A Fast New Algorithm for Solving a Nonlinear Beam Equation under Nonlinear Boundary Conditions

2017 ◽  
Vol 72 (5) ◽  
pp. 397-400 ◽  
Author(s):  
Chein-Shan Liu ◽  
Botong Li
Keyword(s):  
Boundary Conditions ◽  
Numerical Solution ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
Nonlinear Boundary Conditions ◽  
Beam Equation ◽  
Test Functions ◽  
Nonlinear Beam ◽  
Expansion Coefficients ◽  
Sinusoidal Functions

AbstractFor the problem of a nonlinear beam equation under nonlinear boundary conditions of moments, a fast iterative method is developed by transforming the ordinary differential equation into an integral one. The sinusoidal functions are used subtly as test functions as well as the bases of numerical solution in the calculation. Due to the orthogonality of the sinusoidal functions, the expansion coefficients of numerical solution in closed form can be found easily. Hence, the iterative scheme converges very fast to find numerical solutions with high accuracy.

Frequency Response of Fluid Lines With Nonlinear Boundary Conditions

1971 ◽  
Vol 93 (3) ◽  
pp. 365-372 ◽  
Author(s):  
R. D. Strunk
Keyword(s):  
Boundary Conditions ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Frequency Response ◽  
Nonlinear Equations ◽  
Nonlinear Boundary ◽  
Harmonic Distortion ◽  
Nonlinear Boundary Conditions ◽  
Qualitative Information ◽  
Method Of Solution

The harmonic distortion generated when a fluid line is terminated by a nonlinear orifice characteristic is analyzed by using a perturbation method of solution. The perturbation method is shown to be representative of the true phenomenon and to give very good quantitative as well as qualitative information by comparing the results to a numerical solution of the nonlinear equations. The results presented describe the distortion phenomenon as a function of several dimensionless ratios.

scholarly journals Attractor of Beam Equation with Structural Damping under Nonlinear Boundary Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Danxia Wang ◽  
Jianwen Zhang ◽  
Yinzhu Wang ◽  
Sufang Zhang
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Global Solutions ◽  
Nonlinear Boundary Conditions ◽  
Beam Equation ◽  
Rotational Inertia ◽  
Absorbing Set ◽  
Viscous Effect ◽  
Structural Damping ◽  
Longtime Behavior

Simultaneously, considering the viscous effect of material, damping of medium, and rotational inertia, we study a kind of more general Kirchhoff-type extensible beam equationutt-uxxtt+uxxxx-σ(∫0l‍(ux)2dx)uxx-ϕ(∫0l‍(ux)2dx)uxxt=q(x), in [0,L]×R+with the structural damping and the rotational inertia term. Little attention is paid to the longtime behavior of the beam equation under nonlinear boundary conditions. In this paper, under nonlinear boundary conditions, we prove not only the existence and uniqueness of global solutions by prior estimates combined with some inequality skills, but also the existence of a global attractor by the existence of an absorbing set and asymptotic compactness of corresponding solution semigroup. In addition, the same results also can be proved under the other nonlinear boundary conditions.

scholarly journals Monotone Positive Solutions for an Elastic Beam Equation with Nonlinear Boundary Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Jian-Ping Sun ◽  
Xian-Qiang Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Quadratic Function ◽  
Elastic Beam ◽  
Nonlinear Boundary Conditions ◽  
Beam Equation ◽  
Iterative Schemes ◽  
Elastic Beam Equation ◽  
Monotone Positive Solutions

This paper is concerned with the existence of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. By applying monotone iteration method, we not only obtain the existence of monotone positive solutions but also establish iterative schemes for approximating the solutions. It is worth mentioning that these iterative schemes start off with zero function or quadratic function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results obtained.

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