scholarly journals Computational Modelling of Thermal Stability in a Reactive Slab with Reactant Consumption

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
O. D. Makinde
Keyword(s):  
Steady State ◽  
Perturbation Technique ◽  
Nonlinear Partial Differential Equation ◽  
Exothermic Reaction ◽  
Computational Modelling ◽  
Reaction Diffusion ◽  
Lower Surface ◽  
Diffusion Problem ◽  
One Step ◽  
Steady State Problem

This paper investigates both the transient and the steady state of a one-stepnth-order oxidation exothermic reaction in a slab of combustible material with an insulated lower surface and an isothermal upper surface, taking into consideration reactant consumption. The nonlinear partial differential equation governing the transient reaction-diffusion problem is solved numerically using a semidiscretization finite difference technique. The steady-state problem is solved using a perturbation technique together with a special type of the Hermite-Padé approximants. Graphical results are presented and discussed quantitatively with respect to various embedded parameters controlling the systems. The crucial roles played by the boundary conditions in determining the thermal ignition criticality are demonstrated.

Bifurcation and Stability Analysis of a One-Dimensional Diffusion-Autocatalytic Reaction System

1987 ◽  
Vol 42 (9) ◽  
pp. 994-1004 ◽  
Author(s):  
B. De Dier ◽  
F. Walraven ◽  
R. Janssen ◽  
P. Van Rompay ◽  
V. Hlavacek
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Reaction System ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
One Dimensional ◽  
Asymmetric Structures ◽  
Flux Boundary Conditions ◽  
Model Scheme ◽  
Steady State Problem

Results of a numerical analysis of a set of one-dimensional reaction -diffusion equations are presented. The basis of these equations is a model scheme of chemical reactions, involving auto-and cross-catalytic steps (“Brusselator”). The steady state problem is solved numerically, fully exploiting the properties of recently developed continuation codes. Bifurcation diagrams are constructed for zero flux boundary conditions. For a relatively large diffusivity of initial species the Brusselator displays a huge number of dissipative steady state structures. At low system lengths a mechanism of perturbed bifurcation may be percieved. Bifurcations coincide with turning points of asymmetric solution branches. Completely isolated solutions prove to exist as well. For the problem without limited diffusion of the initial species, a careful bifurcation analysis show s the existence of a number of higher order bifurcations. At some of these points asymmetric profiles emanate from other asymmetric structures. Bifurcation points and limit points do not necessarily coincide. Stability analysis shows that relatively few steady states are stable. Especially symmetric solutions are found to be stable.

scholarly journals ASYMPTOTIC BEHAVIOUR AND BLOW-UP FOR A NONLINEAR DIFFUSION PROBLEM WITH A NON-LOCAL SOURCE TERM

2004 ◽  
Vol 47 (2) ◽  
pp. 375-395 ◽  
Author(s):  
N. I. Kavallaris
Keyword(s):  
Steady State ◽  
Blow Up ◽  
Spatial Dimension ◽  
Diffusion Problem ◽  
Local Source ◽  
Primary 35K60 ◽  
Non Local ◽  
Steady State Problem ◽  
Steady State Solutions ◽  
Local Term

AbstractIn this work, the behaviour of solutions for the Dirichlet problem of the non-local equation$$ u_t=\varDelta(\kappa(u))+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^p},\quad \varOmega\subset\mathbb{R}^N,\quad N=1,2, $$is studied, mainly for the case where $f(s)=\mathrm{e}^{\kappa(s)}$. More precisely, the interplay of exponent $p$ of the non-local term and spatial dimension $N$ is investigated with regard to the existence and non-existence of solutions of the associated steady-state problem as well as the global existence and finite-time blow-up of the time-dependent solutions $u(x,t)$. The asymptotic stability of the steady-state solutions is also studied.AMS 2000 Mathematics subject classification: Primary 35K60. Secondary 35B40

scholarly journals ON THE SOLUTION OF THE STEADY‐STATE DIFFUSION PROBLEM FOR FERROMAGNETIC PARTICLES IN A MAGNETIC FLUID

2008 ◽  
Vol 13 (2) ◽  
pp. 233-240 ◽  
Author(s):  
Viktor Polevikov ◽  
Lutz Tobiska
Keyword(s):  
Steady State ◽  
Magnetic Fluid ◽  
Particle Concentration ◽  
Diffusion Problem ◽  
Gradient Magnetic Field ◽  
Concentration Problem ◽  
State Diffusion ◽  
Steady State Problem ◽  
Steady State Diffusion ◽  
Ferromagnetic Particles

A mathematical model for the diffusion process of ferromagnetic particles in a magnetic fluid is described. The unique solvability of the steady‐state particle concentration problem is investigated and an analytical expression for its solution is found. In case that the fluid is under the action of a high‐gradient magnetic field a Stefan‐type diffusion problem can arise. An algorithm for solving the Stefan‐type steady‐state problem is developed.

Existence and asymptotic profiles of the steady state for a diffusive epidemic model with saturated incidence and spontaneous infection mechanism

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xueying Sun ◽  
Renhao Cui
Keyword(s):  
Steady State ◽  
Epidemic Model ◽  
Perturbation Technique ◽  
Degree Theory ◽  
Reaction Diffusion ◽  
Linear Source ◽  
Singular Perturbation Technique ◽  
Infection Mechanism ◽  
Saturated Incidence ◽  
Spontaneous Infection

<p style='text-indent:20px;'>In this paper, we are concerned with a reaction-diffusion SIS epidemic model with saturated incidence rate, linear source and spontaneous infection mechanism. We derive the uniform bounds of parabolic system and obtain the global asymptotic stability of the constant steady state in a homogeneous environment. Moreover, the existence of the positive steady state is established. We mainly analyze the effects of diffusion, saturation and spontaneous infection on the asymptotic profiles of the steady state. These results show that the linear source and spontaneous infection can enhance the persistence of an infectious disease. Our mathematical approach is based on topological degree theory, singular perturbation technique, the comparison principles for elliptic equations and various elliptic estimates.</p>

scholarly journals Unsteady/Steady Free Convective Couette Flow of Reactive Viscous Fluid in a Vertical Channel Formed by Two Vertical Porous Plates

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Basant K. Jha ◽  
Ahmad K. Samaila ◽  
Abiodun O. Ajibade
Keyword(s):  
Steady State ◽  
Free Convection ◽  
Viscous Fluid ◽  
Couette Flow ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Vertical Channel ◽  
Regular Perturbation ◽  
Transient Solutions ◽  
Steady State Problem

This paper presents unsteady as well as steady-state free convection Couette flow of reactive viscous fluid in a vertical channel formed by two infinite vertical parallel porous plates. The motion of the fluid is induced due to free convection caused by the reactive nature of viscous fluid as well as the impulsive motion of one of the porous plates. The Boussinesq assumption is applied, and the nonlinear governing equations of motion and energy are developed. The time-dependent problem is solved using implicit finite difference method, and steady-state problem is solved by applying regular perturbation technique. During the course of computation, an excellent agreement was found between the well-known steady-state solutions and transient solutions at large value of time.

scholarly journals Global Asymptotic Stability of 3-Species Mutualism Models with Diffusion and Delay Effects

2009 ◽  
Vol 2009 ◽  
pp. 1-20 ◽  
Author(s):  
Chang-you Wang ◽  
Shu Wang ◽  
Xiang-ping Yan
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Upper And Lower Solutions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Steady State Solution ◽  
State Problem ◽  
Delay Effects ◽  
Steady State Problem

In this paper, the Lotka-Volterra 3-species mutualism models with diffusion and delay effects is investigated. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the unique positive steady-state solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition. Our approach to the problem is based on inequality skill and the method of the upper and lower solutions for a more general reaction—diffusion system. Finally, some numerical simulations are given to illustrate our results.

scholarly journals Reduced basis approximations of the solutions to spectral fractional diffusion problems

2020 ◽  
Vol 28 (3) ◽  
pp. 147-160
Author(s):  
Andrea Bonito ◽  
Diane Guignard ◽  
Ashley R. Zhang
Keyword(s):  
Computational Cost ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Quadrature Point ◽  
Reduced Basis ◽  
Fast Evaluation ◽  
Bottle Neck ◽  
Diffusion Problems

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

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