scholarly journals Solvability of initial boundary value problems for equations describing motions of linear viscoelastic fluids

2005 ◽  
Vol 2005 (1) ◽  
pp. 59-80 ◽  
Author(s):  
N. A. Karazeeva
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Strain Tensor ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Additional Hypothesis ◽  
Definite Operator ◽  
Linear Viscoelastic ◽  
Initial Boundary Value

The nonlinear parabolic equations describing motion of incompressible media are investigated. The rheological equations of most general type are considered. The deviator of the stress tensor is expressed as a nonlinear continuous positive definite operator applied to the rate of strain tensor. The global-in-time estimate of solution of initial boundary value problem is obtained. This estimate is valid for systems of equations of any non-Newtonian fluid. Solvability of initial boundary value problems for such equations is proved under some additional hypothesis. The application of this theory makes it possible to prove the existence of global-in-time solutions of two-dimensional initial boundary value problems for generalized linear viscoelastic liquids, that is, for liquids with linear integral rheological equation, and for third-grade liquids.

Initial-boundary value problems for linear PDEs with variable coefficients

2007 ◽  
Vol 143 (1) ◽  
pp. 221-242 ◽  
Author(s):  
P. A. TREHARNE ◽  
A. S. FOKAS
Keyword(s):  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Initial Boundary Value Problems ◽  
Linear Evolution ◽  
Higher Derivatives ◽  
Initial Boundary Value

AbstractA new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

Abstract initial boundary value problems

1994 ◽  
Vol 124 (5) ◽  
pp. 879-908 ◽  
Author(s):  
C. Palencia ◽  
I. Alonso Mallo
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Operators ◽  
Initial Boundary Value Problems ◽  
Hyperbolic Problems ◽  
Euler’S Method ◽  
Initial Boundary Value

We consider abstract initial boundary value problems in a spirit similar to that of the classical theory of linear semigroups. We assume that the solution u at time t is given by u(t) = S(t) ξ + V(t)g, where ξ and g are respectively the initial and boundary data and S(t) and V(t) are linear operators. We take as a departing point the functional equations satisfied by the propagators S and V. We discuss conditions under which a pair (S, V) describes the solution of an abstract differential initial boundary value problem. Several examples are provided of parabolic and hyperbolic problems that can be accommodated within the abstract theory. We study the backward Euler's method for the time integration of the problems considered.

Blow-up of error estimates in time-fractional initial-boundary value problems

2020 ◽  
Author(s):  
Hu Chen ◽  
Martin Stynes
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Error Bounds ◽  
Time Domain ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

Abstract Time-fractional initial-boundary value problems of the form $D_t^\alpha u-p \varDelta u +cu=f$ are considered, where $D_t^\alpha u$ is a Caputo fractional derivative of order $\alpha \in (0,1)$ and the spatial domain lies in $\mathbb{R}^d$ for some $d\in \{1,2,3\}$. As $\alpha \to 1^-$ we prove that the solution $u$ converges, uniformly on the space-time domain, to the solution of the classical parabolic initial-boundary value problem where $D_t^\alpha u$ is replaced by $\partial u/\partial t$. Nevertheless, most of the rigorous analyses of numerical methods for this time-fractional problem have error bounds that blow up as $\alpha \to 1^-$, as we demonstrate. We show that in some cases these analyses can be modified to obtain robust error bounds that do not blow up as $\alpha \to 1^-$.

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