Steady Motion of a Thread Over a Rotating Roller

1994 ◽  
Vol 61 (1) ◽  
pp. 16-22 ◽  
Author(s):  
R.-J. Yang
Keyword(s):  
Circular Cylinder ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Analytical Solutions ◽  
Boundary Value ◽  
Steady Motion ◽  
Dynamic Equations ◽  
Initial Value ◽  
Perturbation Result ◽  
The Third

Dynamic equations of steady motion governing the behavior of threads over rotating arbitrary-axisymmetric rollers are derived. Various types of boundary conditions resulting in initial value or boundary value problems are discussed. Analytical solutions for the case of a circular cylinder are found. Two of the integrals obtained are exact. The third one, being a perturbation result, is thus approximate. Comparisons of results for a circular cylinder with those for tapered and parabolic rollers are made.

The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equations

2005 ◽  
Vol 461 (2061) ◽  
pp. 2965-2984 ◽  
Author(s):  
Beatrice Pelloni
Keyword(s):  
Integral Representation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Series Representation ◽  
Point Boundary ◽  
Third Order ◽  
Linear Evolution ◽  
The Third ◽  
Order Case

We use a spectral transform method to study general boundary-value problems for third-order, linear, evolution partial differential equations with constant coefficients, posed on a finite space domain. We show how this method yields a simple characterization of the discrete spectrum of the associated spatial differential operator, and discuss the obstructions that arise when trying to represent the solution of such a problem as a series of exponential functions. We first review the theory for second-order two-point boundary-value problems, and present an alternative way to derive the classical series representation, as well as an equivalent integral representation, which generally involves complex contours. We illustrate the advantages of the integral representation by studying in some detail the case where Robin-type boundary conditions are prescribed. We then consider the third-order case and show that the integral representation is in general not equivalent to a discrete series representation, justifying a posteriori the failure of some of the classical approaches. We illustrate the third-order case in detail, using the example of the equation q t + q xxx =0 for various types of boundary conditions. In contrast with the second-order case, the qualitative properties of the spectrum of the associated spatial differential operator depend in this case not only on the equation but also on the type of boundary conditions. In particular, the solution appears to admit a series representation only when the prescribed boundary conditions couple the two endpoints of the interval.

scholarly journals SOME ANALOGUES OF A THEOREM OF PEANO FOR BOUNDARY VALUE PROBLEM

1991 ◽  
Vol 22 (1) ◽  
pp. 83-98
Author(s):  
RICK BRANTLEY ◽  
JOHNNY HENDERSON
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Initial Value Problems ◽  
Boundary Value ◽  
Boundary Values ◽  
Initial Value ◽  
Boundary Points

Under certain conditions, solutions of boundary value problems for $y'''=f (x,y, y', y'')$ are differentiated with respect to boundary conditions, both boundary points and boundary values. The results obtained are analogues of one of Peano's theorems on initial value problems.

An Iterative Shooting Method for the Solution of Higher Order Boundary Value Problems with Additional Boundary Conditions

2015 ◽  
Vol 235 ◽  
pp. 31-36
Author(s):  
Renata Filipowska
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Shooting Method ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Initial Value ◽  
Unknown Parameters ◽  
Sensitivity Functions ◽  
Fourth Order Differential Equation

This paper treats an iterative shooting method based on sensitivity functions for solving non–linear two–point boundary value problems (BVPs), in the form of a fourth–order differential equation and more than four boundary conditions. The solution of this problem is possible only when the equation includes the required number of unknown parameters. In order to use this method, it is necessary to convert the BVP to an appropriate initial value problem (IVP). The presented method has been illustrated with a numerical example.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

scholarly journals Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ji Lin ◽  
Yuhui Zhang ◽  
Chein-Shan Liu
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Initial Conditions ◽  
Boundary Value ◽  
Accurate Solution ◽  
Point Boundary ◽  
Third Order ◽  
Boundary Shape ◽  
Free Function ◽  
New Algorithms

AbstractFor nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

Sign in / Sign up

Export Citation Format

Share Document