scholarly journals Controlling the dynamics of Burgers equation with a high-order nonlinearity

2004 ◽  
Vol 2004 (62) ◽  
pp. 3321-3332 ◽  
Author(s):  
Nejib Smaoui
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Nonlinear Boundary ◽  
Burgers Equation ◽  
High Order ◽  
Time Stepping ◽  
Backward Euler ◽  
High Order Nonlinearity ◽  
Steady State Solutions

We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e.,ut=νuxx−unux+mu+h(x)). We show existence of an absorbing ball inL2[0,1]and uniqueness of steady state solutions for all integern≥1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. Numerical results using Chebychev collocation method with backward Euler time stepping scheme are presented for both the controlled and the uncontrolled equations illustrating the performance of the controller and supporting the analytical results.

scholarly journals On the global attractors in one mathematical model of antiviral immunity

2020 ◽  
Author(s):  
Alexei Tsygvintsev
Keyword(s):  
Mathematical Model ◽  
Steady State ◽  
Immune System ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Antiviral Immunity ◽  
Global Attractors ◽  
Viral Pathogens ◽  
Basic Reproductive Ratio ◽  
The Mathematical Model

AbstractWe consider the mathematical model introduced by Batholdy et al. [1] describing the interaction between viral pathogens and immune system. We prove the global asymptotic stability of the infection steady-state if the basic reproductive ratio R0 is greater than unity. That solves the conjecture announced in [7].

Steady-State Dynamics of Smooth and Non-Smooth Systems With Strong Nonlinearities

2003 ◽  
Author(s):  
Xiaoai Jiang ◽  
Alexander F. Vakakis
Keyword(s):  
Steady State ◽  
Vibration Isolation ◽  
High Order ◽  
State Behavior ◽  
Energy Pumping ◽  
Nonlinear Energy Pumping ◽  
Clearance Nonlinearity ◽  
High Order Nonlinearity ◽  
Odd Nonlinearity ◽  
Nonlinear Energy

The nonlinear energy sinks (NESs) with strong essential stiffness nonlinearities have been shown to result in vibration isolation in the studied system. In comparison, we also studied the steady-state dynamic response of a system with its smooth high-order odd nonlinearity replaced with the best fitted nonsmooth “clearance nonlinearity”. The analysis was based on the complexification technique and the separation of the dynamic terms into the “slow-varying” and the “fast-varying” components. We found that the steady-state behavior of a system with the non-smooth NES resembles that of the system with the smooth high-order nonlinearity, preserving the nonlinear energy-pumping feature. This finding paves the way for constructing practical NESs and applying them to practical vibration-isolation problems.

Dynamic Stability of a Vibrating Hammer

1969 ◽  
Vol 91 (4) ◽  
pp. 1175-1179 ◽  
Author(s):  
C. C. Fu ◽  
B. Paul
Keyword(s):  
Computer Simulation ◽  
Steady State ◽  
Asymptotic Stability ◽  
Drill Bit ◽  
Asymptotically Stable ◽  
Stability Of Motion ◽  
Rock Drill ◽  
The Stability ◽  
Energy Absorbing ◽  
Steady State Solutions

This paper deals with the stability of motion of an elastically suspended vibrating hammer that impacts upon an energy absorbing surface. The energy absorber could represent, for example, a rock drill bit or drill steel, or a spike being driven by the hammer. The problem is intrinsically nonlinear because the instant of impact depends upon the motion of the hammer. “Simple steady-state solutions” are derived, and their asymptotic stability is examined. Regions in which the analytically constructed simple solutions are asymptotically stable are determined in parameter space. Results have been checked by a digital computer simulation.

scholarly journals On Steady MHD Thermally Radiating and Reacting Thermosolutal Viscous Flow through a Channel with Porous Medium

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Promise Mebine ◽  
Rhoda H. Gumus
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Boundary Value Problem ◽  
Viscous Flow ◽  
Nonlinear Boundary ◽  
Approximate Solutions ◽  
Nonlinear Boundary Value Problem ◽  
Arrhenius Kinetics ◽  
Flow Through ◽  
Steady State Solutions

This paper investigates steady-state solutions to MHD thermally radiating and reacting thermosolutal viscous flow through a channel with porous medium. The reaction is assumed to be strongly exothermic under generalized Arrhenius kinetics, neglecting the consumption of the material. Approximate solutions are constructed for the governing nonlinear boundary value problem using WKBJ approximations. The results, which are discussed with the aid of the dimensionless parameters entering the problem, are seen to depend sensitively on the parameters.

A High Order Well-Balanced Finite Volume WENO Scheme for a Blood Flow Model in Arteries

2017 ◽  
Vol 7 (4) ◽  
pp. 852-866
Author(s):  
Zhonghua Yao ◽  
Gang Li ◽  
Jinmei Gao
Keyword(s):  
Blood Flow ◽  
Steady State ◽  
Finite Volume ◽  
Flow Model ◽  
Source Term ◽  
High Order ◽  
Weno Scheme ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
Steady State Solutions

AbstractThe numerical simulations for the blood flow in arteries by high order accurate schemes have a wide range of applications in medical engineering. The blood flow model admits the steady state solutions, in which the flux gradient is non-zero and is exactly balanced by the source term. In this paper, we present a high order finite volume weighted essentially non-oscillatory (WENO) scheme, which preserves the steady state solutions and maintains genuine high order accuracy for general solutions. The well-balanced property is obtained by a novel source term reformulation and discretisation, combined with well-balanced numerical fluxes. Extensive numerical experiments are carried out to verify well-balanced property, high order accuracy, as well as good resolution for smooth and discontinuous solutions.

Convergence of a homotopy finite element method for computing steady states of Burgers’ equation

2019 ◽  
Vol 53 (5) ◽  
pp. 1629-1644 ◽  
Author(s):  
Wenrui Hao ◽  
Yong Yang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Steady State ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Steady State Solution ◽  
State Problem ◽  
Steady State Problem ◽  
Steady State Solutions ◽  
Element Method

In this paper, the convergence of a homotopy method (1.1) for solving the steady state problem of Burgers’ equation is considered. When ν is fixed, we prove that the solution of (1.1) converges to the unique steady state solution as ε → 0, which is independent of the initial conditions. Numerical examples are presented to confirm this conclusion by using the continuous finite element method. In contrast, when ν = ε →, numerically we show that steady state solutions obtained by (1.1) indeed depend on initial conditions.

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