Study of rigidity of a first-order algebro-differential system with perturbation in the right-hand side

2021 ◽  
pp. 172-181
Author(s):  
Vladimir I. Uskov
Keyword(s):  
Dynamical System ◽  
Boundary Layer ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Bifurcation Equation ◽  
First Order ◽  
Initial Boundary Value ◽  
The Right

The rigidity of a dynamical system described by a first-order differential equationwith an irreversible operator at the highest derivative is investigated. The system is perturbed by an operator addition of the order of the second power of a small parameter. Conditions under which the system is robust with respect to these disturbances are determined as well as conditions under which the influence of disturbances is significant. For this, the bifurcation equation is derived. It is used to set the type of boundary layer functions. As an example, we investigate the initial boundary value problem for a system of partial differential equations with a mixed second partial derivative which occurs in the study of the processes of sorption anddesorption of gases, drying processes, etc.

scholarly journals Boundary layer study of a nonlinear parabolic equation with a small parameter

2021 ◽  
Author(s):  
qin xulong ◽  
xu zhao ◽  
wenshu zhou
Keyword(s):  
Boundary Layer ◽  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Nonlinear Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic ◽  
Initial Boundary Value

This paper is concerned with the initial-boundary value problem for a nonlinear parabolic equation with a small parameter. The existence of a boundary layer as the parameter goes to zero is obtained together with the estimation on the thickness of the boundary layer. The main result extends an earlier work of Frid and Shelukhin (1999).

Numerical Method for Singularly Perturbed Parabolic Equations in Unbounded Domains in the Case of Solutions Growing at Infinity

2009 ◽  
Vol 9 (1) ◽  
pp. 100-110
Author(s):  
G. I. Shishkin
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Maximum Norm ◽  
Lateral Part ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.

scholarly journals Boundary problem on semi-direct for pseudoparabolic equation with a small parametr

2020 ◽  
pp. 29-41
Author(s):  
Б.С. Аблабеков ◽  
А.Т. Муканбетова
Keyword(s):  
Analytical Solution ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Boundary Problem ◽  
Unique Solvability ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Pseudoparabolic Equation ◽  
Explicit Analytical Solution ◽  
Initial Boundary Value

Данная статья посвящена построению классического решения краевой задачи на полупрямой для линейного псевдопараболического уравнения с малым параметром. Для построения явного решения используется метод преобразование Фурье. В работе исследуется вопрос об однозначной разрешимости начально-краевой задачи для псевдопараболического уравнения с малым параметром на полуоси. Получено явное аналитическое решение поставленной задачи. In this paper, we study the question of the unique solvability of the initial-boundary-value problem for a pseudoparabolic equation with a small parameter on the semi-axis. An explicit analytical solution to the problem is obtained.

scholarly journals Initial-boundary value problem for a time-fractional subdiffusion equation on the torus

2021 ◽  
Vol 65 (3) ◽  
pp. 17-24
Author(s):  
Ravshan Ashurov ◽  
◽  
Oqila Muhiddinova
Keyword(s):  
Boundary Value Problem ◽  
Initial Function ◽  
Boundary Value ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Subdiffusion Equation ◽  
Initial Boundary Value ◽  
The Right

An initial-boundary value problem for a time-fractional subdiffusion equation with the Riemann-Liouville derivatives on N-dimensional torus is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. It should be noted, that the condition on the initial function found in this paper is less restrictive than the analogous condition in the case of an equation with derivatives in the sense of Caputo.

scholarly journals Development and correctness analysis of the mathematical model of transport and suspension sedimentation depending on bottom relief variation

2018 ◽  
Vol 18 (4) ◽  
pp. 350-361
Author(s):  
A. I. Sukhinov ◽  
V. V. Sidoryakina
Keyword(s):  
Boundary Value Problem ◽  
Coastal Area ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Quadratic Functional ◽  
Bottom Relief ◽  
Aquatic Medium ◽  
Initial Boundary Value ◽  
The Right

Introduction. The paper is devoted to the study on the three-dimensional model of transport and suspension sedimentation in the coastal area due to changes in the bottom relief. The model considers the following processes: advective transfer caused by the aquatic medium motion, micro-turbulent diffusion, and gravity sedimentation of suspended particles, as well as the bottom geometry variation caused by the particle settling or bottom sediment rising. The work objective was to conduct an analytical study of the correctness of the initial-boundary value problem corresponding to the constructed model.Materials and Methods. The change in the bottom relief aids in solution to the initial-boundary value problem for a parabolic equation with the lowest derivatives in a domain whose geometry depends on the desired function of the solution, which in general leads to a nonlinear formulation of the problem. The model is linearized on the time grid due to the “freezing” of the bottom relief within a single step in time and the subsequent recalculation of the bottom surface function on the basis of the changed function of the suspension concentration, as well as a possible change in the velocity vector of the aquatic medium.Research Results. For the linearized problem, a quadratic functional is constructed, and the uniqueness of the solution to the corresponding initial boundary value problem is proved within the limits of an unspecified time step. On the basis of the quadratic functional transformation, we obtain a prior estimate of the solution norm in the functional space L2 as a function of the integral time estimates of the right side, and the initial condition. Thus, the stability of the solution to the initial problem from the change of the initial and boundary conditions, the right-hand side function, is established.Discussion and Conclusions. The model can be of value for predicting the spread of contaminants and changes in the bottom topography, both under an anthropogenic impact and due to the natural processes in the coastal area.

scholarly journals On linear and nonlinear instability of the incompressible swept attachment-line boundary layer

1998 ◽  
Vol 355 ◽  
pp. 193-227 ◽  
Author(s):  
VASSILIOS THEOFILIS
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Small Amplitude ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Instability Waves ◽  
Linear And Nonlinear ◽  
Initial Boundary Value

The stability of an incompressible swept attachment-line boundary layer flow is studied numerically, within the Görtler–Hämmerlin framework, in both the linear and nonlinear two-dimensional regimes in a self-consistent manner. The initial-boundary-value problem resulting from substitution of small-amplitude excitation into the incompressible Navier–Stokes equations and linearization about the generalized Hiemenz profile is solved. A comprehensive comparison of all linear approaches utilized to date is presented and it is demonstrated that the linear initial-boundary-value problem formulation delivers results in excellent agreement with those obtained by solution of either the temporal or the spatial linear stability theory eigenvalue problem for both zero suction and a layer in which blowing is applied. In the latter boundary layer recent experiments have documented the growth of instability waves with frequencies in a range encompassed by that of the unstable Görtler–Hämmerlin linear modes found in our simulations. In order to enable further comparisons with experiment and, thus, assess the validity of the Görtler–Hämmerlin theoretical model, we make available the spatial structure of the eigenfunctions at maximum growth conditions.The condition on smallness of the imposed excitation is subsequently relaxed and the resulting nonlinear initial-boundary-value problem is solved. Extensive numerical experimentation has been performed which has verified theoretical predictions on the way in which the solution is expected to bifurcate from the linear neutral loop. However, it is demonstrated that the two-dimensional model equations considered do not deliver subcritical instability of this flow; this strengthens the conjecture that three-dimensionality is, at least partly, responsible for the observed discrepancy between the linear theory critical Reynolds number and the subcritical turbulence observed either experimentally or in three-dimensional numerical simulations. Further, the present nonlinear computations demonstrate that the unstable flow has its line of maximum amplification in the neighbourhood of the experimentally observed instability waves, in a manner analogous to the Blasius boundary layer. In line with previous eigenvalue problem and direct simulation work, suction is observed to be a powerful stabilization mechanism for naturally occurring instabilities of small amplitude.

NONLINEAR STABILITY OF PLANAR BOUNDARY LAYER SOLUTIONS FOR DAMPED WAVE EQUATION

2011 ◽  
Vol 08 (03) ◽  
pp. 545-590 ◽  
Author(s):  
LILI FAN ◽  
HONGXIA LIU ◽  
HUIJIANG ZHAO
Keyword(s):  
Boundary Layer ◽  
Wave Equation ◽  
Boundary Value Problem ◽  
Energy Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Damped Wave Equation ◽  
Planar Boundary ◽  
Initial Boundary Value ◽  
Layer Solution

We study the global stability of planar boundary layer solutions to the initial boundary value problem for the damped wave equation, in presence of a nonlinear convection term in the two-dimensional half space R+× R. By employing the energy method and a continuation argument, we establish that such an initial boundary value problem admits a unique global solution for a class of large initial perturbations, and that this global solution converges to the corresponding planar boundary layer solution, uniformly in (x, y) ∈ R+× R as t tends to infinity — provided the strength of the planar boundary layer solution is suitably small. Moreover, by exploiting the space–time weighted energy method and the properties of the planar boundary layer solutions, we also derive a convergence rate (both algebraic and exponential in nature) for the non-degenerate case.

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