be improved significantly in our paper.

Triple Positive Solutions to a Third-order Three-point Boundary Value Problem with p-Laplacian Operator

In this paper, we consider the three-point boundary value problem （φp（uˊˊ（t）））ˊ ＋a（t）f（t, u（t）, uˊ（t）, uˊˊ（t）） = 0, t ∈ [0, 1] subject to the boundary conditions u（0） =βuˊ（0）, uˊ（1） = αuˊ（η）, uˊˊ（0） = 0, where φp（s） = ｜s｜p？2s with p 〉 1, 0 〈 α, η 〈 1 and 0 ≤ β 〈 1. Applying a fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem.

Methodology for Extraction of Tunnel Cross-Sections Using Dense Point Cloud Data

Tunnel deformation monitoring is a crucial task to evaluate tunnel stability during the metro operation period.Terrestrial Laser Scanning(TLS)can collect high density and high accuracy point cloud data in a few minutes as an innovation technique,which provides promising applications in tunnel deformation monitoring.Here,an efficient method for extracting tunnel cross-sections and convergence analysis using dense TLS point cloud data is proposed.First,the tunnel orientation is determined using principal component analysis(PCA)in the Euclidean plane.Two control points are introduced to detect and remove the unsuitable points by using point cloud division and then the ground points are removed by defining an elevation value width of 0.5 m.Next,a z-score method is introduced to detect and remove the outlies.Because the tunnel cross-section’s standard shape is round,the circle fitting is implemented using the least-squares method.Afterward,the convergence analysis is made at the angles of 0°,30°and 150°.The proposed approach’s feasibility is tested on a TLS point cloud of a Nanjing subway tunnel acquired using a FARO X330 laser scanner.The results indicate that the proposed methodology achieves an overall accuracy of 1.34 mm,which is also in agreement with the measurements acquired by a total station instrument.The proposed methodology provides new insights and references for the applications of TLS in tunnel deformation monitoring,which can also be extended to other engineering applications.

An analytical model for the prediction of cross-section profile and mean roll radius in alloy bar rolling

In a round-oval-round pass rolling sequence, the cross-section profile of an outgoing workpiece was predicted first after getting the maximum spread. The concept ＂critical point on the contact boundary＂ was proposed and the coordinates of the critical point were solved. The equivalent contact section area was represented and the mean roll radius was determined. The validity of this model was examined by alloy bar rolling experiment and rigid-plastic FEM simulation. Compared with the existing models, the mean roll radius obtained by this model is similar to experiment data.

Existence of Positive Solutions for a Third-Order Three-Point Boundary Value Problem with Semipositone Nonlinearity *

The existence of positive solutions is investigated for following semipositone nonlinear third-order three-point BVP ω''(t) - λf(t,w(t)) = 0, 0 ≤ t ≤ 1, ω(0) = ω'(n) = ω'(1) = 0.

Third-order point contact approach for five-axis sculptured surface machining using non-ball-end tools(II): Tool positioning strategy

Based on the mathematical model describing the third-order approximation of the cutter envelope surface according to one given cutter location(CL),a tool positioning strategy is proposed for efficiently machining free-form surfaces with non-ball-end cutters.The optimal CL is obtained by adjusting the inclination and tilt angles of the cutter until its envelope surface and the design surface have the third-order contact at the cutter contact(CC)point,which results in a wide machining strip.The strategy can handle the constraints of machine joint angle limits,global collision avoidance and tool path smoothness in a nature way,and can be applied to general rotary cutters and complex surfaces.Numerical examples demonstrate that the third-order point contact approach can improve the machining strip width greatly as compared with the recently reported second-order one.

Third-order point contact approach for five-axis sculptured surface machining using non-ball-end tools (Ⅰ): Third-order approximation of tool envelope surface

In this paper, the geometric properties of a pair of line contact surfaces are investigated. Then, based on the observation that the cutter envelope surface contacts with the cutter surface and design surface along the characteristic curve and cutter contact (CC) path, respectively, a mathematical model describing the third-order approximation of the cutter envelope surface according to just one given cutter location (CL) is developed. It is shown that at the CC point both the normal curvature of the normal section of the cutter envelope surface and its derivative with respect to the arc length of the normal section can be determined by those of the cutter surface and design surface. This model characterizes the intrinsic relationship among the cutter surface, cutter envelope surface and design surface in the neighborhood of the CC point, and yields the mathematical foundation for optimally approximating the cutter envelope surface to the design surface by adjusting the cutter location.

The Existence and Multiplicity of Positive Solutions for a Third-order Three-point Boundary Value Problem

Abstract The existence of n positive solutions for a class of third-order three-point boundary value problems is investigated, where n is an arbitrary natural number. The main tool is Krasnosel'skii fixed point theorem on the cone.

Positive Solutions to a Singular Third-Order Three-Point Boundary Value Problem

In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positive parameter and 0 ≤ η 1 2 .By using the classical Krasnosel’skii’s fixed point theorem in cone,we obtain various new results on the existence of positive solution,and the solution is strictly increasing.Finally we give an example.

MULTIPLE POSITIVE SOLUTIONS TO A SINGULAR THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM

In this paper,we study a singular third-order three-point boundary value problem. By a fixed point theorem of cone expansion-compression type due to Krasnosel’skii,we obtain various new results on the existence of two positive solutions to the problem,whose coefficient is allowed to have suitable singularities. Finally,we give an example to verify our results.

EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR NONLINEAR THIRD-ORDER ROBIN BOUNDARY VALUE PROBLEM WITH A TURNING POINT

The paper aims to obtain existence and uniqueness of the solution as well as asymptotic estimate of the solution for singularly perturbed nonlinear thirdorder Robin boundary value problem with a turning point. In order to achieve this aim, existence and uniqueness of the solution for third-order nonlinear Robin boundary value problem is derived first based on the upper and lower solutions method under relatively weaker conditions. In this manner, the goal of this paper is gained by applying the existence and uniqueness results mentioned above.