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.

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.

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.

scholarly journals A Class of Numerical Methods for the Solution of Fourth-Order Ordinary Differential Equations in Polar Coordinates

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Jyoti Talwar ◽  
R. K. Mohanty
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Methods ◽  
Iterative Method ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Polar Coordinates ◽  
Order Ordinary Differential Equation ◽  
Grid Points ◽  
Discretization Procedure

In this piece of work using only three grid points, we propose two sets of numerical methods in a coupled manner for the solution of fourth-order ordinary differential equation uiv(x)=f(x,u(x),u′(x),u′′(x),u′′′(x)), a<x<b, subject to boundary conditions u(a)=A0, u′(a)=A1, u(b)=B0, and u′(b)=B1, where A0, A1, B0, and B1 are real constants. We do not require to discretize the boundary conditions. The derivative of the solution is obtained as a byproduct of the discretization procedure. We use block iterative method and tridiagonal solver to obtain the solution in both cases. Convergence analysis is discussed and numerical results are provided to show the accuracy and usefulness of the proposed methods.

scholarly journals Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
R. Naz ◽  
F. M. Mahomed
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Power Law ◽  
Fourth Order ◽  
Applied Load ◽  
Mass Density ◽  
Load Case ◽  
Order Ordinary Differential Equation ◽  
General Power

We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass densityg(x), and the applied load denoted byf(u), a function of transverse displacementu(t,x). The complete Lie group classification is obtained for different forms of the variable lineal mass densityg(x)and applied loadf(u). The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms ofg(x). For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature wheng(x)is constant with variable applied loadf(u). For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions.

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