A REACTION INFILTRATION PROBLEM: EXISTENCE, UNIQUENESS, AND REGULARITY OF SOLUTIONS IN TWO SPACE DIMENSIONS

1995 ◽  
Vol 05 (05) ◽  
pp. 599-618 ◽  
Author(s):  
JOHN CHADAM ◽  
XINFU CHEN ◽  
ROBERTO GIANNI ◽  
RICCARDO RICCI
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Elliptic Equation ◽  
Partial Regularity ◽  
Regularity Of Solutions ◽  
Infinite Strip ◽  
Two Dimensional ◽  
Finite Strip ◽  
Two Space Dimensions

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. We establish global existence, uniqueness and regularity of the solution in a two-dimensional finite strip (−M, M)×(0, 1) and the existence and partial regularity of solutions in an infinite strip (−∞, ∞)×(0, 1).

scholarly journals A reaction infiltration problem: classical solutions

1997 ◽  
Vol 40 (2) ◽  
pp. 275-291 ◽  
Author(s):  
John Chadam ◽  
Xinfu Chen ◽  
Roberto Gianni ◽  
Riccardo Ricci
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Bounded Domains ◽  
Global Classical Solution

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. Existence and uniqueness of a global classical solution is proved for bounded domains Ω⊂RN, under suitable boundary conditions.

Numerical Solution of Parabolic Equation on a 2-D Infinite Strip with Composite Hermite-Legendre Spectral Method

2013 ◽  
Vol 647 ◽  
pp. 875-879
Author(s):  
Ting Jing Zhao
Keyword(s):  
Differential Equation ◽  
Parabolic Equation ◽  
Model Problem ◽  
Convergence Result ◽  
Parabolic Differential Equation ◽  
Infinite Strip ◽  
Two Dimensional ◽  
Numerical Tests ◽  
Legendre Spectral Method ◽  
Legendre Galerkin Method

For solving numerically parabolic differential equation on a two-dimensional infinite strip, composite Hermite-Legendre Galerkin method is proposed in this article. By making use of stabilised scaled factor, the proposed method achieves stability. We also establish the convergence result for the proposed method. Numerical tests conduct for the model problem. It is shown that the proposed method is efficient.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

scholarly journals Extreme Analysis of a Random Ordinary Differential Equation

2014 ◽  
Vol 51 (4) ◽  
pp. 1021-1036 ◽  
Author(s):  
Jingchen Liu ◽  
Xiang Zhou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Elliptic Equation ◽  
Gaussian Process ◽  
Stochastic System ◽  
Random Coefficient ◽  
Asymptotic Approximations ◽  
Tail Probabilities ◽  
One Dimensional ◽  
Spatially Varying

In this paper we consider a one dimensional stochastic system described by an elliptic equation. A spatially varying random coefficient is introduced to account for uncertainty or imprecise measurements. We model the logarithm of this coefficient by a Gaussian process and provide asymptotic approximations of the tail probabilities of the derivative of the solution.

scholarly journals Initiation of thermal explosion by intense light: criticality dependence on data and parameters

1987 ◽  
Vol 28 (3) ◽  
pp. 287-295 ◽  
Author(s):  
K. K. Tam
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Steady State Solution ◽  
Linear Ordinary Differential Equation ◽  
Critical Parameters ◽  
Linear Parabolic Equation ◽  
Thermal Ignition ◽  
Non Linear ◽  
Intense Light

AbstractA model for thermal ignition by intense light is studied. The governing non-linear parabolic equation is linearized in a two-step manner with the aid of a non-linear ordinary differential equation which captures the salient features of the non-linear parabolic equation. The critical parameters are computed from the steady-state solution of the ordinary differential equation, which can be obtained without actually solving the equation. Comparison with available data shows that the present method yields good results.

BIFURCATION STRUCTURE IN A MODEL OF A FREQUENCY MODULATED CO2 LASER

1993 ◽  
Vol 03 (01) ◽  
pp. 97-111 ◽  
Author(s):  
C. MIRA ◽  
I. DJELLIT
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Organization Structure ◽  
The Other ◽  
Two Dimensional ◽  
Periodic Excitation ◽  
Bifurcation Structure ◽  
Higher Harmonic ◽  
Bifurcation Structures ◽  
Qualitative Changes

This paper concerns the bifurcation properties of a model of a frequency modulated CO 2 laser in the form of a two-dimensional ordinary differential equation with a parametric periodic excitation. These properties are related to the bifurcation curves organization (structure) in a parameter plane (amplitude, frequency of the modulation). Two basic bifurcation structures appear, one concerning the higher harmonic solutions, the other the subharmonic solutions. Qualitative changes of these structures are considered when a third parameter (pump parameter) is varied.

BEHAVIOR OF HARMONICS GENERATED BY A DUFFING TYPE EQUATION WITH A NONLINEAR DAMPING: PART II

2009 ◽  
Vol 19 (04) ◽  
pp. 1227-1254
Author(s):  
HÉDI KHAMMARI ◽  
CHRISTIAN MIRA
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Study ◽  
Type Equation ◽  
Nonlinear Damping ◽  
Two Dimensional ◽  
Cubic Nonlinearity ◽  
Fold Bifurcation ◽  
Present Part ◽  
Nonautonomous Ordinary Differential Equation

This paper is Part II of an earlier paper dealing with the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, and an external periodical excitation of period τ = 2π/ω (amplitude E). In the absence of this excitation, this equation of Duffing type does not give rise to self-oscillations. Part I was essentially devoted to analyze the harmonics behavior of period τ solutions, more precisely the behavior of rank-p harmonics according to the points of the parameter plane (ω,E). The present Part II deals with period kτ solutions related to a cascade of closed fold bifurcation curves related to fractional harmonics p/k, k = 3, p = 3,4,…. With respect to the organization of bifurcation curves associated with rank-p harmonics of the basic period τ, this study shows that the situation is a lot more complex for the sequence of bifurcation curves related to rank-p/3 harmonics.

scholarly journals Toward a Wong–Zakai Approximation for Big Order Generators

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1893
Author(s):  
Rémi Léandre
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Stochastic Differential Equation ◽  
Finite Energy ◽  
Higher Order ◽  
Semi Group

We give a new approximation with respect of the traditional parametrix method of the solution of a parabolic equation whose generator is of big order and under the Hoermander form. This generalizes to a higher order generator the traditional approximation of Stratonovitch diffusion which put in relation random ordinary differential equation (the leading process is random and of finite energy. When a trajectory of it is chosen, the solution of the equation is defined) and stochastic differential equation (the leading process is random and only continuous and we cannot choose a path in the solution which is only almost surely defined). We consider simple operators where the computations can be fully performed. This approximation fits with the semi-group only and not for the full path measure in the case of a stochastic differential equation.

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