Existence of Solutions for Fourth Order Nonlocal Boundary Value Problems

2006 ◽  
Vol 13 (3) ◽  
pp. 473-484
Author(s):  
Johnny Henderson ◽  
Ding Ma
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Existence Of Solutions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence Results ◽  
Order Ordinary Differential Equation

Abstract Uniqueness implies existence results are obtained for solutions of the fourth order ordinary differential equation, 𝑦(4) = 𝑓(𝑥, 𝑦, 𝑦′, 𝑦″, 𝑦‴), satisfying 5-point, 4-point and 3-point nonlocal boundary conditions.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

scholarly journals Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Manfred Möller ◽  
Bertin Zinsou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Imaginary Axis ◽  
Order Ordinary Differential Equation ◽  
Eigenvalue Parameter ◽  
Upper Half Plane ◽  
Parameter Dependent ◽  
Asymptotics Of The Eigenvalues

Considered is a regular fourth order ordinary differential equation which depends quadratically on the eigenvalue parameterλand which has separable boundary conditions depending linearly onλ. It is shown that the eigenvalues lie in the closed upper half plane or on the imaginary axis and are symmetric with respect to the imaginary axis. The first four terms in the asymptotic expansion of the eigenvalues are provided.

Fourth Order Boundary Value Problems and Comparison Theorems

1961 ◽  
Vol 13 ◽  
pp. 625-638 ◽  
Author(s):  
John H. Barrett
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Comparison Theorems ◽  
Point Boundary ◽  
Image Position

This paper is primarily concerned with the existence of solutions of the fourth-order self-adjoint differential equation(1)(where r(x) > 0, q(x) ≥ 0, p(x) ≥ 0 and all three coefficients are continuous on [a, ∞)) and one of the two-point boundary conditions:(2)or(3)the subscript notation for any solution y(x) denoting:(4)

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