scholarly journals Research of a mathematical model of low-concentrated aqueous polymer solutions

2006 ◽  
Vol 2006 ◽  
pp. 1-27 ◽  
Author(s):  
Mikhail V. Turbin
Keyword(s):  
Mathematical Model ◽  
Weak Solution ◽  
Boundary Value Problem ◽  
Polymer Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Aqueous Polymer ◽  
Initial Boundary Value ◽  
The Mathematical Model

The initial-boundary value problem for the mathematical model of low-concentrated aqueous polymer solutions is considered. For this initial-boundary value problem a concept of a weak solution is introduced and the existence theorem for such solutions is proved.

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

The Initial Boundary-Value Problem for a Mathematical Model for Granular Medium

2015 ◽  
Vol 725-726 ◽  
pp. 863-868
Author(s):  
Vladimir Lalin ◽  
Elizaveta Zdanchuk
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Stress Tensor ◽  
Granular Medium ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cosserat Continuum ◽  
Suitable Model ◽  
Initial Boundary Value

In this work we consider a mathematical model for granular medium. Here we claim that Reduced Cosserat continuum is a suitable model to describe granular materials. Reduced Cosserat Continuum is an elastic medium, where all translations and rotations are independent. Moreover a force stress tensor is asymmetric and a couple stress tensor is equal to zero. Here we establish the variational (weak) form of an initial boundary-value problem for the reduced Cosserat continuum. We calculate the variation of corresponding Hamiltonian to obtain motion differential equation.

Weak solutions to the initial-boundary value problem for the shearing of non-homogeneous thermoviscoplastic materials

1989 ◽  
Vol 113 (3-4) ◽  
pp. 257-265 ◽  
Author(s):  
Nicolas Charalambakis ◽  
François Murat
Keyword(s):  
Weak Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Temperature Dependent ◽  
Initial Boundary Value

SynopsisWe prove the existence of a weak solution for the system of partial differential equations describing the shearing of stratified thermoviscoplastic materials with temperature-dependent non-homogeneous viscosity.

scholarly journals Countable stability of a weak solution of a parabolic differential-difference system with distributed parameters on the graph

2020 ◽  
Vol 16 (4) ◽  
pp. 402-414
Author(s):  
Vyacheslav V. Provotorov ◽  
◽  
Sergey M. Sergeev ◽  
Van Nguyen Hoang ◽  
◽  
...  
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Optimal Control Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Control Problems ◽  
Distributed Parameters ◽  
The Stability ◽  
Initial Boundary Value

The article proposes an analog of E. Rothe’s method (semi-discretization with respect to the time variable) for construction convergent different schemes when analyzing the countable stability of a weak solution of an initial boundary value problem of the parabolic type with distributed parameters on a graph in the class of summable functions. The proposed method leads to the study of the input initial boundary value problem to analyze the boundary value problem in a weak setting for elliptical type equations with distributed parameters on the graph. By virtue of the specifics of this method, the stability of a weak solution is understood in terms of the spectral criterion of stability (Neumann’s countable stability), which establishes the stability of the solution with respect to each harmonic of the generalized Fourier series of a weak solution or a segment of this series. Thus, there is another possibility indicated, in addition to the Faedo—Galerkin method, for constructing approaches to the desired solution of the initial boundary value problem, to analyze its stability and the way to prove the theorem of the existence of a weak solution to the input problem. The approach is applied to finding sufficient conditions for the countable stability of weak solutions to other initial boundary value problems with more general boundary conditions — in which elliptical equations are considered with the boundary conditions of the second or third type. Further analysis is possible to find the conditions under which Lyapunov stability is established. The approach can be used to analyze the optimal control problems, as well as the problems of stabilization and stability of differential systems with delay. Presented method of finite difference opens new ways for approximating the states of a parabolic system, analyzing their stability in the numerical implementation and algorithmization of optimal control problems.

scholarly journals MODELING PROCESS OF DESTRUCTION OF PARTICLES OF CEMENT AT INITIAL STAGE OF HYDRATION

2019 ◽  
Vol 62 (11) ◽  
pp. 138-142
Author(s):  
Olga N. Filimonova ◽  
Marina V. Enyutina ◽  
Anatoly A. Khvostov ◽  
Viktor I. Ryashskih
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Coordinate System ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Spherical Coordinate ◽  
Initial Stage ◽  
Average Size ◽  
Initial Boundary Value

The substantiation the existence of the destruction period of cement particles during the hydration of concrete into fragments, in size significantly larger than the average size of molecules is presented. An idealized structure of the cell representation for the cement grain and the water surrounding it in the form of nested spheres is adopted. The outer sphere does not change its coordinates, and the inner sphere imitating the cement grain decreases in the course of the process due to the transfer of the formed fragments into the inter-spherical space filled with water. The assumption is made that the concentration field is uniform and varies along the radius. The crushing process is described using a physical model of "pseudo-dissolution" and is based on the formulation of Fick's law. By introducing the rate of destruction associated with the movement of the interfacial boundary, a mathematical model is formulated in the form of an initial boundary value problem for the diffusion-type equation in a spherical coordinate system. By introducing a special coordinate system, the formulated initial-boundary value problem is transformed into a problem with fixed boundaries. Integration of the obtained system is performed numerically using an explicit finite-difference scheme. The computational experiment confirmed the efficiency of the proposed algorithm, which allowed to conduct a qualitative analysis of the model, which showed the correctness of the assumptions made in the formulation of the mathematical model. It is found that in a spherical cell containing an average cement grain, the fragments are localized near the interfacial surface due to the weak influence of the diffusion transfer mechanism. The estimation of the average size of the fragments, which are significantly larger than the average molecular size, further confirmed the hypothesis of the existence of the period of destruction of cement grains in the initial stage of concrete hydration.

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