scholarly journals Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Boumedièene Chentouf ◽  
Sabeur Mansouri
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Time Dependent ◽  
Flexible Structure ◽  
Main Concern ◽  
Boundary Stabilization ◽  
Dynamic Boundary Conditions ◽  
Well Posed ◽  
Do So ◽  
Load Mass

<p style='text-indent:20px;'>This article deals with the dynamic stability of a flexible cable attached at its top end to a cart and a load mass at its bottom end. The model is governed by a system of one partial differential equation coupled with two ordinary differential equations. Assuming that a time-dependent delay occurs in one boundary, the main concern of this paper is to stabilize the dynamics of the cable as well as the dynamical terms related to the cart and the load mass. To do so, we first prove that the problem is well-posed in the sense of semigroups theory provided that some conditions on the delay are satisfied. Thereafter, an appropriate Lyapunov function is put forward, which leads to the exponential decay of the energy as well as an estimate of the decay rate.</p>

MODELING OF GOLD DISPERSION IN GLASSY POLYMERS BY SMOLUCHOWSKI EQUATION

2002 ◽  
Vol 13 (09) ◽  
pp. 1301-1312 ◽  
Author(s):  
ZBIGNIEW J. GRZYWNA ◽  
JACEK STOLARCZYK
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Partial Differential Equation ◽  
Smoluchowski Equation ◽  
Glassy Polymers ◽  
Time Dependent ◽  
Acetone Vapor ◽  
Induced Stress ◽  
Theoretical Predictions ◽  
Good Agreement

A unidimensional diffusion in a potential field of induced stress is considered. The way from random walk (RW) to limiting partial differential equation (Smoluchowski equation) for standard and time dependent RW is shown. A technologically important case of gold dispersion in crystallizing polymer swollen by acetone vapor is analyzed. Theoretical predictions based on Smoluchowski equation with time dependent coefficients are found to be in very good agreement with experimental data.

Exponential Stabilization of a Transversely Vibrating Beam by Boundary Control Via Lyapunov’s Direct Method

1999 ◽  
Vol 123 (2) ◽  
pp. 195-200 ◽  
Author(s):  
Mehrdad P. Fard ◽  
Svein I. Sagatun
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Control ◽  
Direct Method ◽  
Nonlinear Partial Differential Equation ◽  
Nonlinear Function ◽  
Control Law ◽  
Boundary Stabilization ◽  
Free Transverse Vibration ◽  
Nonlinear Control Law

This paper discusses the boundary stabilization of a beam in free transverse vibration. The dynamics of the beam is presented by a nonlinear partial differential equation (PDE). Based on this model a nonlinear control law is constructed to stabilize the system. The control law is a nonlinear function of the slopes and velocity at the boundary of the beam. The novelty of this article is that it has been possible to exponentially stabilize a free transversely vibrating beam via boundary control without restoring to truncation of the model. This result is achieved while the coupling between longitudinal and transversal displacements has been taken into account.

scholarly journals Finite Analytic Method for One-Dimensional Nonlinear Consolidation under Time-Dependent Loading

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Dawei Cheng ◽  
Wenke Wang ◽  
Xi Chen ◽  
Zaiyong Zhang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Time Dependent ◽  
Analytic Method ◽  
Partial Differential ◽  
One Dimensional ◽  
Finite Analytic Method ◽  
Nonlinear Consolidation

For one-dimensional (1D) nonlinear consolidation, the governing partial differential equation is nonlinear. This paper develops the finite analytic method (FAM) to simulate 1D nonlinear consolidation under different time-dependent loading and initial conditions. To achieve this, the assumption of constant initial effective stress is not considered and the governing partial differential equation is transformed into the diffusion equation. Then, the finite analytic implicit scheme is established. The convergence and stability of finite analytic numerical scheme are proven by a rigorous mathematical analysis. In addition, the paper obtains three corrected semianalytical solutions undergoing suddenly imposed constant loading, single ramp loading, and trapezoidal cyclic loading, respectively. Comparisons of the results of FAM with the three semianalytical solutions and the result of FDM, respectively, show that the FAM can obtain stable and accurate numerical solutions and ensure the convergence of spatial discretization for 1D nonlinear consolidation.

scholarly journals Fick's second law transformed: one path to cloaking in mass diffusion

2013 ◽  
Vol 10 (83) ◽  
pp. 20130106 ◽  
Author(s):  
S. Guenneau ◽  
T. M. Puvirajesinghe
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Mass Concentration ◽  
Heat Diffusion ◽  
Time Dependent ◽  
Parabolic Partial Differential Equation ◽  
Second Law ◽  
Water Based ◽  
Potential Applications ◽  
Effective Medium Approach

Here, we adapt the concept of transformational thermodynamics, whereby the flux of temperature is controlled via anisotropic heterogeneous diffusivity, for the diffusion and transport of mass concentration. The n -dimensional, time-dependent, anisotropic heterogeneous Fick's equation is considered, which is a parabolic partial differential equation also applicable to heat diffusion, when convection occurs, for example, in fluids. This theory is illustrated with finite-element computations for a liposome particle surrounded by a cylindrical multi-layered cloak in a water-based environment, and for a spherical multi-layered cloak consisting of layers of fluid with an isotropic homogeneous diffusivity, deduced from an effective medium approach. Initial potential applications could be sought in bioengineering.

scholarly journals On an improved computational solution for the 3D HCIR PDE in finance

2019 ◽  
Vol 27 (3) ◽  
pp. 207-230 ◽  
Author(s):  
Fazlollah Soleymani ◽  
Ali Akgül ◽  
Esra Karatas Akgül
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Three Dimensional ◽  
Time Dependent ◽  
Computational Domain ◽  
Order Of Accuracy ◽  
Spatial Variables ◽  
Stability Bound ◽  
Computational Solution

AbstractThe aim of this work is to tackle the three–dimensional (3D) Heston– Cox–Ingersoll–Ross (HCIR) time–dependent partial differential equation (PDE) computationally by employing a non–uniform discretization and gathering the finite difference (FD) weighting coe cients into differentiation matrices. In fact, a non–uniform discretization of the 3D computational domain is employed to achieve the second–order of accuracy for all the spatial variables. It is contributed that under what conditions the proposed procedure is stable. This stability bound is novel in literature for solving this model. Several financial experiments are worked out along with computation of the hedging quantities Delta and Gamma.

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