scholarly journals One-dimensional pattern formation in a model of burning

1993 ◽  
Vol 35 (2) ◽  
pp. 145-173 ◽  
Author(s):  
Lawrence K. Forbes
Keyword(s):  
Thermal Conduction ◽  
Linear Equations ◽  
Series Representation ◽  
Reaction Diffusion ◽  
Temperature Sensitive ◽  
Governing Equations ◽  
Second Stage ◽  
Nonuniqueness Of Solutions ◽  
The Stability ◽  
Non Linear Equations

AbstractWe discuss a model of a burning process, essentially due to Sal'nikov, in which a substrate undergoes a two-stage decay through some intermediate chemical to form a final product. The second stage of the process occurs at a temperature-sensitive rate, and is also responsible for the production of heat. The effects of thermal conduction are included, and the intermediate chemical is assumed to be capable of diffusion through the decomposing substrate. The governing equations thus form a reaction-diffusion system, and spatially inhomogeneous behaviour is therefore possible.This paper is concerned with stationary patterns of temperature and chemical concentration in the model. A numerical method for the solution of the governing equations is outlined, and makes use of a Fourier-series representation of the pattern. The question of the stability of these patterns is discussed in detail, and a linearised solution is presented, which is valid for patterns of very small amplitude. The results of accurate solutions to the fully non-linear equations are discussed, and compared with the predictions of the linearised theory. Parameter regions in which there exists genuine nonuniqueness of solutions are identified.

scholarly journals On Classical Solutions for A Kuramoto–Sinelshchikov–Velarde-Type Equation

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 77 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Thermal Conduction ◽  
Type Equation ◽  
Reaction Diffusion ◽  
Classical Solutions ◽  
Well Posedness ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Cauchy Problem ◽  
The Stability ◽  
Plane Flame

The Kuramoto–Sinelshchikov–Velarde equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking into account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

scholarly journals Time-step limits for stable solutions of the ice-sheet equation

1996 ◽  
Vol 23 ◽  
pp. 74-85 ◽  
Author(s):  
Richard C. A. Hindmarsh ◽  
Antony J. Payne
Keyword(s):  
Linear Equations ◽  
Computational Cost ◽  
Ice Sheet ◽  
Two Dimensions ◽  
Picard Iteration ◽  
Time Step ◽  
Iterated Maps ◽  
A New Technique ◽  
The Stability ◽  
Non Linear Equations

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared.The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

scholarly journals Magnetohydrodynamic go-water nanofluid flow and heat transfer between two parallel moving disks

2018 ◽  
Vol 22 (1 Part B) ◽  
pp. 383-390
Author(s):  
Mohammadreza Azimi ◽  
Rouzbeh Riazi
Keyword(s):  
Similarity Transformation ◽  
Linear Equations ◽  
Squeezing Flow ◽  
Governing Equations ◽  
Nanofluid Flow ◽  
Parallel Disks ◽  
Flow And Heat Transfer ◽  
Runge Kutta Method ◽  
Lower Disk ◽  
Non Linear Equations

The unsteady MHD squeezing flow of nanofluid with different type of nanoparticles between two parallel disks is discussed. The governing equations, continuity, momentum, energy, and concentration for this problem are reduced to coupled non-linear equations by using a similarity transformation. It has been found that for contracting motion of upper disk combined with suction at lower disk, effects of increasing absolute values of squeeze parameter are quite opposite to the case of expanding motion. In this case, radial velocity near upper disk decreases while near the lower disk an accelerated radial flow is observed. The comparison between analytical results and numerical ones achieved by forth order Runge-Kutta method, assures us about the validity and accuracy of problem.

scholarly journals Time-step limits for stable solutions of the ice-sheet equation

1996 ◽  
Vol 23 ◽  
pp. 74-85 ◽  
Author(s):  
Richard C. A. Hindmarsh ◽  
Antony J. Payne
Keyword(s):  
Linear Equations ◽  
Computational Cost ◽  
Ice Sheet ◽  
Two Dimensions ◽  
Picard Iteration ◽  
Time Step ◽  
Iterated Maps ◽  
A New Technique ◽  
The Stability ◽  
Non Linear Equations

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared. The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

Stability and vibration of a non-linear vehicle and passenger system

2002 ◽  
Vol 216 (2) ◽  
pp. 109-116 ◽  
Author(s):  
K Yu ◽  
A C J Luo ◽  
Y He
Keyword(s):  
Equations Of Motion ◽  
Linear Equations ◽  
Dynamic Responses ◽  
Spring Stiffness ◽  
Torsional Spring ◽  
Non Linear ◽  
Linear Equations Of Motion ◽  
Safety Belts ◽  
The Stability ◽  
Non Linear Equations

A non-linear dynamic model to predict the passenger's response in a vehicle travelling on a rough pavement surface (or a rough terrain) is developed. The corresponding equilibrium and stability are investigated through the non-linear equations of motion for a vehicle and passenger system with impacts. The stability with respect to the torsional spring stiffness of safety belts is illustrated. Based on such a stability condition, the dynamic responses for the vehicle and passenger system with and without impacts are simulated numerically. This investigation shows that a strong torsional spring is required in order to reduce the vibration amplitudes of passengers and to avoid impacts between the vehicle and passenger.

scholarly journals Effects of Conduction Variation on MHD Natural Convection Flow Along a vertical Flat Plate

2016 ◽  
Vol 34 ◽  
pp. 63-73
Author(s):  
AKM Safiqul Islam ◽  
MA Alim ◽  
Md Rezaul Karim ◽  
ATM M Rahman
Keyword(s):  
Flat Plate ◽  
Linear Equations ◽  
Convection Flow ◽  
Temperature Profiles ◽  
Governing Equations ◽  
Free Convection Flow ◽  
Boundary Layer Equations ◽  
Implicit Finite Difference ◽  
Non Linear Equations ◽  
Vertical Flat Plate

This paper reports free convection flow along a vertical flat plate with conduction variation on magneto hydrodynamic (MHD) effects. The governing equations with associated boundary conditions reduce to local non-similarity boundary layer equations for this phenomenon are converted to dimensionless forms using a suitable transformation. The transformed non-linear equations are then solved using the implicit finite difference method together with Keller-box technique. Numerical results of the velocity and temperature profiles, skin friction and surface temperature profiles for different values of the magnetic parameter, the Prandtl number and the conduction variation parameters are presented graphically. Detailed discussion is given for the effect of the aforementioned parameters.GANIT J. Bangladesh Math. Soc.Vol. 34 (2014) 63-73

II.—The Stability of Solutions of Non-linear Difference-differential Equations

1950 ◽  
Vol 63 (1) ◽  
pp. 18-26 ◽  
Author(s):  
E. M. Wright
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Linear Perturbation ◽  
Stability Of Solutions ◽  
Essential Step ◽  
Non Linear ◽  
Independent Variable ◽  
The Stability ◽  
Similar Theory ◽  
Non Linear Equations

SynopsisPoincaré, Liapounoff, Perron and others have proved theorems about the order of smallness, as the independent variable tends to + ∞, of solutions of differential equations with non-linear perturbation terms. A similar theory exists for difference equations. By a simple use of transforms, we here extend the theorems, with suitable modifications, to difference-differential equations. The results are an essential step in the development of a general theory of non-linear equations of this type.

scholarly journals Adapted block hybrid method for the numerical solution of Duffing equations and related problems

2021 ◽  
Vol 6 (12) ◽  
pp. 14013-14034
Author(s):  
Ridwanulahi Iyanda Abdulganiy ◽  
◽  
Shiping Wen ◽  
Yuming Feng ◽  
Wei Zhang ◽  
...  
Keyword(s):  
Linear Equations ◽  
Real Life ◽  
Step Length ◽  
Stability And Convergence ◽  
Duffing Equations ◽  
Numerical Integrator ◽  
Non Linear ◽  
Life Phenomena ◽  
The Stability ◽  
Non Linear Equations

<abstract><p>Problems of non-linear equations to model real-life phenomena have a long history in science and engineering. One of the popular of such non-linear equations is the Duffing equation. An adapted block hybrid numerical integrator that is dependent on a fixed frequency and fixed step length is proposed for the integration of Duffing equations. The stability and convergence of the method are demonstrated; its accuracy and efficiency are also established.</p></abstract>

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