A nonlinear system for irreversible phase changes

1990 ◽  
Vol 1 (1) ◽  
pp. 91-100 ◽  
Author(s):  
Dominique Blanchard ◽  
Hamid Ghidouche
Keyword(s):  
Nonlinear System ◽  
Existence And Uniqueness ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Phase Changes ◽  
The Other ◽  
System Modelling ◽  
Mathematical Study ◽  
Uniqueness Of The Solution ◽  
Change Problem

This paper is concerned with the mathematical study of a nonlinear system modelling an irreversible phase change problem. Uniqueness of the solution is proved using the accretivity of the system in (L1)2. Expressing one of the two unknowns as an explicit functional of the other reduces the system to a single nonlinear evolution equation and ultimately leads to an existence theorem.In this paper the existence and uniqueness of the solution of a nonlinear system modelling some irreversible phase changes is established.

scholarly journals Bäcklund Transformations between the KdV Equation and a New Nonlinear Evolution Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Xifang Cao
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
The Other ◽  
Bäcklund Transformations ◽  
Backlund Transformations ◽  
The Kdv Equation ◽  
New Equation

We first give a Bäcklund transformation from the KdV equation to a new nonlinear evolution equation. We then derive two Bäcklund transformations with two pseudopotentials, one of which is from the KdV equation to the new equation and the other from the new equation to itself. As applications, by applying our Bäcklund transformations to known solutions, we construct some novel solutions to the new equation.

Application of Improved (G′/G)–Expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations

2012 ◽  
Vol 4 (1) ◽  
pp. 122-130 ◽  
Author(s):  
Xiaohua Liu ◽  
Weiguo Zhang ◽  
Zhengming Li
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Nonlinear Evolution Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
The Other ◽  
Wave Solutions

AbstractIn this work, the improved (G′/G)-expansion method is proposed for constructing more general exact solutions of nonlinear evolution equation with the aid of symbolic computation. In order to illustrate the validity of the method we choose the RLW equation and SRLW equation. As a result, many new and more general exact solutions have been obtained for the equations. We will compare our solutions with those gained by the other authors.

scholarly journals Theoretical Framework of a Variational Formulation for Nonlinear Heat Transfer with Phase Changes

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Eric Feulvarch ◽  
Jean-Christophe Roux ◽  
Jean-Michel Bergheau
Keyword(s):  
Heat Transfer ◽  
Finite Element ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Theoretical Framework ◽  
Phase Changes ◽  
Time Step ◽  
Numerical Resolution ◽  
Nonlinear Heat Transfer ◽  
Uniqueness Of The Solution

This paper discusses mathematical results for a variational formulation dedicated to heat transfer with phase changes. Practical finite element experiences show that the studied formulation can lead to difficulties for the numerical resolution at each time step. The aim of the paper is to show that such numerical pathologies do not come from the basic variational formulation by showing existence and uniqueness of the solution.

scholarly journals Nonlinear evolution equation for modeling waves at the boundary of a stratified flow of viscous liquids in an inclined channel

2021 ◽  
Vol 2119 (1) ◽  
pp. 012058
Author(s):  
D G Arkhipov ◽  
G A Khabakhpashov
Keyword(s):  
Differential Equation ◽  
Nonlinear Evolution Equation ◽  
Wave Motion ◽  
Nonlinear Evolution ◽  
Stationary Flow ◽  
The Other ◽  
Two Dimensional ◽  
Viscous Liquids ◽  
Inclined Channel ◽  
Small Flow

Abstract The dynamics of perturbations of the interface of a two-layer Poiseuille flow in a flat closed inclined channel is studied. The velocity profiles of wave motion are analytically found neglecting dissipation, dispersion and pumping of perturbations. On the basis of the found solution, a nonlinear evolution integro-differential equation for plane moderately long perturbations of the interface of the liquids is derived. The coefficients of the equation are represented by integrals over the layer thicknesses from functions depending on the stationary flow and perturbation profiles. The equation takes into account viscous dissipation: one of the integrals in this equation corresponds to dissipation in lion-stationary boundary layers, and the other corresponds to the transfer of energy from the flow to the wave. For the case of small flow velocities, the coefficients of the equation are analytically calculated. The equation has also been generalized to the quasi-two-dimensional case when the gradients along the transversal coordinate are small.

scholarly journals THE PHILLIP ISLAND PENGUIN PARADE (A MATHEMATICAL TREATMENT)

2018 ◽  
Vol 60 (1) ◽  
pp. 27-54
Author(s):  
SERENA DIPIERRO ◽  
LUCA LOMBARDINI ◽  
PIETRO MIRAGLIO ◽  
ENRICO VALDINOCI
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Existence And Uniqueness ◽  
The Other ◽  
Mathematical Treatment ◽  
Mathematical Framework ◽  
Stop And Go ◽  
Actual Movement ◽  
Uniqueness Of The Solution ◽  
Return Home

Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviours in their homecoming, which are interesting to observe and to describe analytically. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group, and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account “natural parameters”, such as the eyesight of the penguins and their cruising speed. The model that we propose favours the formation of conglomerates of penguins that gather together, but, on the other hand, it also allows the possibility of isolated and exposed individuals.The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behaviour of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a “stop-and-go” procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins remain isolated). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguin parade.

HOMOGENIZATION OF ELECTRICAL NETWORKS INCLUDING VOLTAGE-TO-VOLTAGE AMPLIFIERS

1999 ◽  
Vol 09 (06) ◽  
pp. 899-932 ◽  
Author(s):  
MICHEL LENCZNER ◽  
GHOUTI SENOUCI-BEREKSI
Keyword(s):  
Network Topology ◽  
Existence And Uniqueness ◽  
General Framework ◽  
The Other ◽  
Scale Transformation ◽  
Electrical Networks ◽  
Convergence Results ◽  
Uniqueness Of The Solution ◽  
Homogenized Model ◽  
The One

We derive the homogenized model of periodic electrical networks which includes resistive devices, voltage-to-voltage amplifiers, sources of tension and sources of current. On the one hand, in considering the homogenized problem, general conditions are stated insuring the existence and uniqueness of the solution. They are formulated in function of the network topology. On the other hand, the two-scale transformation introduced by Arbogast, Douglas and Hornung is adapted to the context of electrical networks. New two-scale. convergence results, inspired by the principle of Allaire's two-scale convergence, are shown in this context. In particular, the two-scale convergence for the tangential derivative on a network is established. Following these results, two models of homogenized networks are derived. The first one belongs to a general framework whereas the second one does not.

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

Sign in / Sign up

Export Citation Format

Share Document