Class Of Linear Parabolic Equations Research Articles

A unified approach to maximum-norm a posteriori error estimation for second-order time discretizations of parabolic equations

Abstract A class of linear parabolic equations is considered. We derive a common framework for the a posteriori error analysis of certain second-order time discretizations combined with finite element discretizations in space. In particular, we study the Crank–Nicolson method, the extrapolated Euler method, the backward differentiation formula of order 2, the Lobatto IIIC method and a two-stage SDIRK method. We use the idea of elliptic reconstructions and certain bounds for the Green’s function of the parabolic operator.

On the invariants of two dimensional linear parabolic equations

We consider the most general two dimensional linear parabolic equations. Motivated by the recent work of Ibragimov et al. [1–3] we construct differential invariants, semi-invariants and invariant equations. These results are achieved with the employment of the equivalence group admitted by this class of parabolic equations. We derive those variable coefficient equations of this class of linear parabolic equations that can be mapped into constant coefficient equations. Further applications are presented.

Admissible conditions for parabolic equations degenerating at infinity

Well-posedness in $L^\infty (\mathbb {R}^n)$ $(n \ge 3)$ of the Cauchy problem is studied for a class of linear parabolic equations with variable density. In view of degeneracy at infinity, some

Discrete maximum principle for the finite element solution of linear non-stationary diffusion–reaction problems

The maximum principle is one of the basic characteristic properties of solutions of second order partial differential equations of parabolic (and elliptic) types. The preservation of this property for solutions of corresponding discretized problems is a very natural requirement in reliable and meaningful numerical modelling of various real-life phenomena (heat conduction, air pollution, etc.). In the present paper we analyse a full discretization of a quite general class of linear parabolic equations and present sufficient conditions for the validity of a discrete analogue of the maximum principle in the case when bilinear finite elements are used for discretization in space.

Estimations optimales en temps petit et près de la frontière pour les solutions de l'équation de la chaleur avec données non compatibles

In this Note, we prove small time estimates near the boundary for solutions of the linear heat equation: u t = Δ u in (0, ∞) x Ω, u = 0 on ∂Ω, u(0) =in Ω, with non-compatible initial dataε C(Ω), where Ω is a bounded domain of ℝ N. The proofs are primarily based on the maximum principle, via the construction of suitable sub and super-solutions, and on a monotonicity argument. We obtain similar results for the associated inhomogeneous problem. A generalization to a large class of linear parabolic equations is announced. © Académie des Sciences/Elsevier, Paris

A mixed bvp for flow in strongly anisotropic media

Initial-boundary value problems for a class of linear parabolic equations are considered. The anisotropy of the medium is characterised by a small parameter. The solution structure is analysed by singular perturbation methods which include the construction of outer solutions and boundary and initial layer terms. The analysis is justified by convergence results

Sufficient conditions for controllability of a class of distributed parameter systems

In the paper we present certain results on the question of controllability for (i) linear evolution equations (parabolic and hyperbolic) with distributed controls and (ii) a class of linear parabolic equations with boundary controls.