scholarly journals A High-Order Iterative Scheme for a Nonlinear Pseudoparabolic Equation and Numerical Results

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Nguyen Huu Nhan ◽  
Tran Trinh Manh Dung ◽  
Le Thi Mai Thanh ◽  
Le Thi Phuong Ngoc ◽  
Nguyen Thanh Long
Keyword(s):  
Iterative Scheme ◽  
Galerkin Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Numerical Results ◽  
Pseudoparabolic Equation ◽  
Unique Existence ◽  
Basic Concepts ◽  
Initial Boundary Value ◽  
Nonlinear Pseudoparabolic Equation

In this paper, by applying the Faedo-Galerkin approximation method and using basic concepts of nonlinear analysis, we study the initial-boundary value problem for a nonlinear pseudoparabolic equation with Robin–Dirichlet conditions. It consists of two main parts. Part 1 is devoted to proof of the unique existence of a weak solution by establishing an approximate sequence u m based on a N -order iterative scheme in case of f ∈ C N 0,1 × 0 , T ∗ × ℝ N ≥ 2 , or a single-iterative scheme in case of f ∈ C 1 Ω ¯ × 0 , T ∗ × ℝ . In Part 2, we begin with the construction of a difference scheme to approximate u m of the N -order iterative scheme, with N = 2 . Next, we present numerical results in detail to show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.

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.

Piecewise continuous solutions of nonlinear pseudoparabolic equations in two space dimensions*

1992 ◽  
Vol 121 (3-4) ◽  
pp. 203-217 ◽  
Author(s):  
Dao-Qing Dai ◽  
Wei Lin
Keyword(s):  
Lipschitz Constant ◽  
Complex Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Lipschitz Continuous ◽  
Two Space Dimensions ◽  
Image Position ◽  
Initial Boundary Value ◽  
Pseudoparabolic Equations ◽  
Nonlinear Pseudoparabolic Equation

SynopsisAn initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variablesThe complex functionHis measurable on ℂ ×I × ℂ5, withIbeing an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.

scholarly journals Global Well-posedness and Asymptotics of Full Compressible Non-resistive MHD System with Large External Potential Forces

2021 ◽  
Author(s):  
Wanrong Yang ◽  
Xiaokui Zhao
Keyword(s):  
Stationary Solution ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Unique Existence ◽  
Background Magnetic Field ◽  
External Forces ◽  
Mhd System ◽  
Initial Boundary Value

We consider the global well-posedness and asymptotic behavior of compressible viscous, heat-conductive, and non-resistive magnetohydrodynamics (MHD) fluid in a field of external forces over three-dimensional periodic thin domain $\Omega=\mathbb{T}^2\times(0,\delta)$. The unique existence of the stationary solution is shown under the adhesion and the adiabatic boundary conditions. Then, it is shown that a solution to the initial boundary value problem with the same boundary and periodic conditions uniquely exists globally in time and converges to the stationary solution as time tends to infinity. Moreover, if the external forces are small or disappeared in an appropriate Sobolev space, then $\delta$ can be a general constant. Our proof relies on the two-tier energy method for the reformulated system in Lagrangian coordinates and the background magnetic field which is perpendicular to the flat layer. Compared to the work of Tan and Wang (SIAM J. Math. Anal. 50:1432–1470, 2018), we not only overcome the difficulties caused by temperature, but also consider the big external forces.

scholarly journals The equations of viscous incompressible non-homogeneous fluids: on the existence and regularity

1991 ◽  
Vol 33 (1) ◽  
pp. 94-110 ◽  
Author(s):  
Rodolfo Salvi
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Galerkin Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Compatibility Conditions ◽  
Initial Boundary Value

AbstractWe prove the existence and regularity of the solution of an initial boundary value problem for viscous incompressible non-homogeneous fluids, using a semi-Galerkin approximation and so-called compatibility conditions.

scholarly journals Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients

2018 ◽  
Vol 21 (2) ◽  
pp. 276-311 ◽  
Author(s):  
Adam Kubica ◽  
Masahiro Yamamoto
Keyword(s):  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Diffusion ◽  
Initial Boundary Value Problems ◽  
Unique Existence ◽  
Fractional Diffusion Equations ◽  
Time Variables ◽  
Initial Boundary Value

Abstract We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is attached. We prove the unique existence of weak and regular solutions.

scholarly journals A numerical method for solving the second initial-boundary value problem for a multidimensional third-order pseudoparabolic equation

2021 ◽  
Vol 31 (3) ◽  
pp. 384-408
Author(s):  
M.Kh. Beshtokov
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Pseudoparabolic Equation ◽  
Third Order ◽  
One Dimensional ◽  
Initial Boundary Value

The work is devoted to the study of the second initial-boundary value problem for a general-form third-order differential equation of pseudoparabolic type with variable coefficients in a multidimensional domain with an arbitrary boundary. In this paper, a multidimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter, and a locally one-dimensional difference scheme by A.A. Samarskii is used. Using the maximum principle, an a priori estimate is obtained for the solution of a locally one-dimensional difference scheme in the uniform metric in the $C$ norm. The stability and convergence of the locally one-dimensional difference scheme are proved.

ASYMPTOTIC BEHAVIORS AND CLASSICAL LIMITS OF SOLUTIONS TO A QUANTUM DRIFT-DIFFUSION MODEL FOR SEMICONDUCTORS

2010 ◽  
Vol 20 (06) ◽  
pp. 909-936 ◽  
Author(s):  
SHINYA NISHIBATA ◽  
NAOTAKA SHIGETA ◽  
MASAHIRO SUZUKI
Keyword(s):  
Diffusion Model ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singular Limit ◽  
Drift Diffusion Model ◽  
Planck Constant ◽  
Drift Diffusion ◽  
One Dimensional ◽  
Unique Existence ◽  
Initial Boundary Value

This paper discusses a time global existence, asymptotic behavior and a singular limit of a solution to the initial boundary value problem for a quantum drift-diffusion model of semiconductors over a one-dimensional bounded domain. Firstly, we show a unique existence and an asymptotic stability of a stationary solution for the model. Secondly, it is shown that the time global solution for the quantum drift-diffusion model converges to that for a drift-diffusion model as the scaled Planck constant tends to zero. This singular limit is called a classical limit. Here these theorems allow the initial data to be arbitrarily large in the suitable Sobolev space. We prove them by applying an energy method.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

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